Lectures on Serre’s conjectures (October 4, 1999)

Kenneth A. Ribet William A. Stein

Contents

Lectures on Serre’s conjectures (October 4, 1999) 1 Lecture 1. Introduction to Serre’s conjecture 4 Lecture 2. The weak and strong conjectures 11 Lecture 3. The weight in Serre’s conjecture 17 Lecture 4. Galois representations from modular forms 27 Lecture 5. Introduction to level lowering 31 Lecture 6. Approaches to level lowering 37 Lecture 7. Level lowering without multiplicity one 43 Lecture 8. Level lowering with multiplicity one 47 Lecture 9. Other directions 51 Lecture 10. Katz modular forms (Appendix) 53 Lecture 11. Exercises (Appendix) 57 Bibliography 65

IAS/Park City Mathematics Series Volume 00, October 4, 1999

Lectures on Serre’s conjectures

Kenneth A. Ribet William A. Stein

1Math Department, MC 3840; Berkeley, CA 94720-3840.

E-mail address: ribet@math.berkeley.edu, was@math.berkeley.edu.

\begin{matrix}c \\ ◯ \end{matrix}October 4, 1999 American Mathematical Society

My plan is to begin by discussing some examples of mod ℓ representations of Gal(\overline{Q}/Q). I’ll try to motivate Serre’s conjectures by referring first to the case of representations that are unramified outside ℓ; these should come from cusp forms on the full modular group SL(2, Z). In another direction, one might think about representations coming from ℓ-division points on elliptic curves, or more generally from ℓ-division points on abelian varieties of “GL_{2}-type.” Amazingly (to me), Serre’s conjectures imply that all odd irreducible two-dimensional mod ℓ representations of Gal(\overline{Q}/Q) may be realized in spaces of ℓ-division points on such abelian varieties. The weak Serre conjecture states that all such representations come from modular forms, and then it takes only a bit of technique to show that one can take the modular forms to have weight two (if one allows powers of ℓ in the level).

Since little work has been done toward proving the weak Serre conjecture, my lectures will focus on the bridge between the weak and the strong conjectures. The latter states that each ρ as above comes from the space of cusp forms of a specific weight and level, with these invariants between determined by the local behavior of ρ at ℓ and at primes other than ℓ (respectively). To motivate the strong conjecture, and to work toward the bridge, we need to discuss the local behavior of those ρ that do come from modular forms. For the most part, we can look only at forms of weight k ≥ 2 whose levels N are prime to ℓ. In that case, the behavior of ρ at ℓ is described in detail in [26], where theorems of Deligne and Fontaine are recalled. (In [26, §6], Edixhoven supplies a proof of Fontaine’s theorem.) Further, the behavior of ρ at primes p ≠ ℓ may be inferred from Carayol’s theorems [10, 11], which relate the behavior at p of the ℓ-adic representations attached to f with the p-adic component of the automorphic representation of GL(2) that one associates with f. (The behavior of ρ at ℓ in the case where ℓ divides N is analyzed in [64].)

In [75], Serre associates to each ρ a level N(ρ) and a weight k(ρ). These invariants are defined so that N(ρ) is prime to ℓ and so that k is an integer greater than 1. As Serre was no doubt aware, if ρ arises from a modular form of weight k and level N, and if k is at least 2 and N is prime to ℓ, then one has k(ρ) ≤ k and N(ρ) | N. To find an f for which N = N(ρ) and k = k(ρ) is to “optimize” the level and weight of a form giving ρ. As Edixhoven explains in his article [26] weight optimization follows in a somewhat straightforward manner from the theorems of Deligne and Fontaine alluded to above, plus the theorem of Gross on companion forms [34]. (See [15] for a partial treatment of the companion form problem when ℓ = 2.) Luckily, we can perform weight optimization after having optimized the level, since the weight optimization machinery does not disturb the level at any point.

In [11], Carayol analyzes the level optimization problem. He shows, in particular, that the problem breaks down into a series of sub-problems, all but one of which he treats by appealing to a single lemma, the lemma of [11, §3]. The remaining subproblem is the one which intervenes in showing the implication “Shimura–Taniyama ⟹ Fermat.” I have written about this sub-problem repeatedly [60, 61, 62, 63]. Most likely, I will explain the principle of [62] early on in these lectures. Then, toward the end of the series, I will explain the more refined level-lowering argument that I introduced in my article [63]. (This argument was extended by Diamond in [21].) One reason for going over the argument now is that the argument hasn’t yet been made to work for ℓ = 2. Perhaps the reader will see how to modify the argument so that it extends to this case.

The case ℓ = 2 is the only remaining case for which the level optimization problem has not been resolved. The proof in [21, 63] of level lowering for ℓ ≥ 3 does not fully exploit multiplicity one results, but appears to completely break down when ℓ = 2. It was recently pointed out to me by Kevin Buzzard that many new cases of multiplicity one are known and that this can be used to obtain new level optimization results when ℓ = 2. In Lecture 8 we extend the arguments of [61] to obtain a level optimization theorem in new cases for which ℓ = 2.

A certain amount of work has been done on the Hilbert modular case, i.e., the case where Q is replaced by a totally real number field F. For this work, the reader may consult articles of Frazer Jarvis [38, 39, 40] and of Kazuhiro Fujiwara [33]. I am especially grateful to Fujiwara for his sending me a preliminary version of his manuscript, “Level optimization in the totally real case.” I suspect, however, that these lectures will treat only the classical case F = Q.

LECTURE 1

Introduction to Serre’s conjecture

Let’s start with an elliptic curve E/Q. Nowadays, it’s a familiar activity to consider the Galois representations defined by groups of division points of E. Namely, let n be a positive integer, and let E[n] be the kernel of multiplication by n on E(\overline{Q}). The group E[n] is free of rank two over Z/nZ; choose a basis. The action of Gal(\overline{Q}/Q) on E[n] is given by a homomorphism

ρ_{n} : Gal(\overline{Q}/Q) → Aut(E[n]) ≈ GL_{2}(Z/nZ).

This homomorphism is unramified at each prime p that is prime to the product of n with the conductor N of E (Exercise 25). For each such p, ρ_{n}(Frob_{p}) is a 2 × 2 matrix that is well defined up to conjugation. Its determinant is p mod n; its trace is a_{p} mod n, where a_{p} is the usual “trace of Frobenius” attached to E and p, i.e., the quantity 1 + p − #E(F_{p}). In his pathbreaking 1966 article [80], G. Shimura studied these representations (for the curve E = J_{o}(11)) and the number fields that they cut out. He noticed that for prime values n = ℓ, the representations ρ_{n} tended to have large images. In [67] J-P. Serre proved that for any fixed elliptic curve E, not having complex multiplication, the indices

(GL_{2}(Z/nZ) : ρ_{n}(Gal(\overline{Q}/Q)))

are bounded independently of n.

In this series of lectures, we will be concerned mainly with two-dimensional representations over finite fields. To that end, we restrict attention to the case where n = ℓ is prime. The representation ρ_{ℓ} is “modular” in the familiar sense that it’s a representation of a group over a field in positive characteristic. The theme of this course is that it’s modular in a different and deeper sense: it comes from a modular form! Indeed, according to a recent announcement by Breuil, Conrad, Diamond and Taylor (see [7, 16, 86, 89]), the Shimura–Taniyama conjecture is now a theorem—all elliptic curves over Q are modular!! Thus there is a weight-two newform f = ∑_{m} c_{m}q^{m} (q = e^{2πiz}) on Γ_{0}(N) with the property that a_{p} = c_{p} for all p prime to N. Accordingly, ρ_{ℓ} is connected up with modular forms via the relation tr(ρ_{ℓ}(Frob_{p})) = c_{p} mod ℓ, valid for all but finitely many primes p.

More precisely, the Shimura–Taniyama conjecture asserts that for each positive integer N there is a bijection between isogeny classes of elliptic curves A over Q of conductor N and weight two rational newforms f for Γ_{0}(N). Given A, the Shimura–Taniyama conjecture produces a modular form f whose Dirichlet series

f = ∑_{n=1}^{∞} c_{n}q^{n}, q = q(z) = e^{2πiz}

is connected to the L-series of A by the relation

∑_{n=1}^{∞} \frac{c_{n}}{n^{s}} = L(f, s)  Shimura–Taniyama  L(A, s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} .

The integers a_{n} encode information about the number of points on A over various finite fields. More precisely, if p is a prime not dividing N then a_{p} = p+1−#A(F_{p}).

If p | N, then

a_{p} = \begin{cases}−1 & if A has non-split multiplicative reduction at p \\ 1 & if A has split multiplicative reduction at p \\ 0 & if A has additive reduction at p. \end{cases}

(See [82, VII.5] for a description of the reduction types.) The integers a_{n} are then obtained from the a_{p} using the following relations:

• a_{p^{r}} = \begin{cases}a_{p^{r−1}}a_{p} − pa_{p^{r−2}} & if p ∤ N, \\ a_{p}^{r} & if p | N. \end{cases}

• a_{nm} = a_{n}a_{m}, if (n, m) = 1.

Serre’s conjectures [75] concern representations ρ : Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{ℓ}). We require always (usually tacitly) that our representations are continuous, the topologies on Gal(\overline{Q}/Q) and on GL_{2}(\overline{F}_{ℓ}) being the Krull topology and the discrete topology, respectively. The continuity condition just means that the kernel of ρ is open, so that it corresponds to a finite Galois extension K of Q. The representation ρ then embeds Gal(K/Q) into GL_{2}(\overline{F}_{ℓ}). Since K/Q is a finite extension, the image of ρ is contained in GL_{2}(F) for some finite field F ⊂ \overline{F}_{ℓ}.

\overline{Q} _{G_{Q}} K Q  \xymatrix{G_{Q} \ar@{->>}[d] \ar[dr]^{ρ} \\ Gal(K/Q) \ar@{^{(}->}[r] & GL(2, \overline{F}_{ℓ})}

For various technical reasons, the original conjectures of Serre insist that ρ be irreducible. It is nevertheless sometimes fruitful to consider the reducible case (see [83] and forthcoming work of C. Skinner and A. Wiles).

The conjectures state (in particular) that each continuous irreducible ρ that satisfies a necessary parity condition “arises from” (or is associated with) a suitable modular form mod ℓ. To explain what’s going on, I would like to start with Δ = ∑_{n} τ(n)q^{n} = q ∏_{i}(1 − q^{i})^{24}, the unique (normalized) cusp form of weight 12 on SL(2, Z). In [66], Serre conjectured the existence of a “strictly compatible” family of ℓ-adic representations of Gal(\overline{Q}/Q) whose L-function is the L-function of Δ, namely ∑_{n} τ(n)n^{−s} = ∏_{p}(1 − τ(p)p^{−s} + p^{11−2s})^{−1}. The conjectured ℓ-adic representations were constructed soon after by Deligne [19]. Specifically, Deligne constructed, for each prime ℓ, a representation

ρ_{ℓ^{∞}} : Gal(\overline{Q}/Q) → GL_{2}(Z_{ℓ}),

unramified outside ℓ, such that

tr(ρ_{ℓ^{∞}}(Frob_{p})) = τ(p), det(ρ_{ℓ^{∞}}(Frob_{p})) = p^{11}

for all primes p ≠ ℓ. On reducing ρ_{ℓ^{∞}} mod ℓ, we obtain a representation

ρ_{ℓ} : Gal(\overline{Q}/Q) → GL_{2}(F_{ℓ})

with analogous properties. (Equalities are replaced by congruences mod ℓ.) In other words, the ρ_{ℓ} for Δ is just like the ρ_{ℓ} for an elliptic curve E, except that the integers a_{p} are replaced by the corresponding values of the τ-function. The determinant of ρ_{ℓ} is the 11th power of the mod ℓ cyclotomic character χ : Gal(\overline{Q}/Q) → F_{ℓ}^{∗}, i.e., the character giving the action of Gal(\overline{Q}/Q) on the group of ℓth roots of 1 in \overline{Q}.

More generally, take a weight k ≥ 12 and suppose that f = ∑_{n} c_{n}q^{n} is a nonzero weight-k cusp form for SL(2, Z) that satisfies f|T_{n} = c_{n}f for all n ≥ 1, T_{n} being the nth Hecke operator on the space of cusp forms of weight k for SL(2, Z). (See, e.g., [70, Ch. VII].) Then the complex numbers c_{n} (n ≥ 1) are algebraic integers. Moreover, the field E := Q(…, c_{n}, …) generated by the c_{n} is a totally real number field (of finite degree over Q). Thus the c_{n} all lie in the integer ring O_{E} of E. For each ring homomorphism ϕ : O_{E} → \overline{F}_{ℓ}, one finds a representation

ρ = ρ_{ϕ} : Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{ℓ}),

unramified outside ℓ, such that

tr(ρ(Frob_{p})) = ϕ(c_{p}), det(ρ(Frob_{p})) = p^{k−1}

for all p ≠ ℓ. We have det ρ = χ^{k−1}. Of course, there is no guarantee that ρ is irreducible. We can (and do) suppose that ρ is semisimple by replacing it by its semisimplification. Then ρ is determined up to isomorphism by the displayed trace and determinant conditions; this follows from the Cebotarev density theorem and the Brauer-Nesbitt theorem [18], which states that semisimple representations are determined by their characteristic polynomials.

It is important to note that k is necessarily an even integer; otherwise the space S_{k}(SL(2, Z)) of weight-k cusp forms on SL(2, Z) is easily seen to be 0. Thus the determinant χ^{k−1} of ρ is an odd power of χ. In particular, det ρ : Gal(\overline{Q}/Q) → F_{ℓ}^{∗} is unramified outside ℓ and takes the value −1 on complex conjugations c ∈ Gal(\overline{Q}/Q). It’s a nice exercise (Exercise 12) to check that, conversely, all continuous homomorphisms with these properties are odd powers of χ.

In the early 1970s, Serre and Tate asked themselves whether all homomorphisms that are “like ρ” might come from cusp forms of some weight on SL(2, Z). Namely, let

ρ : Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{ℓ})

be a continuous, irreducible representation that is (1) unramified outside ℓ and (2) of odd determinant (in the sense that det ρ(c) = −1 ∈ \overline{F}_{ℓ} for complex conjugations c ∈ Gal(\overline{Q}/Q)). Is it true that ρ is of the form ρ_{ϕ}? This would mean that there is a k ≥ 12, an eigenform f ∈ S_{k}(SL(2, Z)), and a homomorphism ϕ : O_{E} → \overline{F}_{ℓ} (where O_{E} is the ring of integers of the field generated by the coefficients of f) so that ρ_{ϕ} and ρ are isomorphic. At the time, there was little evidence that the answer might be “yes”, and no general framework in which to state the question. If I understand correctly, Serre and Tate were willing to admit an affirmative answer to their question as a working hypothesis, but they were reluctant to conjecture that their hypothesis was correct.

To investigate a concrete consequence of an affirmative answer to the question, it is fruitful to consider the operation ρ ↦ ρ ⊗ χ on representations. This “twisting” operation preserves the set of representations that come from modular forms. Indeed, let θ = q \frac{d}{dq} be the operator of Serre and Swinnerton-Dyer [44, 68, 84], ∑ a_{n}q^{n} ↦ ∑ na_{n}q^{n}; if f is a mod ℓ form of weight k, then θf is naturally a mod ℓ form of weight k + ℓ + 1. Then if ρ is associated to f, ρ ⊗ χ is associated with θf. According to a result of Atkin, Serre and Tate [71, Th. 3], if ρ comes from an eigenform in some space S_{k}(SL(2, Z)), then a suitable twist ρ ⊗ χ^{i} of f comes from a form of weight ≤ ℓ + 1.

An affirmative answer to the Serre–Tate question thus has the following consequence: each two-dimensional irreducible odd representation of Gal(\overline{Q}/Q) over \overline{F}_{ℓ} that is unramified outside ℓ has a twist (by a power of χ) coming from an eigenform on SL(2, Z) of weight at most ℓ + 1. In particular, suppose that ℓ < 11. Then the spaces S_{k}(SL(2, Z)) with k ≤ ℓ + 1 are all 0; as a result, they contain no non-zero eigenforms! The conjecture that all ρ are modular (of level 1) thus predicts that there are no representations of the type contemplated if ℓ is 2, 3, 5 or 7. In support of the conjecture, the non-existence statement was proved for ℓ = 2 by J. Tate in a May, 1973 letter to Serre [85]. Soon after, Serre treated the case ℓ = 3 by methods similar to those of Tate. (See [85, p. 155] for a discussion and a reference to a note in Serre’s Œuvres.) Quite recently, S. Brueggeman considered the case ℓ = 5; she proved that the conjectured result follows from the Generalized Riemann Hypothesis [8].

One reason to be skeptical of the conjecture was advanced by Deligne: Suppose that one takes a ρ coming from an eigenform f′ of some weight and of level N > 1. On general grounds, ρ has the right to be ramified at primes p dividing N as well as at the prime ℓ. Suppose that, by accident as it were, ρ turned out to be unramified at all primes p | N. Then the conjecture would predict the existence of a level-1 form f′ (presumably of the same weight as f) whose mod ℓ representation was isomorphic to ρ. How could one manufacture the f′?

The passage f ⇝ f′ comes under the rubric of “level lowering”. When you take a representation ρ that comes from high level N, and it seems as though that representation comes from a lower level N′, then to “lower the level” is to cough up a form of level N′ that gives ρ.

1.1. Background material

1.1.1. The cyclotomic character

Let A be an elliptic curve and ℓ a prime number. The Weil pairing e_{ℓ} (see [82, III.8]) sets up an isomorphism of G_{Q}-modules

e_{ℓ} : ⋀^{2} A[ℓ] ⟶^{≅} μ_{ℓ}. (1.1)

The determinant of the representation ρ_{A,ℓ} is the mod ℓ cyclotomic character χ_{ℓ}. This is the character defined by considering the group μ_{ℓ} of ℓth roots of unity in \overline{Q}; the action of the Galois group G_{Q} on the cyclic group μ_{ℓ} gives rise to a continuous homomorphism

χ_{ℓ} : G_{Q} ⟶ Aut(μ_{ℓ}). (1.2)

Since μ_{ℓ} is a cyclic group of order ℓ, its group of automorphisms is canonically the group (Z/ℓZ)^{∗} = F_{ℓ}^{∗}. We emerge with a map G_{Q} → F_{ℓ}^{∗}, which is the character in question.

Suppose now that c ∈ G_{Q} is the automorphism “complex conjugation.” Then the determinant of ρ_{A,ℓ}(c) is χ_{ℓ}(c). Now c operates on roots of unity by the map ζ ↦ ζ^{−1}, since roots of unity have absolute value 1. Accordingly,

det ρ_{A,ℓ}(c) = −1; (1.3)

one says that ρ_{A,ℓ} is odd. If ℓ ≠ 2, then ρ_{A,ℓ}(c) is conjugate over \overline{F}_{ℓ} to ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) (Exercise 17). If ℓ = 2 then the characteristic polynomial of ρ_{A,ℓ}(c) is (x + 1)^{2} so ρ_{A,ℓ}(c) is conjugate over \overline{F}_{ℓ} to either the identity matrix or ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ).

1.1.2. Frobenius elements

Let A be an elliptic curve over Q and ℓ a prime number. The fixed field of ρ_{A,ℓ} is a finite Galois extension K_{ℓ}/Q whose Galois group G_{ℓ} is a subgroup of GL(2, F_{ℓ}). A key piece of information about the extension K_{ℓ}/Q is that its discriminant is divisible at most by ℓ and primes dividing the conductor of A. In other words, if p ≠ ℓ is a prime number at which A has good reduction, then K_{ℓ}/Q is unramified at ℓ (Exercise 25); one says that the representation ρ_{A,ℓ} is unramified at p. Whenever this occurs, a familiar construction in algebraic number theory produces a Frobenius element σ_{p} in G_{ℓ} that is well defined up to conjugation.

We shall now summarize this construction with K_{n} replaced by an arbitrary finite Galois extension K of Q. The Galois group Gal(K/Q) leaves the ring O_{K} of integers of K invariant, so that one obtains an induced action on the ideals of O_{K}. The set of prime ideals 𝔭 of O_{K} lying over p (i.e., that contain p) is permuted under this action. For each 𝔭, the subgroup D_{𝔭} of Gal(K/Q) fixing 𝔭 is called the decomposition group of 𝔭. Meanwhile, F_{𝔭} := O_{K}/𝔭 is a finite extension of F_{p}. The extension F_{𝔭}/F_{p} is necessarily Galois; its Galois group cyclic, generated by the Frobenius automorphism φ_{p} : x ↦ x^{p} of F_{𝔭}. There is a natural surjective map D_{𝔭} → Gal(F_{𝔭}/F_{p}); its injectivity is equivalent to the assertion that p is unramified in K/Q. Therefore, whenever this assertion is true, there is a uique σ_{𝔭} ∈ D_{𝔭} whose image in Gal(F_{𝔭}/F_{p}) is φ_{p}. The automorphism σ_{𝔭} is then a well defined element of Gal(K/Q) known as the Frobenius automorphism for 𝔭. It is easy to show that the various 𝔭 are all conjugate under Gal(K/Q) and that the Frobenius automorphism for the conjugate of 𝔭 by g is gσ_{𝔭}g^{−1}. In particular, the various σ_{𝔭} are all conjugate; this justifies the practice of writing σ_{p} for any one of them and stating that σ_{p} is well defined up to conjugation.

We prolong this discussion and introduce the concept of Frobenius elements in G = Gal(\overline{Q}/Q). Let p again be a prime and let 𝔭 now be a prime of the ring of integers of \overline{Q} lying over p. To 𝔭 we associate: (1) its residue field F_{𝔭}, which is an algebraic closure of F_{p}, and (2) a decomposition subgroup D_{𝔭} of G. There is again a surjective map D_{𝔭} → Gal(F_{𝔭}/F_{p}). The Frobenius automorphism φ_{p} topologically generates the target group. We shall use the symbol Frob_{p} to denote any preimage of φ_{p} in any D_{𝔭} corresponding to a prime lying over p, and refer to Frob_{p} as a Frobenius element for p in G. This element is doubly ill-defined. The ambiguity in Frob_{p} results from the circumstance that 𝔭 needs to be chosen and from the fact that D_{𝔭} → Gal(F_{𝔭}/F_{p}) has a large kernel, the inertia subgroup I_{𝔭} of D_{𝔭}. The usefulness of Frob_{𝔭} stems from the fact that the various 𝔭 are all conjugate, so that likewise the subgroups D_{𝔭} and I_{𝔭} are conjugate. Thus if ρ is a homomorphism mapping G to some other group, the kernel of ρ contains one I_{𝔭} iff it contains all I_{𝔭}. In this case, one says that ρ is unramified at p; the image of Frob_{p} is then an element of the target that is well defined up to conjugation.

Fix again an elliptic curve A and a prime number ℓ, and let ρ_{A,ℓ} : G → GL(2, F_{ℓ}) be the associated representation. For each prime p not dividing ℓ at which A has good reduction the Frobenius σ_{p} = ρ_{A,ℓ}(Frob_{p}) is well defined only up to conjugation. Nevertheless, the trace and determinant of σ_{p} are well defined. The determinant of ρ_{A,ℓ} is the mod ℓ cyclotomic character χ, so σ_{p} = χ(Frob_{p}) = p ∈ F_{ℓ}. On the other hand (Exercise 24), one has the striking congruence

tr(ρ_{A,ℓ}(Frob_{p})) = p + 1 − #Ã(F_{p}) (mod ℓ).

1.1.3. Modular forms

We now summarize some background material concerning modular forms. Serre’s book [70] is an excellent introduction (it treats only N = 1). One might also read the survey article [22] or consult any of the standard references [47, 48, 56, 81].

The modular group SL(2, Z) is the group of 2 × 2 invertible integer matrices. For each positive integer N, consider the subgroup

Γ_{1}(N) := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL(2, Z) : N | c and a ≡ d ≡ 1 (mod N)\} .

Let 𝔥 be the complex upper half plane. A cuspidal modular form of integer weight k ≥ 1 and level N is a holomorphic function f(z) on 𝔥 such that

f (\frac{az + b}{cz + d}) = (cz + d)^{k}f(z) for all (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{1}(N) (1.4)

and that “vanishes at the cusps” (see [81, §2.1]). We denote by S_{k}(Γ_{1}(N)) the space of weight k cusps forms of level N. It is a finite dimensional complex vector space. When k ≥ 2 a formula for the dimension can be found in [81, §2.6].

Modular forms are usually presented as convergent Fourier series

f(z) = ∑_{n=1}^{∞} a_{n}q^{n}

where q := e^{2πiz}. This is possible because the matrices ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) lie in Γ_{1}(N) so that f(z + b) = f(z) for all integers b. For the forms that most interest us, the complex numbers a_{n} are algebraic integers.

The space S_{k}(Γ_{1}(N)) has an action of (Z/NZ)^{∗}, given by

f(z) ↦ f|⟨\overline{d}⟩(z) = (cz + d)^{−k}f (\frac{az + b}{cz + d})

for any choice of ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, Z) such that d ≡ \overline{d} (mod N). The operator ⟨d⟩ = ⟨\overline{d}⟩ is referred to as “diamond brackets d.”

For each integer n ≥ 1, the nth Hecke operator on S_{k}(Γ_{1}(N)) is an endomorphism T_{n} of S_{k}(Γ_{1}(N)). The action is generally written on the right: f ↦ f|T_{n}. The various T_{n} commute with each other and are interrelated by identities that express a given T_{n} in terms of the Hecke operators indexed by the prime factors of n. If p ∤ N is a prime define the operator T_{p} on S_{k}(Γ_{1}(N)) by

f|T_{p}(z) = ∑_{n=1}^{∞} a_{np}q^{n} + ∑_{n=1}^{∞} a_{n}(f|⟨p⟩)q^{np}.

For p | N prime, define T_{p} by

f|T_{p}(z) = ∑_{n=1}^{∞} a_{np}q^{n}.

The Hecke algebra is the ring

T ⊂ End(S_{k}(Γ_{1}(N)))

generated over Z by all the T_{n} and ⟨d⟩. It is finite as a module over Z (Exercise 30).

An eigenform is a nonzero element f ∈ S_{k}(Γ_{1}(N)) that is an eigenvector for every element of the Hecke algebra T. Writing f = ∑ a_{n}q^{n} we find that a_{n} = a_{1}c_{n} where c_{n} is the eigenvalue of T_{n} on f. Since f is nonzero a_{1} is not zero either, so it is possible to multiply f by a_{1}^{−1}. The resulting normalized eigenform or newform wears its eigenvalues on its sleeve: f = ∑ c_{n}q^{n}.

We now recall the notion of newform. Associated to an eigenform f ∈ S_{k}(Γ_{1}(N)) we have a system (… a_{p} …), p ∤ N, of eigenvalues. We say that f is a newform if this system of eigenvalues does not comes from any level M | N with M ≠ N. Newforms have been extensively studied (see [2, 12, 51, 56]); the idea is to understand where systems of eigenvalues first arise, and then reconstruct the full space from the newforms.

Let f ∈ S_{k}(Γ_{1}(N)) be a newform and ρ the associated representation. By Exercise 17, ε(−1) = (−1)^{k}, so

det(ρ(c)) = χ^{k−1}(−1) · ε(−1) = (−1)^{k−1}(−1)^{k} = (−1)^{k+k−1} = −1.

LECTURE 2

The weak and strong conjectures

[[DEFINITELY SAME SOMETHING ABOUT THE CONFUSION IN THE LITERATURE BETWEEN WEAK AND STRONG.]]

2.1. The weak conjecture

In the mid 1980s, G. Frey began lecturing on the link between Fermat’s Last Theorem and elliptic curves. As you probably know, he proposed that if a^{ℓ} + b^{ℓ} was a perfect ℓth power, then the elliptic curve y^{2} = x(x − a^{ℓ})(x + b^{ℓ}) could be proved to be non-modular [30, 31]. Soon after, Serre pointed out that the non-modularity contemplated by Frey would follow from suitable level-lowering results concerning modular forms [74]. If I understand correctly, Serre went back to the tentative conjecture that he had made 15 years earlier and decided to treat representations ρ : Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{ℓ}) that may be ramified at primes other than ℓ. The result, of course, were the conjectures of [75].

An important element of these conjectures is the so-called “weak conjecture of Serre.” As background, I recall (but really mention for the first time at this point), that Hecke eigenforms on congruence subgroups of SL(2, Z) give rise to two-dimensional representations of Gal(\overline{Q}/Q). If we set things up correctly, we get representations over \overline{F}_{ℓ}. More specifically, take integers k ≥ 2 and N ≥ 1; these are the weight and level, respectively. Let f = ∑ a_{n}q^{n} be a normalized Hecke eigenform in the space S_{k}(Γ_{1}(N)) of complex weight-k cusp forms on the subgroup Γ_{1}(N) of SL(2, Z). Thus f is non-zero and it satisfies f|T_{n} = a_{n}f for all n ≥ 1. Further, there is a character ε : (Z/NZ)^{∗} → C^{∗} so that f|⟨d⟩ = ε(d)f for all d ∈ (Z/NZ)^{∗}; here ⟨d⟩ denotes the so-called “diamond-bracket” operator. (It’s really a Hecke operator, and it lies in the ring generated by the T_{n} as they act on S_{k}(Γ_{1}(N)).) Again, let O be the ring of integers of the field Q(… a_{n} …) generated by the a_{n}; this field is a number field that is either totally real or a CM field. Consider a ring homomorphism φ : O → \overline{F}_{ℓ} as before. Associated to this set-up is a representation ρ : Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{ℓ}) with properties that connect it up with f (and φ): First, the representation is unramified at all p not dividing ℓN. Next, for all such p, we have

tr(ρ(Frob_{p})) = a_{p}, det(ρ(Frob_{p})) = p^{k−1}ε(p);

the numbers a_{p} and p^{k−1}ε(p), a priori in O, are mapped tacitly into \overline{F}_{ℓ} by φ. The representation ρ is determined us to isomorphism by the trace and determinant identities that are displayed, plus the supplemental requirement that it be semisimple. We are interested mainly in the (generic) case in which ρ is irreducible; in that case, it is of course semisimple.

The construction of ρ from f, k and φ was accomplished in principle in [19]. Indeed, Deligne sketches a method in his article that manufactures for each nonarchimedean prime λ of E a representation ρ̃_{λ} : Gal(\overline{Q}/Q) → GL_{2}(E_{λ}), where E_{λ} denotes the completion of E at λ. Given φ, we let λ be ker φ and find a model of ρ̃_{λ} over the ring of integers O_{λ} of E_{λ}. Reducing mod λ, we get a representation over the residue field O/λO of O_{λ}. The map φ embeds this finite field into \overline{F}_{ℓ}.

In retrospect, one remarks, however, that the machinary of [19] can be avoided if one seeks only the mod λ representation attached to f (as opposed to the full λ-adic representation ρ̃_{λ}). As I point out in [63], one can use congruences among modular forms to find a form of weight two and level Nℓ^{2} that gives rise to ρ. Accordingly, one can find ρ concretely by looking within the group of ℓ-division points of a suitable abelian variety over Q: the variety J_{1}(ℓ^{2}N).

Which representations of Gal(\overline{Q}/Q) arise in this way (as k, N, f and φ all vary)? As in the case N = 1 (i.e., that where Γ_{1}(N) = SL(2, Z)), any ρ that comes from modular forms is an odd representation: we have det(ρ(c)) = −1 when c ∈ Gal(\overline{Q}/Q) is a complex conjugation. To see this, we begin with the fact that ε(−1) = (−1)^{k}, which generalizes ( 1.3); this follows from the functional equation that relates f(\frac{az+b}{cz+d}) to f when ( \begin{matrix}a & b \\ c & d \end{matrix} ) is an element of Γ_{0}(N) (see Exercise 17). On the other hand, using the Cebotarev density theorem, we find that det ρ = χ^{k−1}ε, where χ is again the mod ℓ cyclotomic character and where ε is regarded now as a map Gal(\overline{Q}/Q) → \overline{F}_{ℓ}^{∗} in the obvious way, namely by composing ε : (Z/NZ)^{∗} → \overline{F}_{ℓ}^{∗} with the mod N cyclotomic character. The value on c of the latter incarnation of ε is the number ε(−1) = (−1)^{k}. Since χ(c) = −1, we deduce that (det ρ)(c) = −1, as was claimed.

Serre’s weak conjecture states that, conversely, every irreducible odd representation ρ : Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{ℓ}) is modular in the sense that it arises from some f and φ.

This conjecture has a certain number of amazing consequences. Among other things, it implies that all odd irreducible 2-dimensional ρ come from abelian varieties over Q. Given ρ, one should be able to find a totally real or CM number field E, an abelian variety A over Q of dimension [E : Q] that comes equipped with an action of the ring of integers O of E, and a ring homomorphism φ : O → \overline{F}_{ℓ} with the following property: Let λ = ker φ. Then the representation A[λ] ⊗_{O/λ} \overline{F}_{ℓ} is isomorphic to ρ. (In comparing A[λ] and ρ, we use φ : O/λ ↪ \overline{F}_{ℓ} to promote the 2-dimensional A[λ] into a representation over \overline{F}_{ℓ}.)

Another frequently noted consequence of the weak conjecture is that representations ρ over \overline{F}_{ℓ} are required to “lift” to λ-adic representations of Gal(\overline{Q}/Q). In a recent article [58], R. Ramakrishna proved a result in this direction.

Much of the evidence for the weak conjecture concerns representations taking values in GL_{2}(F_{q}) where the finite field F_{q} has small cardinality. In his original article [75, §5], Serre uses theorems of Langlands [50] and Tunnell [87] to establish his weak conjecture for odd irreducible representations with values in GL_{2}(F_{2}) and GL_{2}(F_{3}). Further, he reports on numerical computations of J.-F. Mestre that pertain to representations over F_{4} (and trivial determinant). Additionally, Serre remarks [75, p. 219] that the weak conjecture is true for those representations with values in GL_{2}(\overline{F}_{p}) that are dihedral in the sense that their projective images (in PGL(2, \overline{F}_{p})) are dihedral groups. (See also [23, §5] for a related argument.)

More recently, representations over the fields F_{4} and F_{5} were treated, under somewhat mild hypotheses, by Shepherd-Barron and Taylor [78]. For example, Shepherd-Barron and Taylor show that ρ : Gal(\overline{Q}/Q) → GL_{2}(F_{5}) is modular if it is odd, irreducible and unramified at 3, provided in addition that det ρ is the mod 5 cyclotomic character.

2.2. The strong conjecture

Fix an odd irreducible Galois representation

ρ : G_{Q} ⟶ GL(2, \overline{F}_{ℓ}).

The weak conjecture asserts that ρ is modular, in the sense that there exists integers N and k such that ρ comes from some f ∈ S_{k}(Γ_{1}(N)). The strong conjecture goes further and gives a recipe for integers N(ρ) and k(ρ) and then asserts that ρ comes from S_{k(ρ)}(Γ_{1}(N(ρ))). In particular instances the strong conjecture is verifiable (at least in theory) because the space S_{k(ρ)}(Γ_{1}(N(ρ))) is finite dimensional and can be computed. The relation between the weak and strong conjectures is like the difference between asserting that every elliptic curve is modular of some level N and asserting that every elliptic curve is modular of level equal to the conductor of the curve.

For each prime p, let I_{p} denote an inertia group at p (see §1.1.2). The level N(ρ) and weight k(ρ) associated to the mod ℓ representation ρ satisfy the following two properties.

• First, N(ρ) = ∏_{p≠ℓ} p^{n(p)}, where n(p) depends only on ρ|_{I_{p}}.
• Second, k(ρ) depends only on ρ|_{I_{ℓ}}.

The integer n(p) is a conductor in additive notation. In particular,

n(p) = 0 ⟺ ρ is unramified at p.

View ρ as a homomorphism G_{Q} → Aut(V), where V is a two-dimensional vector space over \overline{F}_{ℓ}. It is natural to consider the subspace

V^{I_{p}} = {v ∈ V : ρ(σ)v = v, all σ ∈ I_{p}}

of inertia invariants. For example, V^{I_{p}} = V exactly when ρ is unramified at p. As a first approximation write n(p) = dim(V/V^{I_{p}}) + (wild term) where the wild term is the Swan conductor; in particular it is nonnegative.

Suppose ρ comes from a newform f ∈ S_{k}(Γ_{1}(N)), but do not make the assumption that k and N are Serre’s optimal k(ρ) and N(ρ). Since N(ρ) is optimal, N(ρ) | N. It is productive to inquiry about the quotient N/N(ρ). Let O be the ring of integers of the field generated by the Fourier coefficients of f and let φ : O → \overline{F}_{ℓ} be the map such that φ(a_{p}) = tr(ρ(Frob_{p})). Let λ be a prime of O lying over ℓ and E_{λ} be the completion of Frac(O). Deligne [19] attached to f and λ a representation

ρ_{λ} : G_{Q} ⟶ GL(2, E_{λ}) = Aut(Ṽ)

where Ṽ is a two-dimensional vector space over E_{λ}. The representation ρ_{λ} can be chosen in such a way that it stabilizes O_{λ}, and so that the reduction modulo λ is ρ. The following diagram summarizes the set up.

\xymatrix{ &  & GL(2, E_{λ}) = Aut(Ṽ) \\ G_{Q} \ar[urr]^{ρ_{λ}} \ar[drr]_{ρ} \ar[r]^{ρ_{λ}} & GL(2, O_{λ}) \ar[dr]^{φ} \ar@{^{(}->}[ur]^{ι} \\  &  & GL(2, \overline{F}_{ℓ}) = Aut(V)}

[[CITE CARAYOL, Duke 1989: “Have ord_{p} N = n_{p}(ρ̃) by Carayol. Have n_{p}(ρ̃) = dim(Ṽ /Ṽ^{I_{p}}) + (wild term). Then wild=wild.”]]

Though Ṽ and V are vector spaces over different fields, we can compare the dimensions of their inertia invariant subspaces. These satisfy [[SAY WHY]] dim Ṽ^{I_{p}} ≤ dim V^{I_{p}}, but we need not have equality. For example, it could be that Ṽ is highly ramified that Ṽ^{I_{p}} = 0, but V is unramified so dim V^{I_{p}} = dim V = 2. The formula

(2.1) ord_{p}(N) = n(p) + (dim V^{I_{p}} − dim Ṽ^{I_{p}})

indicates how this difference measures the deviation of N from the optimal level. This is the description of n(p) that is used in attempting to prove that if ρ is modular at all, then it is possible to refine N and k to eventually discover that ρ indeed does come from S_{k(ρ)}(Γ_{1}(N(ρ))). After much work [21, 63]^{1} it has been shown that for ℓ > 2 the weak and strong conjectures are equivalent.

Rearranging (2.1) into

n(p) = ord_{p}(N) − (dim V^{I_{p}} − dim Ṽ^{I_{p}})

provides us with a way to read off N(ρ) from ord_{p}(N), dim V^{I_{p}}, and dim Ṽ^{I_{p}}. If f ∈ S_{k}(Γ_{1}(N)) gives rise to ρ and ℓ ∤ N, then k(ρ) ≤ k. In contrast, if we allow powers of ℓ in the level then the weight k can always be made equal to 2. Amazingly (to me), Serre’s conjectures imply that all odd irreducible two-dimensional mod ℓ representations of G_{Q} may be realized in spaces of ℓ-division points on abelian varieties. The weak Serre conjecture states that all such representations come from modular forms, and then it takes only a bit of technique to show that one can take the modular forms to have weight two (if one allows powers of ℓ in the level).

2.3. Representations arising from elliptic curves

Equations for elliptic curves can be found in tables such as [4, 17]. Consider the elliptic curve B of given by the equation

y^{2} + y = x^{3} + x^{2} − 12x + 2. (2.2)

This is the curve labeled 141A1 in [17]; it has conductor N = 141 = 3 · 47 and discriminant Δ = 3^{7} · 47. Let f ∈ S_{2}(Γ_{0}(141)) be the newform attached to B. The curve B is isolated in its isogeny class, equivalently, for every ℓ, the representation

ρ_{ℓ} = ρ_{B,ℓ} : G_{Q} → GL(2, F_{ℓ}) ≈ Aut(B[ℓ])

1More references?

is irreducible (see Exercise 14 and Exercise 15). It follows from results of Serre that each ρ_{ℓ} is surjective [67] [[MORE REFS, DISCUSSION ABOUT ℓ = 2]]. We will return to this example periodically in the course of these lectures.

We determine N(ρ_{ℓ}) and k(ρ_{ℓ}) for each prime ℓ. At primes of bad reduction, after a possibly quadratic extension, B becomes a Tate curve. This can be used [76]^{2} to show that for p ≠ ℓ,

ρ_{ℓ} is unramified at p ⟺ ord_{p}(Δ) ≡ 0 (mod ℓ).

[[THE FOLLOWING IS INCOMPREHENSIBLY WRITTEN.]] Take p = 3, ℓ ≠ 3: then ρ_{ℓ} is unramified at 3 iff ℓ | ord_{3}(Δ) = 7, i.e., when ℓ = 7. Furthermore, ρ_{ℓ} is always ramified at 47. What does this say about the optimal level, in the sense of Serre’s conjecture? Take ℓ ≠ 3, 47. Then N(ρ_{ℓ}) | N = 3 · 47. If ℓ ≠ 7 then N(ρ_{ℓ}) = N = 3 · 47, and the strong case is verified. If ℓ = 7 then N(ρ_{ℓ}) = 47, and k(ρ_{ℓ}) = 2 since ℓ ∤ 3 · 47.

Still taking p = 3, the remaining cases are ℓ = 3, 47. If ℓ = 47 then N(ρ_{ℓ}) = 3. Because the cyclotomic character has order ℓ − 1, k(ρ_{ℓ}) ≡ 2 (mod 47 − 1). Serre’s recipe gives k(ρ_{ℓ}) = 2 + (47 − 1) = 48. In order to verify the strong conjecture we need a mechanism for relating eigenforms in S_{2}(Γ_{0}(141); F_{ℓ}) to eigenforms in S_{48}(Γ_{0}(3); F_{ℓ}). In Serre [4] he constructs such a map for level one. Koike and others generalized this map to higher level.

Let us return to case ℓ = 7, in which the optimal level N(ρ_{ℓ}) = 47 is a proper divisor of N = 3·47. The strong conjecture predicts the existence of g ∈ S_{2}(Γ_{0}(47)) giving rise to ρ. Our initial instinct is to look for an elliptic curve A of conductor 47 such that A[ℓ] = B[ℓ]. In fact, there is no elliptic curve of conductor 47; the space S_{2}(Γ_{0}(47)) is four-dimensional, having basis the Galois conjugates of a single eigenform f = ∑ c_{n}q^{n}. The Fourier coefficients c_{n} of f generate the full ring of integers in the field K obtained from Q by adjoining a root of h(x) = x^{4} − x^{3} − 5x^{2} + 5x − 1. The discriminant 1957 = 19 · 103 of K equals the discriminant of h, therefore a root of h generates the full ring of integers. The eigenvalue c_{2} satisfies h(c_{2}) = 0. Since h(x) ≡ (x + 2)(x^{3} + 4x^{2} + x + 3) (mod 7), there is a prime λ lying over 7 such that O/λ ≅ F_{7}; the isomorphism is given by c_{2} ↦ −2 mod 7. As a check, note that #B(F_{2}) = 5 so a_{2} = 3 − 5 = −2 = φ(c_{2}). More generally, for p ∤ 7 · 141, we have φ(c_{p}) ≡ a_{p} mod 7.

We just stripped 3 from the level. The representation ρ_{7} starts at level 141 = 3 · 47 and is unramfied at 3. We verified that there is a form of level 47 giving rise to ρ. In general the easier case, due to Mazur, is when p ≢ 1 (mod ℓ). The hard case, due to the first author, is when p ≡ 1 (mod ℓ).

2Where in here?

LECTURE 3

The weight in Serre’s conjecture

In [75, §2], Serre associated to a continuous, odd, irreducible Galois representation ρ : G_{Q} → GL(2, \overline{F}_{ℓ}) two integers N(ρ) and k(ρ), meant to be the minimal level and weight of a form giving rise to ρ. He then made the following conjecture:

Conjecture 1 (Weak conjecture of Serre). Let ρ : G_{Q} → GL(2, \overline{F}_{ℓ}) be an odd irreducible Galois representation arising from a modular form. Then ρ arises from a modular form of level N(ρ) and weight k(ρ).

In this lecture we are concerned with k(ρ). We consider a representation ρ that arises from an eigenform of level N = N(ρ) and weight k ≥ k(ρ). [[EDIXHOVEN DOESN’T ASSUME N = N(ρ); he assumes ℓ ∤ N.]] Using various arguments involving the Eichler-Shimura correspondence and Tate’s θ-cycles, Edixhoven showed in [25] that there must exist another form of weight k(ρ), still of level N(ρ), that gives rise to ρ. When ℓ = 2, Edixhoven’s result relies on unchecked compatibilities assumed in [34]. In this lecture we sketch his arguments to give a flavor of the subject.

3.1. Representations arising from forms of low weight

We first consider Galois representations associated to newforms of low weight: Fix a prime ℓ and suppose f = ∑ a_{n}q^{n} is a newform of weight k and level N, such that ℓ ∤ N and 2 ≤ k ≤ ℓ + 1. (We avoid weight one forms in this lecture.) Fix a homomorphism φ from the ring of integers O of Q(…, a_{n}, …) to \overline{F}_{ℓ}. To abbreviate notation, we often write a_{n} for φ(a_{n}); whereby thinking of a_{n} as an element \overline{F}_{ℓ}. Let

ρ = ρ_{f,φ} : G_{Q} → GL(2, \overline{F}_{ℓ})

be the two-dimensional odd Galois representation attached to f and φ. The recipe for N(ρ) depends on the local behavior of ρ at primes p other than ℓ; the recipe for k(ρ) depends on the restriction ρ|_{I_{ℓ}} of ρ to the inertia group at ℓ. Motivated by Serre’s questions, Fontaine and Deligne described the Galois representations ρ|_{I_{ℓ}} in many situations. We distinguish two cases, the ordinary case and the supersingular case.

3.1.1. The ordinary case

Deligne (see [34, Prop. 12.1]) considered the ordinary case, in which ρ arises from a newform f with a_{ℓ}(f) ≠ 0 ∈ \overline{F}_{ℓ}. He showed that ρ has a one-dimensional unramified quotient β; more precisely,

ρ|_{D_{ℓ}} ∼ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

with β(I_{ℓ}) = 1 and αβ = χ^{k−1}ε. The mod N character ε is also unramified; this is because ℓ does not divide N, but the conductor of ε does. Since the mod ℓ cyclotomic character χ has order ℓ − 1 and

ρ|_{I_{ℓ}} ∼ (\begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix}) ,

the value of k modulo ℓ − 1 is determined by ρ|_{I_{ℓ}}. When k is neither 2 nor ℓ + 1, this determines k. Choosing between 2 and ℓ + 1 is discussed in Lecture 4.

3.1.2. The supersingular case and fundamental characters

Fontaine (see [26, §6]) investigated the supersingular case, in which ρ arises from a newform f with a_{ℓ}(f) = 0 ∈ \overline{F}_{ℓ}. To describe the restriction ρ|_{I_{ℓ}} of ρ to the inertia group at ℓ, it is necessary to introduce the fundamental characters of the tame inertia group. Fix an algebraic closure \overline{Q}_{ℓ} of the field Q_{ℓ} of ℓ-adic numbers; let Q_{ℓ}^{nr} ⊂ \overline{Q}_{ℓ} denote the maximal unramified extension of Q_{ℓ}, and Q_{ℓ}^{tm} ⊂ \overline{Q}_{ℓ} the maximal tamely ramified extension of Q_{ℓ}^{nr}. The extension Q_{ℓ}^{tm} is the compositum of all finite extensions of Q_{ℓ}^{nr} in \overline{Q}_{ℓ} of degree prime to ℓ. Letting D_{ℓ} denote the decomposition group, I_{ℓ} the inertia group, I_{t} the tame inertia group, and I_{w} the wild inertia group, we have the following diagram:

\overline{Q}_{ℓ}_{D_{ℓ}}  _{I_{ℓ}}_{I_{w}} Q_{ℓ}^{tm} _{I_{t}} Q_{ℓ}^{nr} Ẑ Q_{ℓ}

It is a standard fact (see e.g., [32, §8]) that the extensions Q_{ℓ}^{nr}( \sqrt[n]{ℓ}), for all n not divisible by ℓ, generate Q_{ℓ}^{tm}. For n not divisible by ℓ, the nth roots of unity μ_{n} are contained in Q_{ℓ}^{nr}. Kummer theory (see [3]) gives, for each n, a canonical isomorphism

Gal(Q_{ℓ}^{nr}( \sqrt[n]{ℓ})/Q_{ℓ}^{nr}) ⟶ μ_{n}, σ ↦ \frac{σ( \sqrt[n]{ℓ})}{\sqrt[n]{ℓ}}.

Each such isomorphism lifts to a map I_{ℓ} → μ_{n}, which factors through the tame quotient I_{t} of I_{ℓ}. The groups μ_{n} = μ_{n}(\overline{Q}_{ℓ}) are contained in the ring of integers \overline{Z}_{ℓ} of \overline{Q}_{ℓ}. Composing any of the maps I_{t} → μ_{n} with reduction modulo the maximal ideal of \overline{Z}_{ℓ} gives a mod ℓ character I_{t} → \overline{F}_{ℓ}^{∗}, as illustrated:

\xymatrix{I_{t} \ar[rrr] \ar[dr] &  &  & \overline{F}_{ℓ}^{∗} \\  & μ_{n}(\overline{Q}_{ℓ}) \ar[r]^{≅} & μ_{n}(\overline{F}_{ℓ}). \ar@{^{(}->}[ur]}

Consider n = ℓ^{ν} − 1 for ν > 0. The map I_{t} → μ_{n} defines a character ε : I_{ℓ} → F_{ℓ^{ν}}^{∗}. Composing with each of the ν field embeddings F_{ℓ^{ν}} → \overline{F}_{ℓ} gives the ν fundamental character of level ν:

[figure: I_{t}; \overline{F}_{ℓ}.; F_{ℓ^{ν}}^{∗} ⊂ F_{ℓ^{ν}}; ν maps]

For example, the unique fundamental character of level 1 is the mod ℓ cyclotomic character (see Exercise 26). When ν = 2, there are two fundamental characters, denoted Ψ and Ψ′; these satisfy Ψ^{ℓ} = Ψ′ and (Ψ′)^{ℓ} = Ψ.

Consider a supersingular elliptic curve A over Q_{ℓ}. In [67], Serre proved that the representation I_{t} → Aut(A[ℓ]) ⊂ GL(2, \overline{F}_{ℓ}) is the direct sum of the two fundamental characters Ψ and Ψ′. More precisely, one of the characters is

I_{t} → F_{ℓ^{2}}^{∗} ⊂ GL(2, F_{ℓ})

where F_{ℓ^{2}}^{∗} sits inside GL(2, F_{ℓ}) via the action of the multiplicative group of a field on itself after a choice of basis. In analogy, Fontaine proved that if f is supersingular of weight k ≤ ℓ, then ρ|_{I_{ℓ}} : I_{ℓ} → GL(2, \overline{F}_{ℓ}) factors through I_{t} and is a direct sum of the two character Ψ^{k−1} and (Ψ′)^{k−1}. This determines k modulo ℓ^{2} − 1, hence determines k.

[[WAIT! WE NEVER GAVE ρ|_{I_{ℓ}} in general!!!!!]]

3.2. Representations arising from forms of high weight

The previous section described how to determine the weight (modulo ℓ − 1) of a form f of weight 2 ≤ k ≤ l + 1 purely from properties of the corresponding Galois representation. Consider a modular representation ρ : D_{ℓ} → GL(2, \overline{F}_{ℓ}) of a decomposition group at ℓ that arises from a form f of possibly large weight k. Let ρ^{ss} denote the semisimplification of ρ; so ρ^{ss} = ρ if ρ is irreducible, otherwise ρ^{ss} is a direct sum of two characters α and β. The following lemma is due to Serre (see [67, Prop. 4]).

Lemma 2. The semisimplification ρ^{ss} of ρ is tame, in the sense that ρ^{ss}(I_{w}) = 0.

Proof. The wild inertia group I_{w} is the profinite Sylow ℓ-subgroup of I_{ℓ}: it is a Sylow ℓ-subgroup because each finite Galois extension of Q_{ℓ}^{tm} has degree a power of ℓ, and the order of I_{t} is prime to ℓ; it is unique, because it is the kernel of Gal(\overline{Q}_{ℓ}/Q_{ℓ}) → Gal(\overline{Q}_{ℓ}/Q_{ℓ}^{tm}), hence normal. Because ρ is continuous, the image of D_{ℓ} is finite and we can view ρ as a representation on a two-dimensional vector space W over a finite extension of F_{ℓ}. The subspace

W^{I_{w}} = {w ∈ W : σ(τ)w = w for all τ ∈ I_{w}}

is invariant under D_{ℓ}. It is nonzero, as can be seen by writing the finite set W as a disjoint union of its orbits under I_{w}. Since I_{w} is an ℓ-group, each orbit has size either 1 or a positive power of ℓ. The orbit {0} has size 1, and #W is a power of ℓ, so there must be at least ℓ − 1 other singleton orbits {w}; for each of these, w ∈ W^{I_{w}}.

If I_{w} does not act trivially, then W^{I_{w}} ≠ W so W contains a D_{ℓ}-invariant subspace of dimension one; semisimplicity of σ then forces every σ(τ) to be diagonalizable. Any τ ∈ I_{w} has order a power ℓ^{n} of ℓ, so there are α, β ∈ F_{ℓ^{ν}} with σ(τ) = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ), and α^{ℓ^{n}} = β^{ℓ^{n}} = 1. Since α, β ∈ F_{ℓ^{ν}}^{∗}, their orders divide |F_{ℓ^{ν}}^{∗} | = ℓ^{ν} − 1, hence α = β = 1. Thus I_{w} acts trivially.

The restriction ρ^{ss}|_{I_{ℓ}} is abelian and semisimple, so it is given by a pair of characters

α, β : I_{ℓ} → \overline{F}_{ℓ}^{∗}.

Since ρ^{ss}|_{I_{ℓ}} is the restriction of a representation of the full decomposition group, these characters exhibit stability properties. Let n be an integer not divisible by ℓ, and consider the tower of fields

Q_{ℓ}^{nr}( \sqrt[n]{ℓ}) G  _{μ_{n}} Q_{ℓ}^{nr} Ẑ=⟨Frob_{ℓ}⟩ Q_{ℓ}

in which G = Gal(Q_{ℓ}^{nr}( \sqrt[n]{ℓ})/Q_{ℓ}), μ_{n} ≅ Gal(Q_{ℓ}^{nr}( \sqrt[n]{ℓ})/Q_{ℓ}^{nr}), and Gal(Q_{ℓ}^{nr}/Q_{ℓ}) is topologically generated by Frob_{ℓ}. Choose a lift g ∈ G of Frob_{ℓ}, and suppose h ∈ μ_{n} corresponds to σ ∈ Gal(Q_{ℓ}^{nr}( \sqrt[n]{ℓ})/Q_{ℓ}^{nr}). Then

\frac{gσg^{−1}( \sqrt[n]{ℓ})}{\sqrt[n]{ℓ}} = \frac{gσ(ζ_{g^{−1}} \sqrt[n]{ℓ})}{\sqrt[n]{ℓ}} = \frac{g(ζ_{g^{−1}}h \sqrt[n]{ℓ})}{\sqrt[n]{ℓ}} = \frac{g(h) \sqrt[n]{ℓ}}{\sqrt[n]{ℓ}} = h^{ℓ},

since g acts as ℓth power map on roots of unity. Applying the conjugation formula ghg^{−1} = h^{ℓ} to ρ^{ss} gives ρ^{ss}(ghg^{−1}) = ρ^{ss}(h^{ℓ}) = ρ^{ss}(h)^{ℓ}, so

ρ^{ss}(g)ρ^{ss}(h)ρ^{ss}(g^{−1}) = ρ^{ss}(ghg^{−1}) = ρ^{ss}(h)^{ℓ}.

The two representations h ↦ ρ^{ss}(h)^{ℓ} and h ↦ ρ^{ss}(h) of I_{t} are equivalent via conjugation by ρ^{ss}(g). Consequently, the pair of characters {α, β} is stable under ℓth power map; as a set {α, β} = {α^{ℓ}, β^{ℓ}}. There are two possibilities:

— The ordinary case: α^{ℓ} = α, β^{ℓ} = β, so α, β : I_{t} → F_{ℓ}^{∗}.

— The supersingular case: α^{ℓ} = β ≠ α, β^{ℓ} = α ≠ β, so α, β : I_{t} → F_{ℓ^{2}}^{∗}. By theorems of Fontaine and Deligne this terminology is consistent with the terminology introduced above.

First consider the supersingular case. Let Ψ denote one of the fundamental characters of level 2, and write α = Ψ^{n}, β = Ψ^{nℓ}, with n an integer modulo ℓ^{2} − 1. Write n in base ℓ as n = a+ℓb with 0 ≤ a, b ≤ ℓ−1. Then ℓn ≡ b+ℓa (mod ℓ^{2} −1).

Switching α, β permutes a, b so, relabeling if necessary, we may assume that a ≤ b. If a = b, then α = Ψ^{a}(Ψ′)^{a} = χ^{a}, so α takes values in F_{ℓ}^{∗}, which is not the supersingular case; thus we may assume that 0 ≤ a < b ≤ ℓ − 1. Factoring out by a power of the cyclotomic character gives

α = Ψ^{n} = Ψ^{a}(Ψ′)^{b} = Ψ^{a}(Ψ′)^{a}(Ψ′)^{b−a} = χ^{a}(Ψ′)^{b−a}
β = χ^{a}Ψ^{b−a}.

Put another way,

ρ^{ss} ∼ χ^{a} ⊗ (\begin{matrix}Ψ^{b−a} & 0 \\ 0 & (Ψ′)^{b−a} \end{matrix}) .

The untwisted representation is ( \begin{matrix}Ψ^{k−1} & 0 \\ 0 & (Ψ′)^{k−1} \end{matrix} ), where k = 1 + b − a. Since 2 ≤ 1+b−a ≤ ℓ−1, the weight of the untwisted representation is in the range considered above.

Before giving k(ρ), it is necessary to understand how the weight changes upon twisting by a power of the cyclotomic character χ. This problem is addressed by the theory of mod ℓ modular forms, first developed by Serre and Swinnerton-Dyer, then generalized by Katz. In [44], Katz defined spaces of mod ℓ modular forms (see Appendix 10), and a q-expansion map

α : ⨁_{k≥0} M_{k}(Γ_{1}(N); F_{ℓ}) → F_{ℓ}[[q]].

This map is not injective, because both the Hasse invariant of weight ℓ − 1 and the constant 1 have the same q-expansion.

The minimal weight filtration w(f) of a mod ℓ modular form f is the smallest integer k such that the q-expansion of f comes from a modular form of weight k; if no such k exists, do not define w(f). Define an operator θ = q \frac{d}{dq}, on q-expansions, by θ(∑ a_{n}q^{n}) = ∑ na_{n}q^{n}. If f is an eigenform of weight k, then there is a mod ℓ eigenform θf of weight k + ℓ + 1, still of level N, whose q-expansion is θ(∑ a_{n}q^{n}).

Theorem 3. Let f be a mod ℓ eigenform. Then

ℓ ∤ w(f) ⟺ w(θf) = w(f) + ℓ + 1.

If ℓ | w(f) then w(θf) < w(f) + ℓ + 1.

We are now ready to give the recipe for k(ρ) in the supersingular case. The minimal weight, before twisting, is 1 + b − a, which is positive and not divisible by ℓ. Each twist by χ adds ℓ + 1 to the weight, so in the supersingular case, one defines

k(ρ) := (1 + b − a) + a(ℓ + 1) = 1 + ℓa + b.

To justify this definition, we observe that the minimal weight does not drop during each of the a twists by χ; since 1 < 1 + b − a < ℓ and (1 + b − a) + a(ℓ + 1) ≤ (ℓ−1)+(ℓ−2)(ℓ+1) < ℓ^{2}, the weight can only drop if there exists c with 1 ≤ c < a such that

(1 + b − a) + c(ℓ + 1) ≡ 0 (mod ℓ).

If this occurred, then c ≡ a − b − 1 (mod ℓ). But 1 ≤ c < a ≤ ℓ − 2, so either c = a−b−1, which implies c ≤ 0 since a < b, or c = ℓ+a−b−1 = a+ℓ−1−b ≥ a, a contradiction.

3.2.1. Supersingular case

Assume that ρ is supersingular in the sense that tr ρ(Frob_{ℓ}) = 0. We now sketch Edixhoven’s proof that ρ arises from a mod ℓ eigenform of weight k(ρ) and level N(ρ).

Let ρ^{ss} denote the semisimplification of the restriction ρ|_{D_{ℓ}} of ρ to a decomposition group at ℓ. The restriction of ρ^{ss} to the inertia group at ℓ is

ρ^{ss}|_{I_{ℓ}} ∼ (\begin{matrix}Ψ^{n} & 0 \\ 0 & (Ψ′)^{n} \end{matrix}) ,

where Ψ and Ψ′ = Ψ^{ℓ} are the two fundamental characters of level 2. If necessary, reorder Ψ and Ψ′, so that n = a + bℓ with 0 ≤ a < b ≤ ℓ − 1. Then

Ψ^{n} = Ψ^{a+bℓ} = Ψ^{a}(Ψ′)^{b} = Ψ^{a}(Ψ′)^{a}(Ψ′)^{b−a} = χ^{a}(Ψ′)^{b−a},

and

ρ^{ss}|_{I_{ℓ}} ∼ χ^{a} ⊗ (\begin{matrix}(Ψ′)^{b−a} & 0 \\ 0 & Ψ^{b−a} \end{matrix}) .

Motivated by Fontaine’s theorem on Galois representations arising from supersingular eigenforms, we have defined

k(ρ) := a(ℓ + 1) + (b − a + 1) = 1 + ℓa + b.

The first step in showing that ρ arises from a form of weight k(ρ), is to prove the well known result that, up to twist, all systems of mod ℓ eigenvalues occur in weight at most ℓ + 1.

Theorem 4. Suppose ρ is modular of level N and some weight k, and that ℓ ∤ N. Then some twist ρ ⊗ χ^{−α} is modular of weight ≤ ℓ + 1 and level N.

This is a general theorem, applying to both the ordinary and supersingular cases. It was discovered by Serre [71] when N = 1; further work was done by Jochnowitz [41] and Ash-Stevens [1, Thm. 3.5] when ℓ ≥ 5. Two proofs are given in [26, Thm. 3.4 and §7].

As a digression, we pause to single out some of the tools involved in the proof of Theorem 4. The group Γ_{1}(N) acts by matrix multiplication on the real vector space R^{2}. The Eichler-Shimura correspondence (see [81, §8.2]) is an isomorphism of real vector spaces

S_{k}(Γ_{1}(N)) ⟶^{≅} H_{P}^{1} (Γ_{1}(N), Sym^{k−2}(R^{2})).

The parabolic (or cuspidal) cohomology group H_{P}^{1} is the intersection, over all cusps α ∈ P^{1}(Q), of the kernels of the restriction maps

res_{α} : H^{1}(Γ_{1}(N), Sym^{k−2}(R^{2})) → H^{1}(Γ_{α}, Sym^{k−2}(R^{2})),

where Γ_{α} denotes the stabilizer of α. For fixed z_{0} in the upper half plane, the Eichler-Shimura isomorphism sends a cuspform f to the class of the cocycle c : Γ_{1}(N) → Sym^{k−2}(R^{2}) induced by

γ ↦ ∫_{z_{0}}^{γ(z_{0})} Re (f(z) (\begin{matrix}z \\ 1 \end{matrix})^{k−2} dz) ,

where (\begin{matrix}z \\ 1 \end{matrix})^{k−2} denotes the image of (\begin{matrix}z \\ 1 \end{matrix}) ⊗ ⋯ ⊗ (\begin{matrix}z \\ 1 \end{matrix}) ∈ Sym^{k−2}(C^{2}), and inte-

gration is coordinate wise. There is a natural action of the Hecke algebra T on H_{P}^{1} (Γ_{1}(N), Sym^{k−2}(R^{2})), and the Eichler-Shimura correspondence is an isomorphism of T-modules.

The forms whose periods are integral correspond to a lattice

H_{P}^{1} (Γ_{1}(N), Sym^{k−2}(Z^{2})) ⊂ H_{P}^{1} (Γ_{1}(N), Sym^{k−2}(R^{2})).

Reducing this lattice modulo ℓ, we obtain a relationship between mod ℓ modular forms and H_{P}^{1} (Γ̃_{1}(N), Sym^{k−2}(F_{ℓ}^{2})), where Γ̃_{1}(N) is the image of Γ_{1}(N) in SL(2, F_{ℓ}). Serre and Hida [[REF??]] observed that, for k − 2 ≥ ℓ, the Γ̃_{1}(N) representations Sym^{k−2}(F_{ℓ}^{2}) are sums of representations arising in Sym^{k′−2}(F_{ℓ}^{2}) for k′ < k. This is the essential idea used in proving that all systems of eigenvalues occur in weight at most ℓ + 1.

We again consider a Galois representation ρ that is supersingular and modular. By Theorem 4 there is a form f of weight k ≤ ℓ + 1 such that χ^{−α} ⊗ ρ ∼ ρ_{f}. In fact, we may assume that 2 ≤ k ≤ ℓ: when k = ℓ + 1 a theorem of Mazur (see [26, Thm. 2.8]) implies that there is a form of weight 2 giving rise to ρ_{f}; when k = 1, multiply f by the weight ℓ − 1 Hasse invariant. To show that w(θ^{α}f) = k(ρ) we investigate how application of the θ-operator changes the minimal weight. We have (ρ_{f} ⊗ χ^{α})|_{I_{ℓ}} ∼ ( \begin{matrix}Ψ^{n} & 0 \\ 0 & (Ψ′)^{n} \end{matrix} ) with n = a + bℓ and a < b. Fontaine’s theory

(see Lecture 3) identifies the characters corresponding to ρ_{f}|_{I_{ℓ}} as powers Ψ^{k−1} and (Ψ′)^{k−1} of the fundamental characters. This gives an equality of unordered sets

{Ψ^{k−1}χ^{α}, (Ψ′)^{k−1}χ^{α}} = {Ψ^{n}, (Ψ′)^{n}}.

It is now possible to compute w(θ^{α}f) by considering two cases, corresponding to the ways in which equality of unordered pairs can occur.

Case 1 Suppose that Ψ^{k−1}χ^{α} = (Ψ′)^{n}. Since χ = Ψ^{ℓ+1}, we have

Ψ^{k−1+α(ℓ+1)} = Ψ^{k−1}χ^{α} = (Ψ′)^{n} = (Ψ′)^{a+bℓ} = Ψ^{b+aℓ}.

Comparing exponents of Ψ gives

k − 1 + α(ℓ + 1) ≡ b + aℓ (mod ℓ^{2} − 1), (3.1)

which reduces modulo ℓ + 1 to k − 1 ≡ b − a (mod ℓ + 1); because 2 ≤ k ≤ ℓ, this implies that k = 1 + b − a. Reducing (3.1) modulo ℓ − 1 and substituting k = 1 + b − a gives b − a + 2α ≡ b + a (mod ℓ − 1); we find the possible solutions α = a + m(ℓ − 1)/2 with m an integer. No solution α = a + m(ℓ − 1)/2, with m odd, satisfies (3.1), so α = a as an integer mod ℓ − 1. Finally, apply Theorem 3 and argue as in the end of Lecture 3, to show that

w(θ^{a}f) = w(f) + a(ℓ + 1) = 1 + b − a + aℓ + a = 1 + b + aℓ = k(ρ).
Case 2 Suppose that Ψ^{k−1}χ^{α} = Ψ^{n}. Then
Ψ^{k−1+α(ℓ+1)} = Ψ^{k−1}χ^{α} = Ψ^{n} = Ψ^{a+bℓ}.

Comparing powers of Ψ, we obtain

k − 1 + α(ℓ + 1) ≡ a + bℓ (mod ℓ^{2} − 1), (3.2)

which reduces modulo ℓ + 1 to k − 1 ≡ a − b (mod ℓ + 1); thus k = ℓ + 2 − (b − a). The difference b − a must be greater than 1; otherwise k = ℓ + 1, contrary to our assumption that 2 ≤ k ≤ ℓ. Reducing (3.2) modulo ℓ − 1 gives

k − 1 + 2α ≡ a + b (mod ℓ − 1);

substituting k = ℓ + 2 − (b − a) we find that α = b − 1 + m(ℓ − 1)/2 with m an integer. If m is odd, then α does not satisfy (3.2), so α = b − 1 as an integer modulo ℓ − 1. It remains to verify the equality w(θ^{b−1}f) = w(ρ). Unfortunately, k = ℓ + 2 − (b − a) is not especially telling. The argument of Case 1 does not apply to compute w(θ^{α}f); instead we use θ-cycles.

Because f is supersingular, Fermat’s Little Theorem implies that θ^{ℓ−1}f = f. We use Tate’s theory of θ-cycles (see [26, §7] and [41]) to compute w(θ^{b−1}f). The θ-cycle associated to f is the sequence of integers

w(f), w(θf), w(θ^{2}f), …, w(θ^{ℓ−2}f), w(f).

The θ-cycle for any supersingular eigenform must behave as follows:

[figure: •; drop; •; drop; •; go up …; •; go up …; •]

go up …, drop once, go up …, drop to original weight

Knowing this, we can deduce the exact θ-cycle. List ℓ numbers starting and ending with k:

k, k + (ℓ + 1), k + 2(ℓ + 1), …, k + (ℓ − k)(ℓ + 1),
ℓ + 3 − k, (ℓ + 3 − k) + (ℓ + 1), …, (ℓ + 3 − k) + (k − 3)(ℓ + 1),
k

The first and second lines contain ℓ + 1 − k and k − 2 numbers, respectively. All told, ℓ numbers are listed; this must be the θ-cycle.

It is now possible to compute w(θ^{b−1}f). If

b − 1 ≤ ℓ − k = ℓ − (ℓ + 2 − b + a) = −2 + b − a,

then a ≤ −1, a contradiction; thus b − 1 > ℓ − k. It follows that

w(θ^{b−1}f) = ℓ + 3 − k + (ℓ + 1)(b − 2 − (ℓ − k)) = 1 + b + aℓ = k(ρ),

verifying Serre’s conjecture in this case.

3.3. Ordinary case

Consider the ordinary case, in which

ρ|_{I_{ℓ}} ∼ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

with α, β : I_{ℓ} → F_{ℓ}^{∗} powers of the cyclotomic character. View ρ_{I_{ℓ}} as the twist of a representation in which the lower right entry is 1:

(\begin{matrix}α & ∗ \\ 0 & β \end{matrix}) ∼ β ⊗ (\begin{matrix}αβ^{−1} & ∗ \\ 0 & 1 \end{matrix}) .

To determine the minimal weight of a form giving rise to ρ|_{I_{ℓ}} it is necessary to develop an ordinary version of θ-cycles. In general this can be complicated, so make the simplifying assumption that β = 1; then ρ|_{I_{ℓ}} ∼ (\begin{matrix}χ^{i} & ∗ \\ 0 & 1 \end{matrix}) with 1 ≤ i ≤ ℓ − 1.

Deligne showed (see Lecture 3) that if f is of weight k and β = 1, then the associated representation is ( \begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix} ) with 2 ≤ k ≤ ℓ + 1. Motivated by this, our first reaction is to define k(ρ) to be i + 1. This definition does not distinguish between the extreme weights 2 and ℓ + 1 because they are congruent modulo ℓ − 1. Given a representation ρ arising from a form of weight either 2 or ℓ + 1, it is incorrect, in general, to set k(ρ) = 2. For example, suppose f = Δ is the level 1 cusp form of weight 12 and ρ = ρ_{f}. It is nonsense to set k(ρ) = 2—there is no cuspform of weight 2 and level 1.

LECTURE 4

Galois representations from modular forms

Consider a representation ρ : G_{Q} → GL(2, \overline{F}_{ℓ}) that arises from a modular form of level N = N(ρ), and some weight k. Assume that ρ is ordinary in the sense that tr ρ(Frob_{ℓ}) ≠ 0. In the simplest case, ρ|_{I_{ℓ}} ∼ ( \begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix} ) with 2 ≤ k ≤ ℓ + 1. Thus ρ|_{I_{ℓ}} determines k(ρ) modulo ℓ − 1, which determines k(ρ) when k ∉ {2, ℓ + 1}. The key to choosing between 2 and ℓ + 1 is good reduction.

The representation ρ = ρ_{f} attached to a normalized eigenform f = ∑ a_{n}q^{n} of weight 2 and level N can be constructed as follows. Consider the totally real or CM field E = Q(…, a_{n}, …). In [81, Thm. 7.14], Shimura associated to f an abelian variety A = A_{f} over Q, furnished with an action O ↪ End_{Q} A of the ring of integers of E. The dimension of A is the degree [E : Q], and A has good reduction at primes ℓ not dividing N. The Riemann surface

X_{1}(N) := \overline{Γ_{1}(N)\𝔥} = Γ_{1}(N)\𝔥 ∪ \{cusps\}

has a natural structure of algebraic curve over Q; it is called the modular curve of level N. Its Jacobian J_{1}(N) := Jac(X_{1}(N)) is an abelian variety over Q of dimension equal to the genus of X_{1}(N), and having good reduction at all primes ℓ ∤ N. When N = 1, the curve X_{1}(1) is isomorphic over Q to the projective line and J_{1}(1) = 0. There is a functorial action of the Hecke algebra T on J_{1}(N), and A = J_{1}(N)/IJ_{1}(N) where I is the kernel of the map T → E sending T_{p} to a_{p}.

The representation ρ = ρ_{f} is found inside of A; let λ = ker(O → \overline{F}_{ℓ}) and set

A[λ] = \{P ∈ A(\overline{Q}) : xP = 0 all x ∈ λ\}.

Since O is the full ring of integers [[ADD PRECISE REF TO SHIMURA]], dim_{O/λ} A[λ] = 2, so A[λ] affords ρ_{f,λ}:

\xymatrix{G_{Q} \ar[rrr]^{ρ_{f,λ}} \ar[dr] &  &  & GL(2, \overline{F}_{ℓ}). \\  & Aut_{O/λ} A[λ] \ar[r]^{≈} & GL(2, O/λ) \ar@{^{(}->}[ur]}

A finite group scheme G over Q_{ℓ}^{nr} is said to have good reduction, or to be finite flat, if it extends to a finite flat group scheme over O_{Q_{ℓ}^{nr}}. The group scheme A[λ] has good reduction; the finite flat group scheme extending A[λ] is the scheme theoretic closure of A[λ] in a good model A/O_{Q_{ℓ}^{nr}} of A, which exists because ℓ ∤ N = N(ρ) so that A has good reduction at ℓ. In general, if ρ is a representation and k(ρ) ≡ 2 (mod ℓ − 1), then we define

k(ρ) := \begin{cases}2 & if ρ is finite flat at ℓ, \\ ℓ + 1 & otherwise. \end{cases}

4.1. Representations arising from elliptic curves

Let A be an elliptic curve over Q with minimal discriminant Δ_{A} and conductor N_{A}. Suppose that ℓ^{2} ∤ N_{A}, so that A has semistable reduction at ℓ, and consider the representation ρ = ρ_{A,ℓ} given by the action of G_{Q} on A[ℓ]. Assume that A[ℓ] is irreducible.

Theorem 5. The representation ρ_{A,ℓ} is finite flat at ℓ if and only if ℓ | ord_{ℓ} Δ_{A}. If p ≠ ℓ, then ρ_{A,ℓ} is unramified at p if and only if ℓ | ord_{p} Δ_{A}.

[[REF TO SERRE]]

4.1.1. Frey curves

We can apply Theorem 5 to deduce Fermat’s Last theorem from the Shimura-Taniyama conjecture and the weak conjecture of Serre. Suppose (a, b, c) is a solution to the Fermat equation a^{ℓ} + b^{ℓ} = c^{ℓ} with ℓ ≥ 11 and abc ≠ 0. Consider the Frey curve A given by the equation y^{2} = x(x − a^{ℓ})(x + b^{ℓ}); it is an elliptic curve with discriminant Δ_{A} = \frac{((abc)^{2})^{ℓ}}{2^{8}} . By [67, §4.1, Prop. 6] the representation A[ℓ] is irreducible. Theorem 5 implies that ρ_{A,ℓ} is unramified away from 2 and ℓ. Thus N(ρ) | 2, and k(ρ) = 2 since ℓ | ord_{ℓ}(Δ_{A}). But there are no cuspforms of level 2 and weight 2. The modularity of A (proved in [86, 89]), together with the weak conjecture of Serre (enough of which is proved in [61]), leads to a contradiction.

4.1.2. Elliptic curves of conductor 33

There is a unique weight 2 normalized newform

f = ∑ a_{n}q^{n} = q + q^{2} − q^{3} − q^{4} − 2q^{5} − q^{6} + 4q^{7} − 3q^{8} + q^{9} + ⋯

on Γ_{0}(33). One of the elliptic curves associated to f is the curve A given by the equation y^{2} + xy = x^{3} + x^{2} − 11x. The discriminant of A is Δ_{A} = 3^{6} · 11^{2} and the conductor is N_{A} = 3 · 11. Because A is semistable and there are no elliptic curves 3-isogeneous to A, the associated mod 3 representation ρ = ρ_{A,3} : G_{Q} → Aut(A[3]) is surjective. [[REF TO ARG EARLIER IN NOTES.]] Since 3 | ord_{3} Δ_{A}, the Serre weight and level are k(ρ) = 2 and N(ρ) = 11. As expected, there is a weight 2 newform on Γ_{0}(11). If B is one of the three elliptic curves of conductor 11 (it does not matter which), then B[3] ≈ A[3] as representations of G_{Q}. Placing the eigenforms corresponding to A and B next to each other, we see that they their Fourier coefficients are congruent modulo 3:

\begin{aligned}f_{A} & = & q & +q^{2} & −q^{3} & −q^{4} & −2q^{5} & −q^{6} & +4q^{7} & −3q^{8} & +q^{9} & + ⋯ \\ f_{B} & = & q & −2q^{2} & −q^{3} & +2q^{4} & +q^{5} & +2q^{6} & −2q^{7} & & −2q^{9} & + ⋯ \end{aligned}

4.1.3. Elliptic curves of conductor 141

Let A be the elliptic curve y^{2} +y = x^{3} +x^{2} −12x+2 of conductor N_{A} = 141 = 3·47 and discriminant Δ_{A} = 3^{7}·47. First consider the mod ℓ = 7 representation ρ = ρ_{A,7} of G_{Q}. Since 7 | ord_{3}(Δ), the Serre invariants of ρ are k(ρ) = 2 and N(ρ) = 47. There is a form f ∈ S_{2}(Γ_{0}(47)), giving rise to ρ, whose Fourier coefficients generate a quartic field. Next consider the mod 3 representation ρ = ρ_{A,3}; the Serre invariants are N(ρ) = 47 and, since 3 ∤ ord_{3}(Δ), k(ρ) = ℓ + 1 = 4. The space S_{4}(Γ_{0}(47)) contains two conjugacy classes of eigenforms, one eigenform defined over a field of degree 3, and the other over a field of degree 8. Sure enough, one of the eigenforms, the degree 3 one, gives rise to ρ_{A,3}.

4.2. Companion forms

Suppose f is ordinary of weight k, that 2 ≤ k ≤ ℓ + 1, and that

ρ_{f}|_{I_{ℓ}} ∼ (\begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix}) .

Is this split or not; put another way, does ∗ = 0? How frequently does this split? The split primes ℓ are in the minority. How can one quantify the number of split primes?

If ∗ = 0, then

ρ|_{I_{ℓ}} ∼ (\begin{matrix}1 & 0 \\ 0 & χ^{k−1} \end{matrix}) , so ρ|_{I_{ℓ}} ⊗ χ^{ℓ−k} ∼ (\begin{matrix}χ^{ℓ−k} & 0 \\ 0 & 1 \end{matrix}) ,

so k(χ^{ℓ−k} ⊗ρ) = 1+ℓ−k. Using the θ-operator we see that ρ⊗χ^{ℓ−k} is modular, of some weight and level. To say that it is modular of Serre’s conjectured weight is to make a much strong statement. If ρ ⊗ χ^{ℓ−k} is indeed modular of weight 1 + ℓ − k, then there exists an eigenform g of weight 1 + ℓ − k with ρ_{g} ∼ ρ_{f} ⊗ χ^{ℓ−k}. Such an eigenform g, if it exists, is called a companion of f. The existence of g is far from obvious. Starting with an arbitrary ordinary form f, it can be shown that if a companion form g does exist, then the representation must split—except perhaps when the two characters are equal.

4.2.1. Special cases

If k(ρ) = ℓ, then the companion has weight 1, i.e., if ρ is unramified at ℓ, then ρ comes from a weight 1 eigenform. If k(ρ) = ℓ + 1 then ρ must be ramified at ℓ; otherwise ρ comes from a form of weight ℓ + 1 − k = 0, a contradiction.

The existence of a companion form was proved (assuming unchecked compatibilities) in most cases in which k < ℓ by Gross in [34], and in a few cases when k = ℓ. Using new methods, Coleman and Voloch [15] proved all cases except k = ℓ = 2, and didn’t use any unchecked compatibilities. The arguments of Coleman and Voloch do not require verification of Gross’s unchecked compatibilities.

LECTURE 5

Introduction to level lowering

Consider an irreducible representation ρ : G_{Q} → GL(2, \overline{F}_{ℓ}) that arises from an eigenform of weight k and level N. Serre associated to ρ integers k(ρ) and N(ρ), and conjectured that ρ arises from a newform of weight k(ρ) and level N(ρ). In Lecture 3 we sketched Edixhoven’s proof that if ℓ ∤ N then ρ arises from an eigenform of weight k(ρ) and level N. In this lecture, we introduce some of the techniques used in proving that ρ arises from a newform of level N(ρ). For more details, see [61, 63].

In [75, §1.2] Serre defined the optimal level N(ρ) as the prime-to-ℓ part of the Artin conductor of ρ. Recall that N(ρ) is a product ∏ p^{n(p)} over prime numbers p ≠ ℓ. The integer n(p) is defined by restricting ρ to a decomposition group D_{p} at p. Consider the sequence of ramification groups G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{i} ⊃ ⋯ where G_{0} is the inertia subgroup I_{p} of D_{p}. Let V be a vector space over \overline{F}_{ℓ} affording the representation ρ, and for each i ≥ 0 let V_{i} be the subspace of V consisting of those v ∈ V that are fixed by G_{i}. Then

n(p) := ∑_{i=0}^{∞} \frac{1}{(G_{0} : G_{i})} dim V/V_{i}.

5.1. Reduction to the salient case

The optimal level N(ρ) is not divisible by ℓ. The first step in level lowering is to strip the power of ℓ from N. When ℓ is odd, this is done explicitly in [63, §2]; for the case ℓ = 2 see [9, §1]. Many of the arguments and key ideas are due to Serre [68]. The proof that ℓ can be stripped from the level uses concrete techniques of Serre [69, §3], [72, Thm. 5.4], and Queen [57, §3]; it involves multiplying f by suitable Eisenstein series and taking traces. Katz’s theory of ℓ-adic modular forms suggests an alternative method: a classical form of weight 2 and level Mℓ^{m} is an ℓ-adic form of level M; the mod ℓ reduction of this form is classical of level M and some weight, and is congruent to f. See the appendices of [43] and the discussions in [35, §1] and [36, §1].

The next step is to replace f be a newform of weight between 2 and ℓ + 1 that gives rise to a twist of ρ. Twisting ρ by the mod ℓ cyclotomic character χ preserves N; this is because ρ⊗χ arises from θ(f) = q \frac{d}{dq} (f) which also has level N. Theorem 4 asserts that some twist ρ ⊗ χ^{i} of ρ arises from a form g of weight between 2 and ℓ + 1. If ρ ⊗ χ^{i} arises from a newform of level N, then ρ also arises from a newform of the same level, so we can replace f by g and k by the weight of g. By results discussed in Lecture 3, we may assume that k = k(ρ ⊗ χ^{i}). For the case ℓ = 2 see [9, Prop. 1.3(a)].

We have reduced to considering a representation ρ that arises from a newform f of weight k(ρ) and level N not divisible by ℓ. The weight satisfies 2 ≤ k(ρ) ≤ ℓ + 1, but N might well be a large multiple of N(ρ). The fact that N is a multiple of N(ρ) is a theorem of Carayol [11], which was also proved independently by Livné [52, Prop. 0.1].

In order to lower N it is convenient to use the crutch of reduction to weight 2. Paradoxically, even though we have just taken all powers of ℓ out of N, we are now going to put one power of ℓ back into N. This allows us to reduce to weight 2 and realize ρ as a group of torsion points on an abelian variety. An alternative approach (see [29, 42]) is to avoid this crutch and work directly with representations coming from arbitrary weights between 2 and ℓ + 1; these are realized in étale cohomology groups. This later approach has the advantage that X_{0}(N) has good reduction at ℓ.

Reduction to weight 2 is accomplished using a general fact that originates with ideas of Serre and Koike. In characteristic ℓ, eigenforms of level N whose weights satisfy 2 < k ≤ ℓ + 1 correspond to eigenforms of weight 2 and level ℓN (see [63, Thm. 2.2]):

[figure: { 2 < k ≤ ℓ + 1, level N }; { k = 2, level ℓN }] .

Thus we work with weight 2 and level

N^{∗} := \begin{cases}N & if k = 2, \\ Nℓ & otherwise. \end{cases}

5.2. Realizing Galois representations geometrically

To understand representations arising from modular forms, it is helpful to realize these representations inside of geometric objects such as J := J_{1}(N^{∗}). These representations are constructed geometrically with the help of the Hecke algebra T := Z[…, T_{n}, …]. Recall that T is a commutative subring of End_{Q} J that is free as a module over Z, and that its rank is equal to the dimension of J. When N is cube free, T is an order in a product of integer rings of number fields; this is a result of Coleman and Edixhoven (see [14, Thm. 4.1]). In contrast, the Hecke operators T_{p}, for p^{3} | N, are usually not semisimple (see Exercise 2).

It is fruitful to view a newform f as a homomorphism

T ⟶ O = Z[…, a_{n}, …] ⊂ C, T_{n} ↦ a_{n}.

Letting φ : O → \overline{F}_{ℓ} be the map sending a_{p} to tr(ρ(Frob_{p})) ∈ \overline{F}_{ℓ} we obtain an exact sequence 0 → 𝔪 → T → \overline{F}_{ℓ} with 𝔪 a maximal ideal. Thus together f and φ embed T/𝔪 in \overline{F}_{ℓ}.

Our next step, after having attached a maximal ideal 𝔪 to the given newform f that gives rise to ρ, is to find a vector space affording ρ inside of J(\overline{Q}). Following [53, §II.7]we consider the T/𝔪-vector space

J[𝔪] := \{P ∈ J(\overline{Q}) : tP = 0 all t ∈ 𝔪\} ⊂ J(\overline{Q})[ℓ] ≈ (Z/ℓZ)^{2g}.

Since the endomorphisms in T are Q-rational, J[𝔪] comes equipped with an action of G_{Q} := Gal(\overline{Q}/Q).

Since tr(ρ(Frob_{p})) ∈ T/𝔪 ↪ \overline{F}_{ℓ} and det(ρ(Frob_{p})) ∈ (T/𝔪)^{∗} ⊂ \overline{F}_{ℓ}^{∗} we suspect that ρ has a model over T/𝔪. Such a model exists^{1} for the rather coarse reason that Br(F_{ℓ}) = 0 (see [73, X.7, Ex. a]). Alternatively, when ℓ is odd, observe that complex conjugation acts through ρ as a matrix with distinct rational eigenvalues; a well known theorem of I. Schur [65, IX a] (cf. [88, Lemme I.1]) then implies that ρ can be conjugated into a representation with values in GL(2, T/𝔪). ^{2} Choose such a model ρ_{𝔪} : G_{Q} → GL(2, T/𝔪) for ρ.

5.3. Multiplicity one

Under the assumption that ρ_{𝔪} is absolutely irreducible Boston, Lenstra, and Ribet (see [6]) proved that J[𝔪] is a sum of copies of ρ_{𝔪}:

J[𝔪] ≈ ⨁_{i=1}^{t} ρ_{𝔪}.

Note that when ℓ is odd, irreducibility of ρ_{𝔪} is equivalent to absolute irreducibility.

In the 1940s Weil showed (see, for example, [55, §12]) that End(J) ⊗ Z_{ℓ} operators faithfully on the Tate module Tate_{ℓ} J. Because T ⊗ Q_{ℓ} is Artinian, it decomposes as a product of local rings: T ⊗ Q_{ℓ} = ⨁_{λ|ℓ} T_{λ} ⊗ Q_{ℓ} where λ runs through the maximal ideals of T lying over ℓ, and T_{λ} denotes the completion of T at λ. Since Tate_{ℓ} J ⊗ Q_{ℓ} is an Artinian T ⊗ Q_{ℓ}-module, it breaks up as a sum

Tate_{ℓ} J ⊗ Q_{ℓ} = ⨁_{λ|ℓ} Tate_{λ} J ⊗ Q_{ℓ}

If Tate_{𝔪} J = 0 then T_{𝔪} ⊗Q_{ℓ} acts trivially on Tate_{ℓ} J ⊗Q, which contradicts Weil’s theorem, so there is some integer r such that J[𝔪^{r}] ≠ 0. Let r be the smallest such integer. Following [53, pg. 112], note that given any generating set of elements (a_{1}, …, a_{t}) of the T/𝔪-vector space 𝔪^{r−1}/𝔪^{r}, the map x ↦ a_{1}x ⊕ ⋯ ⊕ a_{t}x is an injection of the module J[𝔪^{r}]/J[𝔪^{r−1}] into the direct sum of t copies of J[𝔪]. In particular, J[𝔪] ≠ 0. ^{3}

When t = 1 we say that ρ_{𝔪} occurs in J with multiplicity one. A detailed summary of multiplicity one results can be found in [26, §9], and some supplementary results are contained in [89, Thm. 2.1]. In our situation, in which ℓ^{2} ∤ N, we are aware of no examples in which multiplicity one fails. When we admit arbitrary powers of ℓ into the level, then there are counterexamples (see [46]).

Many important techniques for proving multiplicity one results were pioneered by Mazur in [53] who considered J_{0}(N) with N prime. For each maximal ideal 𝔪 of T he considered ρ_{𝔪} and asked, “is J[𝔪] ≅ ρ_{𝔪}?” The answer is yes, if ℓ ≠ 2; and yes—in some sense—if ℓ = 2 and ρ_{𝔪} is reducible. Mazur did not treat the case in which ℓ = 2 and ρ_{𝔪} is ordinary in the sense that T_{2} ∉ 𝔪. The paper [54] elaborates on Mazur’s results in various directions, and [34, §12] proves multiplicity one in many cases for J_{1}(N^{∗}) where N^{∗} = Nℓ.

Consider, as above, a representation ρ = ρ_{𝔪} at level N^{∗} and of weight k = k(ρ) satisfying 2 ≤ k ≤ ℓ + 1. Multiplicity one is known, except perhaps when all of the following hypothesis on ρ are satisfied:

— ℓ = k(ρ); — ρ is unramified at ℓ;

1I don’t see why.

2I hope to record Wallspurger’s proof.

3Ken, believe it or not this is the best argument I can think of! Do you something better?

— ρ is ordinary at ℓ in the sense that T_{ℓ} ∉ 𝔪;

— ρ|_{D_{ℓ}} ∼ ( \begin{matrix}α & ∗ \\ 0 & β \end{matrix}) with α = β.

In [34, §12] Gross proves multiplicity one when α ≠ β, k ≤ ℓ, and ρ is ordinary; he uses this result in his proof of the existence of companion forms. In contrast, Coleman and Voloch [15] prove the existence of companion forms when α = β and ℓ > 2 using a method that avoids the need for multiplicity one.

We now discuss the above hypothesis when ℓ = 2. The optimal weight satisfies 2 ≤ k ≤ ℓ + 1 = 3. If k = 3 then ρ is ramified at 2; this can be used to deduce multiplicity one^{4}. Assume ρ is unramified at 2, so k = 2. When ρ is supersingular, i.e., T_{2} ∈ 𝔪, the inertia group I_{2} operates through the 2 fundamental characters of level 2. These both have order ℓ^{2} − 1 = 3 ≠ 1, so ρ is ramified and we can again deduce multiplicity one. Finally, assume ρ is ordinary; then ρ|_{D_{2}} ∼ ( \begin{matrix}α & ∗ \\ 0 & β \end{matrix}) with β unramified and β(Frob_{2}) = T_{2} mod 𝔪. Since k = 2, the determinant αβ of ρ|_{D_{2}} is the mod 2 cyclotomic character χ^{k−1} = χ, which is unramified, so α is unramified. Let f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{1}(N^{∗}), ε) be a form giving rise to ρ. Since χ(Frob_{2}) = 1 and αβ = χε, we have α(Frob_{2}) = β^{−1}(Frob_{2})ε(2) = a_{2}^{−1}ε(2) (mod 𝔪). The further condition, under which we might not know multiplicity one, is α|_{D_{2}} = β|_{D_{2}}; reexpressed in terms of the image of Frobenius, this becomes a_{2} = a_{2}^{−1}ε(2) ∈ O/𝔪, or equivalently, a_{2}^{2} = ε(2) ∈ O/𝔪.

5.4. The salient case

We have set our problem up so that level lowering pertains to weight 2 forms of appropriate level, and takes place on Jacobians of modular curves. This problem was described and partially treated in a long paper of Carayol [11].

Assume that ρ arises from a newform f of weight 2 and level N^{∗} = pM, with p ∤ M and p ≠ ℓ. Moreover, as could easily happen, assume that the character ε : (Z/N^{∗}Z)^{∗} → O^{∗} of f has “nothing to do with p” in the sense that ε factors through the natural map (Z/N^{∗}Z)^{∗} → (Z/MZ)^{∗}. In the salient case, f, a priori “on” Γ_{1}(N), is also “on” the bigger group Γ_{1}(M) ∩ Γ_{0}(p). The word “on” in this context is special to the area of modular forms; it means that f satisfies the appropriate invariance with respect to the group. Finally, make the extra hypothesis that p ∤ N(ρ). Since precisely the ramified primes (except ℓ) divide Serre’s minimal level, this is the same as assuming that ρ is unramified at p.

Example 6. Consider the representation ρ = A[7] arising from the 7-division points of the modular elliptic curve A of conductor N_{A} = 3 · 47 and minimal discriminant Δ_{A} = 3^{7} · 47. The newform f corresponding to A is on Γ_{0}(3 · 47). As in Section 2.3, since ord_{3}(Δ_{A}) = 7, the representation ρ is unramified at 3 and N(ρ) = 47. We must get p = 3 out of the level.

Example 7 (Frey curves). The elliptic curves that Frey associated in [30] to hypothetical solutions of the Fermat equation x^{ℓ} + y^{ℓ} = z^{ℓ} give rise to mod ℓ Galois representations. According to Wiles’s theorem, these representations fall into the salient case. Since k = 2, N = N^{∗} = 2L with L big and square free. At the same time, N(ρ) = 2. Our strategy is to take p to be any prime dividing L and to try to lower the level. If we succeed, we repeat the process at the lower level, and eventually reach a contradiction.

4why? how? reference?

The salient case divides into two cases. The hard case occurs when the following two conditions are both satisfied:

— p ≡ 1 (mod ℓ);

— ρ(Frob_{p}) is a scalar matrix. The second condition makes sense because p ∤ N(ρ); since det(ρ(Frob_{p})) = χ^{k−1}ε, we know the scalar up to ±1. The easy case, often referred to as “Mazur’s principle”, is the complementary case. Example 6 falls into the easy case because 3 ≢ 1 (mod 7), but treating Example 7 requires a proof in the hard case.

5.5. Level lowering in terms of commutative algebra

Before concluding this lecture, we set up some of the commutative algebra that is required in order to lower levels in the easy case of the salient case. There are two injective maps

\xymatrix{S_{2}(Γ_{1}(M)) \ar@{^{(}->}[r] \ar@{^{(}->}[r] & S_{2}(Γ_{1}(M) ∩ Γ_{0}(p))} .

One is the inclusion f(q) ↦ f(q) and the other is f(q) ↦ f(q^{p}) (see Exercise 28). The p-new subspace S_{2}(Γ_{1}(M) ∩ Γ_{0}(p))^{p-new} is the complement, with respect to the Petersson inner product, of the subspace S generated by the two images of S_{2}(Γ_{1}(M)). The p-new subspace can also be defined algebraically as the kernel of the natural map from S_{2}(Γ_{1}(M) ∩ Γ_{0}(p)) to the direct sum of two copies of S_{2}(Γ_{1}(M)).

Let T denote the Hecke algebra acting on S_{2}(Γ_{1}(M)∩Γ_{0}(p)). If p ∤ M, then T_{p} acts on S as a direct sum of two copies of its action on S_{2}(Γ_{1}(M)); otherwise, T_{p} usually does not act diagonally (see Exercise 1). The action of T on S is through a quotient \overline{T}. Given a representation ρ associated to a maximal ideal 𝔪 of T we ask whether or not the image of 𝔪 in \overline{T} is contained in a maximal ideal. Lowering the level is equivalent to showing that 𝔪 arises by pullback from a maximal ideal of \overline{T} (see Exercise 32).

Example 8. Figure 1 represents the inclusion Spec(\overline{T}) ↪ Spec(T) of Example 6. In this diagram, \overline{T} acts on the image of

S_{2}(Γ_{0}(47)) ⊕ S_{2}(Γ_{0}(47)) ↪ S_{2}(Γ_{0}(47) ∩ Γ_{0}(3))

and is represented by a bold curve.

36 K. A. RIBET, SERRE’S CONJECTURES (October 4, 1999) Figure 1. Level lowering

LECTURE 6

Approaches to level lowering

6.1. Carayol’s reduction to the salient case

Consider a modular representation ρ : G_{Q} → GL(2, \overline{F}_{ℓ}) that arises from a newform of level N and weight k(ρ), and assume that ℓ ∤ N. The goal is to show that there is a newform of level N(ρ) that gives rise to ρ. Set

N^{∗} = \begin{cases}N & if k = 2, \\ Nℓ & otherwise. \end{cases}

Then ρ arises from a newform f = ∑ a_{n}q^{n} of weight 2 and character ε on Γ_{1}(N^{∗}). Let 𝔪 be the maximal ideal of T associated to ρ and f; thus 𝔪 is the kernel of the map T → \overline{F}_{ℓ} given by T_{p} ↦ \overline{a}_{p} = tr(ρ(Frob_{p})) ∈ \overline{F}_{ℓ}, for p ∤ ℓN^{∗}. In particular, ℓ ∈ 𝔪 and the representation ρ is realized geometrically inside the subspace J[𝔪] ⊂ J[ℓ] of the ℓ-torsion on the Jacobian J of X_{1}(N^{∗}). Let O = Z[…, a_{n}, …], and denote by φ the map a_{p} ↦ \overline{a}_{p}.

Fix a divisor p of \frac{N^{∗}}{N(ρ)}. Our goal is to find a newform of level a divisor of \frac{N^{∗}}{p} that also gives rise to ρ. The full tree of possibilities is discussed in papers of Carayol and Livne (see [11, 52]). When ℓ ≥ 5, Carayol reduced the other cases to the salient case (see Section 5.4)^{1}.

Suppose that p | N^{∗} but p ∤ N(ρ), so ρ is unramified at p. In particular, det(ρ) = χ_{ℓ}^{k−1}\overline{ε} = χ_{ℓ}\overline{ε} is unramified at p, so \overline{ε} is unramified at p. The character ε is initialized defined as a map (Z/N^{∗}Z)^{∗} → O^{∗}; the reduction \overline{ε} is obtained by composing ε with φ : O → \overline{F}_{ℓ}.

What does it mean for \overline{ε} to be unramified at p? Let M = \frac{N^{∗}}{p} and write (Z/N^{∗}Z)^{∗} = (Z/pZ)^{∗} × (Z/MZ)^{∗}. Restricting ε to each factor allows us to write ε as a product of two characters: ε = ε_{p} · ε^{(p)} where ε_{p} is a character of (Z/pZ)^{∗} and ε^{(p)} is a character of (Z/MZ)^{∗}. The character ε^{(p)} has conductor dividing M so it is unramified at p. Since ε_{p} is totally ramified at p, the reduction \overline{ε} is unramified at p precisely when \overline{ε}_{p} = 1; equivalently, when ε_{p} has order a power of ℓ. If ε_{p} is nontrivial, then, since the order divides the order p − 1 of a generator of (Z/pZ)^{∗}, a power of ℓ divides p − 1, so p ≡ 1 (mod ℓ). Carayol showed, except in a few cases when ℓ < 5, that there is another newform that also gives rise to ρ whose character is trivial at p. We thus reduce to the salient case.

1Did he also do ℓ = 3, or did someone else; who?

6.2. Level lowering in the salient case

Carayol’s result reduces to the salient case. In the salient case, we are given a form f on Γ_{1}(M) ∩ Γ_{0}(p) of weight 2 that gives rise to ρ such that

— p || N^{∗} = pM;

— p ∤ N(ρ), so ρ is unramified at p. The goal is to show that ρ arises from a newform on Γ_{1}(M). This has been achieved in several cases:

— Using Mazur’s principle: If either ρ(Frob_{p}) is not scalar or p ≢ 1 (mod ℓ), then an argument of Mazur, which we present in Section 6.3, can be used.

— Using multiplicity one: If ρ occurs inside J_{1}(N^{∗}) with multiplicity one:

J[𝔪] = ⨁ ρ_{𝔪} = ρ_{𝔪} (one copy!)

then the argument in [61] can be used. We outline this argument in Lecture 8. The cases in which multiplicity one is known were reviewed in Section 5.3; in particular, we do not know multiplicity one in some cases when k(ρ) = ℓ and the eigenvalues of Frob_{p} are not distinct.

— When ℓ is odd: When ℓ is odd and ε = 1, level lowering in the salient case was proved by Ribet in [63] using an argument that does not require multiplicity one. The fact that −1 ≢ 1 mod ℓ is used in the proof of Proposition 7.8 of [63] to force splitting of a short exact sequence. In [21], Diamond extended the argument to cover the case of arbitrary character, but still under the assumption that ℓ is odd. These arguments are described in Lecture 7. One encounters seemingly insurmountable difficulties in trying to push the

third argument through when ℓ = 2. K. Buzzard observed that the multiplicity one approach goes through when ℓ = 2; for more, see [9] and Lecture 8.

6.3. Mazur’s principle

In this section we outline Mazur’s principle. This argument is used to lower the level in the salient case when either ρ(Frob_{p}) is not scalar or p ≢ 1 (mod ℓ); for more details see [61, §6] and [21, §4].

Let ρ be a representation arising from a modular form f as in the beginning of Section 6.2. First we describe the local representation ρ|_{G_{p}} of a decomposition group G_{p} ≈ Gal(\overline{Q}_{p}/Q_{p}). Suppose that ρ arises from a form f of weight 2, level Mp with (M, p) = 1, and character ε of conductor dividing M. Then ρ|_{G_{p}} resembles the mod ℓ representation attached to a Tate curve^{2}: ρ|_{G_{p}} ∼ ( \begin{matrix}αχ & ∗ \\ 0 & α \end{matrix}). Because we are considering the salient case, the two characters αχ and α are unramified at p. Thus α(Frob_{p}) makes sense; we have α(Frob_{p}) = \overline{a}_{p}(f) and αχ(Frob_{p}) = p\overline{a}_{p}(f). Since det(ρ|_{G_{p}}) = α^{2}χ = \overline{ε}χ, we see that

\overline{a}_{p}^{2} = \overline{ε}(p).

For example, if f is a form on Γ_{0}(M) then we have ε = 1 so \overline{a}_{p}^{2} = 1. When f really does correspond to an elliptic curve, it corresponds to a Tate curve with either split or nonsplit multiplicative reduction at p, depending on the sign of a_{p} (see Exercise 33). This local analysis was vastly generalized by Langlands in [49] which extends it to include many ℓ-adic representations of possibly higher weight.

2What is the canonical reference for this?

Let T be the Hecke algebra associated to Γ_{1}(M) ∩ Γ_{0}(p), and let 𝔪 be the kernel of the map T → \overline{F}_{ℓ}:

0 ⟶ 𝔪 ⟶ T ⟶^{T_{n}↦\overline{a}_{n}, ⟨d⟩↦\overline{ε}(d)} \overline{F}_{ℓ}.

The determinants and traces of ρ lie in T/𝔪 ⊂ \overline{F}_{ℓ}, so we can form a model for ρ over T/𝔪. Let V be the representation space of ρ over T/𝔪; we have

ρ_{𝔪} : G_{Q} ⟶ Aut_{T/𝔪} V, and V ≈ (T/𝔪)^{⊕2}.

Next we realize ρ_{𝔪} as a group of division points in a Jacobian. The curve X_{1}(Mp) corresponding to Γ_{1}(Mp) covers the curve X_{1}(M, p) corresponding to Γ_{1}(M) ∩ Γ_{0}(p). The induced map J = Jac(X_{1}(M, p)) → J_{1}(Mp) = Jac(X_{1}(Mp)) has a finite kernel on which the Galois action is abelian^{3}.

The Hecke algebra associated to Γ_{1}(M) ∩ Γ_{0}(p), can be constructed as a ring of correspondence and viewed as a subring T ⊂ End_{Q}(J). The action of T on modular forms then comes by thinking of S_{2}(Γ_{1}(M) ∩ Γ_{0}(p)) as the tangent space^{4} to the identity of J. Inside of J we find the nonzero G_{Q}-module J[𝔪] ≈ ⊕_{i=1}^{t}V. For the purposes of this discussion, we do not need to know that J[𝔪] is a direct sum of copies of V. The following weaker assertion, known long ago to Mazur [53, §14, pg. 112], will suffice: J[𝔪] is a successive extension of copies of V. In particular, V ⊂ J[𝔪]. A weaker conclusion, true since ℓ ∈ 𝔪, is that V ⊂ J[ℓ],

Our hypothesis that ρ is unramified at p translates into the inclusion V ⊂ J[ℓ]^{I_{p}}. By [77, Lem. 2], if A is an abelian variety over Q with good reduction at p, then A[ℓ]^{I_{p}} ≅ A_{F_{p}}[ℓ]. However, the modular curve X_{1}(M, p) has complicated bad reduction at p, so J is likely to have bad reduction at p—in this case it does. We are led to consider the Néron model J of J, whose construction and main properties are described in [5]. The Néron model is a smooth commutative group scheme over Z with generic fiber J that satisfies the following property: the restriction map Hom_{Z}(S, J) ⟶ Hom_{Q}(S_{Q}, J) is bijective for all smooth schemes S over Z. Passing to the scheme theoretic closure, we have, inside of J, a two-dimensional T/𝔪-vector space scheme V. We know a great deal about the complicated bad reduction of X_{1}(M, p) (see [20, 45]). The Deligne-Rapoport model is often illustrated by the following fairly standard squiggly diagram:

X_{1}(M, p) X_{1}(M, p)

The reduction X_{1}(M, p)_{F_{p}} is the union of 2 copies of X_{1}(M), intersecting transversely at the supersingular points; these are the points E ∈ X_{1}(M) that represent a supersingular elliptic curve enhanced by Γ_{1}(M)-structure. Raynaud thoroughly studied J using a well-understood model of X_{1}(M, p) over Z_{p}.

The subspace S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M)) of S_{2}(Γ_{1}(M) ∩ Γ_{0}(p)) is stable under the Hecke algebra T, so there is a map T → End(S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M))). Recall that the p-old quotient of T is the image \overline{T}. Since the map T → \overline{T} is surjective, the image of 𝔪 in \overline{T} is an ideal \overline{𝔪}. To lower the level in the salient case amounts to showing that \overline{𝔪} is not the unit ideal.

3Where is this discussed in the literature?

4I think it’s more natural to think of cusp forms as the cotangent space, even though the Petersson inner product provides a canonical isomorphism between the tangent and cotangent space.

As is well known (cf. [53, Appendix, Prop 1.4]), the results of Michèle Raynaud [59] and Deligne-Rapaport [20] combine to produce an exact sequence

0 ⟶ T ⟶ J_{F_{p}}^{0} ⟶ J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}} ⟶ 0, (6.1)

where T is a torus, i.e., T_{\overline{F}_{p}} ≈ G_{m} × ⋯ × G_{m}, and J_{F_{p}}^{0} is the identity component of J_{F_{p}}. Furthermore, there is also a precise description of T and of the maps in the exact sequence. Each object in the sequence is equipped with a functorial action of the Hecke operators, and the exact sequence is T-invariant. The p-old quotient \overline{T} can be viewed as coming from the action of T on J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}}.

We next observe that V, which sits inside J, in fact lies in J^{0}. The component group Φ = J /J^{0} is “Eisenstein”, in the sense that it does not contain irreducible representations arising from eigenforms. Since V is irreducible, Φ, as a Galois module, does not contain an isomorphic copy of V. Thus V ⊂ J^{0} and we have the following diagram:

[figure: 0; T; J_{F_{p}}^{0}; J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}}; 0.; V = V_{F_{p}}]

Since 𝔪 acts as 0 on V, the image of V in J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}} is acted on as 0 by the image \overline{𝔪} of 𝔪. If \overline{𝔪} ≠ (1) then we can lower the level, so assume \overline{𝔪} = (1). Then V must go to 0 in J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}}, so by exactness, V ↪ T .

In the following paragraph we will make explicit the action of T on the torus; we discover that Frob_{p} acts as a scalar on V, and that p ≡ 1 (mod ℓ). By hypothesis, one of these conditions fails, so \overline{𝔪} is not the unit ideal, a contradiction. Thus we lower the level.

It remains to make explicit the action of T. Let X_{p}(J) = Hom(T , G_{m}) be the character group of T . Since T is equipped with an action of T, so is X_{p}(J). Furthermore, X_{p}(J) supports an action of Gal(\overline{F}_{p}/F_{p}) which, because tori split over a quadratic extension, factors through the Galois group of F_{p^{2}}. Think of the Galois action as an action of Frob_{p} ∈ D_{p} ≈ Gal(\overline{Q}_{p}/Q_{p}).

In the next section we verify that on X_{p}(J) the following relations hold:

(6.2)  T_{p} = ⟨p⟩ ∘ Frob_{p},
(6.3)  T_{p}^{2} = ⟨p⟩.

We find that T ≅ Hom_{Z}(X_{p}(J), G_{m}) and

V ⊂ T (\overline{F}_{p})[ℓ] = Hom_{Z}(X_{p}(J), μ_{ℓ}).

Since Frob_{p} acts as p on μ_{ℓ}, we find that Frob_{p} acts as pT_{p} on T (\overline{F}_{p}). In particular, Frob_{p} acts on V ⊂ T (\overline{F}_{p}) as pa_{p} ∈ T/𝔪, i.e., as a scalar. Because Frob_{p} acts as a scalar and det(ρ) = χε, we have simulatenously

det(ρ(Frob_{p})) = \begin{cases}pε(p) & and \\ (pa_{p})^{2}. \end{cases}

But on V we have, by (6.3), that a_{p}^{2} = T_{p}^{2} = ⟨p⟩ = ε(p), so p^{2} ≡ p (mod ℓ). Since p ≠ ℓ, this can only happen if p ≡ 1 (mod ℓ).

6.3.1. The action of Frobenius on the torus.

In this section we let Frob denote Frob_{p}.

Proposition 9. On X_{p}(J) the relation ⟨p⟩ ∘ Frob = T_{p} holds.

Proof. First we prove that on D = Div(X_{0}(M, p)^{ss}) we have the relation

⟨p⟩ ∘ T_{p} = Frob_{p} on D. (6.4)

In what follows, we denote by L the level (M, p) structure. First, recall the interpretation of T_{p} over the complex numbers that arises from the following correspondence:

\xymatrix{ & X(Lp) \ar[dl]_{α} \ar[dr]^{β} \\ X(L) &  & X(L)}

The fiber of α over a point x arises from the set of all isogenies f : y → x of degree p (of enhanced elliptic curves). Thus x maps to the sum over all y coming from such an isogeny, equipped with the level structure got by pullback via f from x, followed by a pushforward.

Now consider enhanced elliptic curves in characteristic p; an enhanced elliptic curve is a pair E = (E, C) where C is a level (M, p)-structure. The action of Frobenius on an enhanced curve (E, C) in D is by Frob(E, C) = (E^{(p)}, C^{(p)}). Note that Frob is an isogeny Frob : E → E^{(p)} that is purely inseparable of degree p and C^{(p)} = Frob(C).

The action of T_{p} is got by considering all elliptic curves F equipped with an isogeny F → E of degree p, and then pulling back level structure, followed by a pushforward. When E is supersingular, the only such isogeny is Ver : E^{(p)} → E, where Ver is the isogeny dual to Frob, in the sense that

Ver ∘ Frob = Frob ∘ Ver = p.

We have T_{p}(E, C) = (E^{(p)}, Ver^{−1}(C)) since Ver^{−1}(C) is the pullback of the level structure. The diamond brackets operator is defined by ⟨p⟩(E, C) = (E, pC). Since Frob ∘ Ver = p and E is supersingular, we have p Ver^{−1}(C) = Frob(C) = C^{(p)}, so

(⟨p⟩ ∘ T_{p})(E, C) = (E^{(p)}, p Ver^{−1}(C))
= (E^{(p)}, C^{(p)})
= Frob(E, C).

By [61, Prop. 3.8], the action of T_{p} on the character group, identified as the subgroup of degree 0 divisors in D, is the map induced by Frobenius. This is because T_{p} = −w_{p} and w_{p} swaps the components, hence changes the orientation of the corresponding graph. Thus (6.4) translates into the relation ⟨p⟩ ∘ Frob = T_{p} on the character group X_{p}(J).

Corollary 10. The action of Frob on T = Hom(X_{p}(J), μ_{ℓ}) is by pT_{p}.

42 K. A. RIBET, SERRE’S CONJECTURES (October 4, 1999)

Proof. The action on Hom(X_{p}(J), μ_{ℓ}) is as follows. If φ ∈ Hom(X_{p}(J), μ_{ℓ}), then

(Frob .φ)(x) = Frob(φ(Frob^{−1}(x))).

For example, with this action

H^{0}(Gal(\overline{F}_{p}/F_{p}), Hom(X_{p}(J), μ_{ℓ})) = Hom_{Gal(\overline{F}_{p}/F_{p})}(X_{p}(J), μ_{ℓ}).

Since T_{p}^{2} = ⟨p⟩ and ⟨p⟩ ∘ Frob = T_{p}, we have

Frob ∘T_{p} = ⟨p⟩^{−1} ∘ T_{p} ∘ T_{p} = 1,

so Frob^{−1} = T_{p}. Hence (Frob .φ)(x) = pφ(T_{p}x) = φ(pT_{p}x).

LECTURE 7

Level lowering without multiplicity one

7.1. Easy case

First we give a proof of level lowering in a special case. This proof was given in [62] and has the key feature that it does not require any “multiplicity one” results. Assume that M = K · q with q ∤ K and that ε is defined mod K, so that ρ arises from a from on Γ_{1}(K) ∩ Γ_{0}(pq). Make the

Key assumption: ρ is ramified at q. The trick is to use q as a pivot. Edixhoven pointed out to me the possibility

of doing this, in the context of Fermat. Associated to a (hypothetical) solution a^{ℓ} + b^{ℓ} + c^{ℓ} = 0 is a modular Galois representation V = E[ℓ] attached to an elliptic curve E. Using Tate’s algorithm one discovers that Δ_{E} = \frac{(abc)^{2ℓ}}{2^{8}} , which is a perfect ℓ-th power away from 2, and that the conductor is N_{E} = rad(abc). Take q = 2. Then E[ℓ] is definitely ramified at 2. Use 2 as a pivot to successively remove primes dividing the level N_{E}. So in real life^{1} this setup can occur, though one doesn’t always have this luxury. To complete the proof of Fermat we use q = 2 as a pivot and inductively remove each odd factor from N. One difficulty that may arise (the second case of Fermat Last Theorem) is that ℓ | N. In this case, upon removing ℓ from the level, the weight may initially go up to ℓ + 1. If this occurs, since k(ρ) = 2 we can use [26] to lowe the weight back down to 2.

In order to get started we must make an unavoidable construction: Shimura curves. Denote by X(K, pq) the modular curve associate to Γ_{1}(K) ∩ Γ_{0}(pq) and let J = Jac(X(K, pq)) be its Jacobian. Likewise, denote by X^{pq}(K) the Shimura curve associated to the quaternion algebra of discriminant pq. The curve X^{pq}(K) is constructed as follows.^{2} Let B be an indefinite quaternion algebra over Q of discriminant pq. (Up to isomorphism, B is unique.) Let O be an Eichler order of level K (i.e., reduced discriminant Kpq) in B. Let Γ_{∞} be the group of elements of O with (reduced) norm 1. After fixing an embedding B → M(2, R) (an embedding exists because B is indefinite), we obtain in particular an embedding Γ_{∞} ↪ SL(2, R) and therefore an action of Γ_{∞} on the upper half-plane 𝔥. Let X^{pq}(K) be the standard canonical model, over Q, of the compact Riemann surface Γ_{∞}\𝔥, and let J′ = Jac(X^{pq}(K)) denote its Jacobian. The curve X^{pq}(K) is furnished with Hecke correspondences T_{n} for n ≥ 1. We write T_{n} for the endomorphism of J induced by the T_{n} via Pic functoriality.

1Brian Conrad remarks that counterexamples to Fermat are not real life.

2Copied from your Invent. 100 paper.

By work of [27, 37, 79] there is a correspondence

[figure: traces of automorphic forms; traces of cusp forms]

Upon understanding this, and applying Faltings’s isogeny theorem [28], we find (noncanonically!) an isogeny J′ → J = J(K, pq) = Jac(X(K, pq)). The pq-new part of J is

J_{pq-new} = ker(J(K, pq) ⟶ J(K, p)^{4})

where the maps are induced by Albanese functoriality from the four maps

\xymatrix{X(K, pq) \ar[r] \ar[r] & X(K, p) & and & X(K, pq) \ar[r] \ar[r] & X(K, q).}

The kernel of J′ → J is finite and the image is the pq-new part of J.

Amazingly, there seems to be no canonical map J′ → J. Amazingly, upon looking at the corresponding character groups, you do see some canonical relationship between J′ and J. The Čerednik-Drinfeld theory [13, 24] gives a description of X^{pq}(K) in characteristic p. Using this one finds the following relation between character groups: there is a canonical T-equivariant exact sequence

0 ⟶ X_{p}(J′) ⟶ X_{q}(J) ⟶ X_{q}(J′′) ⟶ 0 (7.1)

where J′′ = Jac(X(K, q))^{2}. Amazingly, this exact sequence relates a character group “in characteristic p” to two character groups “in characteristic q.”

Since ρ is ramified at q (the key assumption), 𝔪 ⊂ T is not q-old. We may as well suppose we are in a situation where we can’t lower the level using Mazur’s principle, so we assume that 𝔪 is not p-old either (and hope for a contradiction). Localization is an exact functor, so the localization

0 ⟶ X_{p}(J′)_{𝔪} ⟶ X_{q}(J)_{𝔪} ⟶ X_{q}(J′′)_{𝔪} ⟶ 0 (7.2)

of (7.1) is also exact. The Hecke algebra T acts on X_{q}(J′′) through a quotient \overline{T}. Since 𝔪 is not q-old, the image of 𝔪 in \overline{T} generates the unit ideal. Therefore X_{q}(J′′)_{𝔪} = 0 and we obtain an isomorphism X_{p}(J′)_{𝔪} ≈ X_{q}(J)_{𝔪}. If M is a Tmodule then M/𝔪M = M_{𝔪}/𝔪M_{𝔪} so

X_{q}(J)/𝔪X_{q}(J) ≈ X_{p}(J′)/𝔪X_{p}(J′). (7.3)

Switching p and q and applying the same argument we find that

X_{p}(J)/𝔪X_{p}(J) ≈ X_{q}(J′)/𝔪X_{q}(J′). (7.4)

Each of these is an isomorphism of T/𝔪-vector spaces. Recall [6] that J[𝔪] ≈ ⨁_{i=1}^{λ} V, with λ > 0 and J′[𝔪] ≈ ⨁_{i=1}^{ν} V. It follows from [37] that ν > 0, but we will not use this here. Our hypothesis that V is unramified automatically propagates to all of J[𝔪] ≈ ⨁ V. Since V is irreducible and we are assuming that 𝔪 is not p-old, the same argument as in the previous lecture shows that J[𝔪] ⊂ T [𝔪] where T is the toric part of J_{F_{p}}. This means that dim(X_{p}(J)/𝔪X_{p}(J)) ≥ 2λ. Using the same argument with J replaced by J′ gives that dim(X_{p}(J′)/𝔪X_{p}(J′)) ≥ 2μ.

Since V is ramified at q (by hypothesis) and there is an unramified line (?), dim(V^{I_{q}}) = 1. Thus dim J[𝔪]^{I_{q}} = λ so, since Hom(X_{q}(J)/𝔪X_{q}(J), μ_{ℓ}) ⊂ J[𝔪]^{I_{q}} we find that dim X_{q}(J)/𝔪X_{q}(J) ≤ λ. A similar argument bound the dimension of X_{q}(J′)/𝔪X_{q}(J′). We obtain the following quadruple of inequalities:

dim X_{q}(J)/𝔪X_{q}(J) ≤ λ,
dim X_{q}(J′)/𝔪X_{q}(J′) ≤ μ,
dim X_{p}(J)/𝔪X_{p}(J) ≥ 2λ,
dim X_{p}(J′)/𝔪X_{p}(J′) ≥ 2μ.

Combining these with (7.3, 7.4) we find that

2λ ≤ dim X_{p}(J)/𝔪X_{p}(J)
= dim X_{q}(J′)/𝔪X_{q}(J′) ≤ μ

and simulatenously that 2μ ≤ λ. Combining gives that 4λ ≤ λ, so λ = 0. This is contrary to fact: the multiplicity of ρ in J[𝔪] is strictly positive.

This completes the proof.

LECTURE 8

Level lowering with multiplicity one

The hard case is illustrated in the following diagram:

[figure: Mpq; pivot; Mq; easy; Mp; level addition; M]

The pivot step is potentially the hardest. This is because the symmetry is broken. Before we knew that q could not be removed from the level. But here, q can be removed. The argument has been made to work under the assumption that ℓ > 2 [63]. When the hypothesis of the multiplicity one theorems hold [26, §9], the following argument works, even when ℓ = 2.

We start with the data of a mod ℓ representation ρ arising from a weight 2 newform f on Γ_{1}(M) ∩ Γ_{0}(p). The prime p does not divide N(ρ) and our goal is to get p out of the level.

For all we know, ρ is unramified at each prime q | M. For example we could have M = 1. In particular, there is no reason to expect that we are in the situation of §7.1; there is no reason for the key assumption to hold for some choice of q | M. We are thus forced to manufacture q as follows. Pick q to be one of the (infinitely many) primes not dividing Mpℓ such that both of the following conditions hold:

1. \begin{cases}ρ(Frob_{q}) is not a scalar, or \\ q ≢ 1 (mod ℓ). \end{cases}

2. \begin{cases}the ratio of the eigenvalues of \\ ρ(Frob_{q}) is either q or 1/q. \end{cases}

To be more precise, in the second condition we mean that the characteristic polynomial of ρ(Frob_{q}) is of the form (x − a)(x − qa) for some a ∈ \overline{F}_{ℓ}^{∗}.

Proposition 11. There are infinitely many q satisfying the two conditions enumerated above.

Proof. First suppose that ℓ > 2. Using the Cebotarev density theorem, find infinitely many prime numbers q such ρ(Frob_{q}) = ρ(c) where c denotes complex conjugation. The eigenvalues of ρ(c) are 1 and −1 (Exercise 18), which have ratio −1. This ratio is equal to q because in characteristic ℓ,

−1 = χ(c) = det(ρ(Frob_{q})) = χ(Frob_{q}) = q.

In the case ℓ = 2 we first observe [by an argument to be referenced and sketched here involving classifying subgroups of GL(2, \overline{F}_{2})] that the image of ρ has even order. After a possible change of basis we find ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ ρ(G_{Q}). Using Cebotarev density we find infinitely many q with ρ(Frob_{q}) conjugate to ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). For such q, condition (1) is satisfied. Condition (2) is also satisfied because the ratio of the eigenvalues is 1 which, because q is an odd prime, is congruent to q modulo ℓ = 2.

With q chosen as above we can raise the level. More precisely, there exists a pq-new form at level Mpq. [[Say a lot more here.]] We illustrate this as follows.

[figure: (M, \underline{pq}); goal; (M, p); raise; (M, q); Mazur (condition 1); M]

We underline pq to emphasize that the situation at level (M, pq) is symmetrical in p and q.

Let J = J(M, pq). Inside of T = Z[… T_{n} …] ⊂ End J there is a maximal ideal 𝔪 attached to the pq-newform f giving rise to ρ. Applying the multiplicity one theorem at level Mpq (when possible–see [26, §9]) we have

J[𝔪] = V (multiplicity one assumption)

where V supports ρ. In all of this, M can be divisible by 2. The distinction between whether or not M is even comes into play mainly in the multiplicity one argument, not in this level lowering argument.

Let J′ = J^{pq}(M) be the Shimura curve analogue of J_{1}(M). This is an analogue of J_{1}(M) with M_{2}(Q) replaced by a quaternion algebra. One has the very important:

J′[𝔪] = ⨁_{i=1}^{ν} V (at least one copy: ν ≥ 1).

This follows morally because ρ arises from a pq-new form. [[The actual argument is more involved.]]

Assume that we cannot lower the level. We have an exact sequence of character groups

0 ⟶ X_{p}(J′) ⟶ X_{q}(J) ⟶ X_{q}(J(M, q)^{2}) ⟶ 0.

Upon localizing at 𝔪 as in (7.2) we discover, taking dimensions over the field T/𝔪, that

dim X_{p}(J′)/𝔪X_{p}(J′) = dim X_{q}(J)/𝔪X_{q}(J). (8.1)

Further, since the component group of J′ at p is a quotient of X_{q}(J(M, q)^{2}), we find that

V ↪ (J′[𝔪]^{I_{p}})^{toric}.

Thus dim X_{p}(J′)/𝔪 ≥ 2, so (8.1) implies that dim X_{q}(J)/𝔪 ≥ 2. The endomorphism Frob_{q} acts as a scalar on J[𝔪]^{toric} ⊂ J[𝔪], and Frob_{q}^{2} = q^{2}⟨q⟩ (?? is it true, do we need this?). But both J[𝔪]^{toric} and J[𝔪] have dimension 2, hence Frob_{q} acts as a scalar on J[𝔪]. If q ≢ 1 (mod ℓ) then we could use Mazur’s principle to lower

LECTURE 8. LEVEL LOWERING WITH MULTIPLICITY ONE 49

the level, so by condition 1 we may assume that ρ(Frob_{q}) is not a scalar. This contradiction completes the proof.

More details.

LECTURE 9 Other directions

I looked through Ash and Sinnott.  There should be a paragraph somewhere
in your PCMI notes indicating this direction.  But where: in the
introduction, or in a new section that would also discuss Ramakrishna's
work and maybe even Serre mod pq?
Perhaps there should be a final section called "other directions".
It could summarize the many topics that we don't touch on in the
notes.
-ken

LECTURE 10

Katz modular forms (Appendix)

In this section we briefly introduce modular forms modulo ℓ from a geometric point of view^{1} Our purpose is to give a brief overview, rather than a complete (or even well motivated) account.

Modular forms as rules. Fix a base ring R. Consider pairs (E, ω)/S where S is a ring over R, E is an elliptic curve over S enhanced with level structure, and ω is a differential which does not vanish modulo any prime ideal of S. A modular form f of weight k is a certain type of rule which takes any such pair (E, ω)/S to an element of S. The two main properties of f are:

1. Weight: For all λ ∈ R^{∗}, f(E, λω) = λ^{−k}f(E, ω).

2. Holomorphicity: f commutes with base change in the evident sense.

3. Tate curves: Condition to insure that q-expansions do not have negative powers of q.

This definition is difficult to think about because we have only

(rules) + (functoriality)

A priori, we do not know what is going on. Fortunately, we know what elliptic curves are, not in the sense of Shimura–Taniyama, but in the moduli sense. Supposing N ≥ 5, there exists a model for X_{1}(N) over Z[1/N] and a scheme E_{1}(N) over X_{1}(N) such that for all rings R in which N is invertible, and generalized elliptic curves E enhanced with level structure, there is a unique map Spec(R) → X_{1}(N) making the following diagram Cartesian, i.e., a fiber product:

\xymatrix{E \ar[r] \ar[d] & E_{1}(N) \ar[d] \\ Spec(R) \ar[r] & X_{1}(N)}

Glueing together differentials on each elliptic curve E corresponding to a point of X_{1}(N) gives a line bundle \underline{ω} on X_{1}(N). Weight k modular forms on Γ_{1}(N)/Z[1/N] are the same thing as elements of H^{0}(X_{1}(N), \underline{ω}^{⊗k}). This is a finite module over Z[1/N] which gives all rules of weight k and level N.

1This section records a lecture of Mathew Emerton. This lecture should be an appendix, and might be replaced/supplemented by notes that Brian Conrad is writing up on his lectures.

Modular forms mod ℓ. Suppose ℓ is a prime with ℓ ∤ N. The space of weight k modular forms on Γ_{1}(N)/F_{ℓ} is the same as

H^{0}(X_{1}(N)/F_{ℓ}, \underline{ω}^{⊗k}).

To relate this space to modular forms over Z[1/N] first consider the diagram:

[figure: E/F_{ℓ}; E/Z[1/N]; \underline{ω}/F_{ℓ}; \underline{ω}/Z[1/N]; X_{1}(N)/F_{ℓ}; X_{1}(N)/Z[1/N]; Spec(F_{ℓ}); Spec(Z[1/N])]

Do all the modular forms over F_{ℓ} come via reduction from modular forms over Z[1/N]? Yes, provided that the appropriate cohomology commutes with base change. This is a subtle question because, in general, cohomology need not commute with base change. Via some trickery, and consideration of the exact sequence

0 → \underline{ω}^{⊗k} ⟶^{ℓ} \underline{ω}^{⊗k} ⟶ \underline{ω}^{⊗k}/F_{ℓ} → 0

one sees that a condition which insures that all forms come from characteristic 0 is that

H^{1}(X_{1}(N)/F_{ℓ}, \underline{ω}^{⊗k}) = 0.

The Riemann-Roch theorem often gives this vanishing, the essential ingredient being knowledge of deg(\underline{ω}). For example, when k ≥ 2 and N ≥ 5, the H^{1} does vanish! There are problems when N ≤ 5 because an appropriate representing object E_{1}(N) → X_{1}(N) does not exist.

Flat base change commutes with cohomology. For example Z[1/N] ↪ C is flat because C is torsion free, whereas Z[1/N] → F_{ℓ} is not flat. Observe that H^{0}(X_{1}(N), ω^{⊗k}/Z[1/N]) is just a choice of Z[1/N] lattice in our favorite complex vector space of cusp forms. The following diagram is very helpful (forgetting it is a common source of confusion.)

\xymatrix{C ⊕ ⨁_{k≥2} M_{k}(C) \ar@{^{(}->}[rr] &  & C[[q]] \\ Z[\frac{1}{N}] ⊕ ⨁_{k≥2} M_{k}(Z[\frac{1}{N}]) \ar@{^{(}->}[rr] \ar@{^{(}->}[u] \ar[d] &  & (Z[\frac{1}{N}])[[q]] \ar@{^{(}->}[u] \ar[d] \\ F_{ℓ} ⊕ ⨁_{k≥2} M_{k}(F_{ℓ}) & not injective \ar[r] & F_{ℓ}[[q]]}

There exists modular forms over F_{ℓ} of different weights which have the same q-expansions. Consider the Hasse invariant A ∈ H^{0}(X_{1}(N)/F_{ℓ}, ω^{⊗ℓ−1}). Let A/F_{ℓ} be an elliptic curve and let Φ denote the Frobenius endomorphism. The map H^{1}(E, O) → H^{1}(E, O) induced by Frobenius is either 0 or not; if not [[fill in details here.]] The q-expansion of A is 1 ∈ F_{ℓ}[[q]]. This is the same as the q-expansion of 1.

In characteristic 0, weight 1 forms are very special so Serre didn’t allow them in his conjecture. In characteristic p, weight 1 forms behave in pretty much the same way as other weights.

The references are: Serre, Bourbaki, 1969 Serre, Swinnerton-Dyer, LNM 350 “Galois representations” Katz “very geometric” Gross “modular forms mod ℓ” Edixhoven Jochnowitz, Duke, Trans. of AMS

LECTURE 11 Exercises (Appendix)

The following exercises were used in the problem sessions^{1}. Solutions to selected exercises are provided at the end.

Exercise 12. Suppose ρ : Gal(\overline{Q}/Q) → F_{ℓ}^{∗} is a one-dimensional continuous odd Galois representation.

1. Give an example to show that ρ need not be a power of the mod ℓ cyclotomic character.

2. Assume that ρ is unramified outside ℓ. Deduce that ρ is a power of the cyclotomic character.

Exercise 13. The principal congruence subgroup Γ(N) of level N is the subgroup of SL(2, Z) consisting of matrices congruent to the identity modulo N. Further, Γ_{1}(N) is the subgroup of SL(2, Z) of matrices that are upper triangular mod N and have diagonal entries ≡ 1 mod N. A congruence subgroup is a subgroup of SL(2, Z) that contains Γ(N) for some N. Let Γ ⊂ SL(2, Z) be a congruence subgroup. Show that there exits g ∈ GL(2, Q) such that g^{−1}Γg ⊂ GL(2, Q) contains Γ_{1}(N) for some N.

Exercise 14. Let A/Q be an elliptic curve. Show that End_{Q}(A) = Z, that is, integer multiplications are the only Q-rational endomorphisms of A. Assume further that A is isolated in its isogeny class, by which we mean that if B/Q is an elliptic curve that is isogeneous to A over Q, then A and B are isomorphic over Q. Show that, for every prime number ℓ, the representation

ρ_{ℓ} : Gal(\overline{Q}/Q) → Aut(A[ℓ]) ≈ GL(2, F_{ℓ})

is irreducible. Must ρ_{ℓ} be absolutely irreducible (i.e., remain irreducible after composing with the embedding GL(2, F_{ℓ}) ↪ GL(2, \overline{F}_{ℓ}))?

Exercise 15. Let A/Q be an elliptic curve, and assume that, for all ℓ, the representation ρ_{A,ℓ} : Gal(\overline{Q}/Q) → Aut(A[ℓ]) is irreducible. Show that A is isolated in its isogeny class. This is the converse of Exercise 14.

Exercise 16. Suppose ρ : Gal(\overline{Q}/Q) → GL(2, F_{ℓ}) arises from the ℓ-torsion of an elliptic curve. Verify, using standard properties of the Weil pairing, that det(ρ) is the mod ℓ cyclotomic character.

1The authors would like to thank D. Savitt, K. Kedlaya, and B. Conrad for contributing problems.

Exercise 17. Let f ∈ S_{k}(Γ_{1}(N)) be a modular form that is an eigenform for all the Hecke operators T_{p} and for the diamond bracket operators ⟨d⟩. Let

ε : (Z/NZ)^{∗} → C^{∗}

be the character of f, so ⟨d⟩f = ε(d)f.

1. Show that f satisfies the following functional equation:

for any ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N),

f(z) = ε(d)(cz + d)^{−k}f(\frac{az + b}{cz + d}).

2. Conclude that ε(−1) = (−1)^{k}.

3. Choose a mod ℓ Galois representation ρ associated to f. Its determinant is det(ρ) = ε · χ^{k−1} where χ is an appropriate cyclotomic character. If c is the complex conjugation, deduce that that ρ is odd: det(ρ(c)) = −1.

Exercise 18. Let ρ : G → GL(2, \overline{F}_{ℓ}) be an odd Galois representation. Let c ∈ G denote the complex conjugation.

1. Assume ℓ ≠ 2. Prove that ρ(c) is conjugate over \overline{F}_{ℓ} to the matrix ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ).

2. Assume ℓ = 2. Give an example to show that ρ(c) need not be conjugate to

( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ).

Exercise 19. Construct a non-continuous homomorphism

ρ : Gal(\overline{Q}/Q) → \{±1\}

where \{±1\} has the discrete topology. You must produce a map ρ : Gal(\overline{Q}/Q) → {±1} such that

1. ρ is a homomorphism, and

2. ρ does not factor through Gal(K/Q) for any finite Galois extension K/Q.

Exercise 20. One difficulty is that the representation ρ arising from a modular form sometimes takes values in a slightly smaller field than O/λ. For example, let f be one of the two conjugate newforms of level 23, weight 2, and trivial character. Then

f = q + αq^{2} + (−2α − 1)q^{3} + (−α − 1)q^{4} + 2αq^{5} + ⋯

with α^{2} + α − 1 = 0. The coefficients of f lie in O = Z[α] = Z[\frac{1+\sqrt{5}}{2}]. Take λ to be the unique prime of O lying over 2, then O/λ ≅ F_{4} and so \overline{ρ}_{f,λ} is a homomorphism to GL(2, F_{4}). Show that if p ≠ 2 then a_{p} ∈ Z[\sqrt{5}], so that \overline{ρ}_{f,λ} has a model over GL(2, F_{2}).

Exercise 21. Let A/Q be an elliptic curve and ℓ ≠ 2 a prime.

1. Prove that the field Q(A[ℓ]) generated by the coordinates of the points in A[ℓ] is not equal to Q.

2. Given an example of an elliptic curve A so that Q(A[2]) = Q.

Exercise 22. Let A be the elliptic curve defined by

y^{2} = x^{3} + ax + b with a, b ∈ Q.

1. Describe the Galois representation

ρ = ρ_{A,2} : Gal(\overline{Q}/Q) ⟶ GL(2, F_{2}).

2. Give necessary and sufficient conditions for ρ to be reducible.

3. Given an example in which ρ is ramified at 23.

Exercise 23. Let ε and ρ be continuous homomorphisms ε, ρ : Gal(\overline{Q}/Q) → F_{ℓ}^{∗}. Suppose further that for all primes p, for which both ρ and ε are unramified, we have

ρ(Frob_{p}) = ε(Frob_{p})p^{i} ∈ F_{ℓ}.

Deduce that ρ = ε · χ^{i} where χ is the mod ℓ cyclotomic character.

Exercise 24. Let A/Q be an elliptic curve of conductor N. Let p be a prime number not dividing N. Denote by Ã the elliptic curve A considered over the finite field F_{p}. There is a Frobenius endomorphism Φ = Φ_{p} : A → A that sends an affine point (x, y) to the point (x^{p}, y^{p}) and fixes ∞. Let X^{2} +aX +b be the characteristic polynomial of Φ acting on the Tate module of A, for some (any) prime ℓ ≠ p. Define tr(Φ) = −a and deg(Φ) = b.

1. Show that deg(Φ).

2. Show that tr(Φ) = p + 1 − #A(F_{p}), i.e., tr(Φ) = a_{p}.

3. Let ℓ ∤ pN be a prime. Then Ã[ℓ] is a vector space of dimension two over F_{ℓ}, and Φ induces a map Ã[ℓ] → Ã[ℓ]. Show that this is exactly the same map as that induced by some choice of Frob_{p} ∈ Gal(\overline{Q}/Q).

4. Conclude that

tr(ρ_{A,ℓ}(Frob_{p})) = p + 1 − #A(F_{p}) (mod ℓ).

Exercise 25. Suppose A/Q is an elliptic curve of conductor N, and let ℓ be a prime. Show that any prime p not dividing ℓN is unramified in Q(A[ℓ])/Q. You may use the following fact proved using formal groups: The map A[ℓ] → Ã(\overline{F}_{p}) is injective, where Ã is the reduction of A modulo p. (Using formal groups one can show that all torsion in the kernel is of p-power order.

Exercise 26. Show that the fundamental character of level 1 is the cyclotomic character χ|_{I}.

Exercise 27. For each of the following semistable elliptic curves A, and each ℓ at which ρ_{A,ℓ} is irreducible, compute the Serre minimal weight k(ρ_{A,ℓ}) and level N(ρ_{A,ℓ}).

N  |Δ|  reducible ℓ A
30  2^{4} · 3^{5} · 5  2, 3  y^{2} + xy + y = x^{3} + x + 2
210  2^{12} · 3^{3} · 5 · 7  2, 3  y^{2} + xy = x^{3} − 41x − 39
330 2^{4} · 3^{2} · 5^{4} · 11^{2}  2  y^{2} + xy = x^{3} + x^{2} − 102x + 324
455  5^{3} · 7^{4} · 13  2  y^{2} + xy = x^{3} − x^{2} − 50x + 111
2926 2^{8} · 7^{3} · 11^{4} · 19^{2}  2  y^{2} + xy + y = x^{3} − x^{2} + 1934x − 1935

Exercise 28. Let M be a positive integer and p a prime. Show that there is injective linear map

S_{2}(Γ_{1}(M)) ↪ S_{2}(Γ_{1}(pM))

sending f(q) to f(q^{p}).

Exercise 29. Let M be such that S_{2}(Γ_{1}(M)) has positive dimension and let p be a prime.^{2} 1. Let f ∈ S_{2}(Γ_{1}(M)) be an eigenvector for T_{p} with eigenvalue λ. Show that T_{p} acting on S_{2}(Γ_{1}(Mp)) preserves the two-dimensional subspace generated by f and T_{p}f. Show furthermore that if λ^{2} ≠ 4p^{k−1} then T_{p} is diagonalizable on this 2-dimensional space. What are the eigenvalues of T_{p} on this space? Remark: I believe there is no example known where λ^{2} = 4p^{k−1} and it’s a theorem that it can never happen if k = 2.

2This problem is based on a Cambridge handout of Kevin Buzzard’s.

2. Show that for any r > 2, the Hecke operator T_{p} on S_{2}(Γ_{1}(Mp^{r})) is not diagonalizable. Conclude that the Hecke algebra T associated to S_{2}(Γ_{1}(Mp^{r})) can not be an order in a product of rings of integers of number fields.

Exercise 30. Let N be a positive integer. Show that the Hecke algebra T = Z[… T_{n} …] ⊂ End_{Q}(J_{1}(N)) is of finite rank as a Z-module.

Exercise 31. Let Spec(T) be the affine scheme associated to the Hecke algebra T. Give a careful interpretation of the geometry of Spec(T) in terms of eigenforms and Galois representations. Draw pictures to illustrate your ideas.

Exercise 32. Suppose N = pM with (p, M) = 1. There is an injection

S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M)) ↪ S_{2}(Γ_{1}(N) ∩ Γ_{0}(p))

given by (f, g) ↦ f(q)+g(q^{p}). The Hecke algebra T = T_{N} acts through a quotient \overline{T} on the image of S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M)). Suppose 𝔪 ⊂ T is a maximal ideal that arises by pullback from a maximal ideal in \overline{T}. Show that ρ_{𝔪} arises from a modular form of level M.

Exercise 33. An exercise on Tate curves.

Solutions.

Solution 12. 1. Let p be a prime different than ℓ and

ρ : Gal(\overline{Q}/Q) → Gal(Q(\sqrt{p})/Q) = \{±1\} ↪ F_{ℓ}^{∗}.

2. Let K = Q^{ker(ρ)}. Then K/Q is abelian and ramified only at ℓ, so K ⊂ Q(ζ_{ℓ^{∞}}). But [K : Q] | ℓ − 1 so K ⊂ Q(ζ_{ℓ}).

Solution 13. Conjugate using g = ( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} ) .

Solution 14. Suppose φ ∈ End_{Q}(E) is a nonzero endomorphism. The induced map dφ on the differentials H^{0}(A, Ω) ≈ Q is multiplication by an integer n. Then d(φ − n) = 0 so φ = n.

Suppose that ρ_{ℓ} is reducible, so that there is a one-dimensional Galois stable subspace V ⊂ A[ℓ]. The quotient B = A/V is then an elliptic curve over Q and there is an isogeny π : A → B of degree ℓ. By the assumption that A is isolated in its isogeny class, we have in fact that B = A and so there is an endomorphism of A of degree ℓ. But all Q-rational endomorphisms are multiplication by an integer, and multiplication by an integer has degree a perfect square.

Solution 15. Suppose all ρ_{A,ℓ} are irreducible, yet there exists an isogeny φ : A → B with B ≅ A̸. Choose φ to have minimal possible degree and let d = deg(φ) > 1. Let ℓ be the smallest prime divisor of d and choose a point x ∈ ker(φ) of exact order ℓ. If the order ℓ cyclic subgroup generated by x is Galois stable, then ρ_{A,ℓ} is reducible, contrary to assumption. Thus ker(φ) contains the full ℓ-torsion subgroup A[ℓ] of A. In particular, φ factors as

\xymatrix{A \ar[rr]^{φ} \ar[dr]^{ℓ} &  & B \\  & A/A[ℓ] \ar@{-->}[ur]}

Since A/A[ℓ] ≅ A, there is an isogeny from A to B of degree equal to d/ℓ^{2}, which is contrary to our assumption that d is minimal.

Solution 16. The Weil pairing ( , ) : A[ℓ] × A[ℓ] → μ_{ℓ} can be viewed as a map

Λ^{2}A[ℓ] ⟶^{≅} μ_{ℓ}

sending P ∧ Q to (P, Q). For any σ ∈ Gal(\overline{Q}/Q), (P^{σ}, Q^{σ}) = (P, Q)^{σ}. With the action (P ∧ Q)^{σ} = P^{σ} ∧ Q^{σ}, the map Λ^{2}A[ℓ] → μ_{ℓ} is a map of Galois modules. To compute det(ρ(σ)) observe that if e_{1}, e_{2} is a basis for A[ℓ], and ρ(σ) = ( \begin{matrix}a & b \\ c & d \end{matrix} ), then

σ(e_{1} ∧ e_{2}) = (ae_{1} + ce_{2}) ∧ (be_{1} + de_{2})
= (ad − bc)e_{1} ∧ e_{2} = det(ρ(σ))e_{1} ∧ e_{2}

Thus Λ^{2}A[ℓ] gives the one-dimensional representation det(ρ). Since Λ^{2}A[ℓ] is isomorphic to μ_{ℓ} it follows that det(ρ) = χ.

Solution 17. The definition of ⟨d⟩ is as follows: choose any matrix σ_{d} ∈ Γ_{0}(N) such that σ_{d} ≡ ( \begin{matrix}d & 0 \\ 0 & d^{−1} \end{matrix} ) (mod N), then ⟨d⟩f = f|_{σ_{d}}. Observe that Γ_{1}(N) is a normal subgroup of Γ_{0}(N) and the matrices σ_{d}, for (d, N) = 1, d < N, are a system of coset representatives. Thus any ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) can be written in the form σ_{d} · g for some g ∈ Γ_{1}(N). We have

f = f|_{σ_{d}g} = (f|_{σ_{d}})|_{g} = (ε(d)f)|_{g} = ε(d)f|_{g} = ε(d)(cz + d)^{−k}f (\frac{az + b}{cz + d}) .

Solution 18.

1. Since c^{2} = 1, the minimal polynomial f of ρ(c) divides x^{2} − 1. Thus f is either x + 1, x − 1, or x^{2} − 1. If f = x + 1 then ρ(c) = −1 = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ). This implies det(ρ(c)) = (−1)^{2} = 1, a contradiction since det(ρ(c)) = −1 and ℓ > 2. If f = x − 1, then ρ(c) = 1; again a contradiction. Thus the minimal polynomial of ρ(c) is x^{2} − 1 = (x − 1)(x + 1). Since −1 ≠ 1 there is a basis of eigenvectors for ρ(c) such that matrix of ρ(c) with respect to this basis is

( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ).

2. The following example shows that when ℓ = 2 the matrix of ρ_{A,ℓ} need not be conjugate to ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). Let y^{2} = x(x^{2} − a) with a ∈ Q not a square. Then

A[2] = \{∞, (0, 0), (\sqrt{a}, 0), (−\sqrt{a}, 0)\}.

The action of c on the basis (0, 0), (\sqrt{a}, 0) is represented by the matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), since c(−\sqrt{a}, 0) = (\sqrt{a}, 0) = (0, 0) + (−\sqrt{a}, 0).

Solution 19.

1. Choose a sequence p_{1}, p_{2}, … of distinct prime numbers. Define ρ_{1} : G → ∏ F_{2} by

ρ_{1}(σ)_{i} = \begin{cases}0 & if σ acts trivially on Q(\sqrt{p_{i}}), \\ 1 & otherwise \end{cases}

Thus ρ_{1} is just

G → Gal(Q(\sqrt{p_{1}}, \sqrt{p_{2}}, …)/Q) ≈ ∏ F_{2}.

2. Let ⊕F_{2} ⊂ ∏ F_{2} be the subgroup of elements having only finitely many nonzero coordinates. Then ∏ F_{2}/⊕F_{2} is a vector space over F_{2} of dimension > 0. By Zorn’s lemma, there is a basis B of ∏ F_{2}/ ⊕ F_{2}. Let b ∈ B and and let W be the space spanned by B − {b}. Then V = (∏ F_{2}/ ⊕ F_{2})/W is an F_{2} vector space of dimensional 1.

3. Let ρ be the composite map

\xymatrix{Gal(\overline{Q}/Q) \ar[d] \ar[drr]^{ρ} \\ ∏ F_{2} \ar[r] & V \ar[r]_{≅} & \{±1\}}

4. Let H = ker(ρ) ⊂ Gal(\overline{Q}/Q). If σ(\sqrt{p_{i}}) = −\sqrt{p_{i}} and σ(\sqrt{p_{j}}) = \sqrt{p_{j}} for i ≠ j, then σ ∈ H. Thus H does not fix any Q(\sqrt{p_{i}}), so the fixed field of H is Q. The largest Galois group quotient through which ρ factors is then Gal(Q/Q) = {1}, a contradiction since ρ ≠ 1.

Solution 20. We have f = f_{1} + αf_{2} with

f_{1} = q − q^{3} − q^{4} + ⋯
f_{2} = q^{2} − 2q^{3} − q^{4} + 2q^{5} + ⋯

Because S_{2}(Γ_{0}(23)) has dimension 2, it is spanned by f_{1} and f_{2}. Let η(q) = q^{\frac{1}{24}} ∏_{n≥1}(1 − q^{n}). Then g = (η(q)η(q^{23}))^{2} ∈ S_{2}(Γ_{0}(23)). Expanding we find that g = q^{2} − 2q^{3} + ⋯, so g = f_{2}. Next observe that g is a power series in q^{2}, modulo 2, since

g = q^{2} ∏(1 − q^{n})^{2}(1 − q^{23n})^{2}
≡ q^{2} ∏(1 − q^{2n})(1 − q^{46n}) (mod 2)
≡ q^{2} ∏(1 + q^{2n} + q^{46n} + q^{48n}) (mod 2)

Thus the coefficient in f_{2} of q^{p}, with p ≠ 2 prime, is even and the proposition follows.

Solution 21.

1. Let ζ ∈ μ_{ℓ} be a primitive ℓth root of unity. Since Λ^{2}A[ℓ] ≅ μ_{ℓ}, there exists P, Q ∈ A[ℓ] such that P ∧Q = ζ. Since ℓ > 2 there exists σ such that ζ^{σ} ≠ ζ, hence P^{σ} ∧ Q^{σ} ≠ P ∧ Q. This is impossible if all ℓ-torsion is rational, since then P^{σ} = P and Q^{σ} = Q.

2. y^{2} = (x − a)(x − b)(x − c) where a, b, c are distinct rational numbers.

Solution 22.

1. Let K be the splitting field of x^{3} + ax + b. Then ρ embeds Gal(K/Q) in GL(2, F_{2}):

\xymatrix{Gal(\overline{Q}/Q) \ar[rr]^{ρ} \ar[dr] &  & GL(2, F_{2}) \\  & Gal(K/Q) \ar@{^{(}->}[ur]}

2. ρ is reducible exactly when the polynomial x^{3} + ax + b has a rational root.

3. y^{2} = x(x^{2} − 23), y^{2} = x^{3} + x − 1. [[Insert Ravi’s geometric comments here.]]

Solution 23. Consider the character τ = εχ/ρ. By assumption τ(Frob_{p}) = 1 for all unramified p. Furthermore, τ factors through a finite abelian extension of K/Q. For any σ ∈ Gal(K/Q), the Cebotarev density theorem insures that there are infinitely many primes p such that Frob_{p} = σ. In particular, τ(σ) = τ(Frob_{p}) = 1, so τ = 1 and hence ρ = εχ.

Solution 24.

1. deg(Φ) = p, [82, 2.11].

2. By [82, 5.5], Φ−1 is separable, so #A(F_{p}) = deg(Φ−1). Since Φ has degree p, there exists an isogeny \overline{Φ} (the dual isogeny [82, III.6]), such that Φ\overline{Φ} = p. Letting bars denote the dual isogeny, we have

#A(F_{p}) = deg(Φ − 1) = (Φ − 1)\overline{(Φ − 1)}
= Φ\overline{Φ} − Φ − \overline{Φ} + 1
= p − tr(Φ) + 1

3. Both maps are pth powering on coordinates.

Solution 25. Because ℓ ≠ p and A has good reduction at p, the natural map A[ℓ] → Ã[ℓ] is an isomorphism. We have the following commutative diagram

\xymatrix{Gal(Q_{p}(A[ℓ])/Q_{p}) \ar@{^{(}->}[r] \ar[d] & Aut(A[ℓ]) \ar[d]^{≅} \\ Gal((O/λ)_{F_{p}}) \ar[r] & Aut(Ã[ℓ])}

It follows that the first vertical map must be injective, which is the same as Q_{p}(A[ℓ])/Q_{p} being unramified.

Solution 26. The level one character Ψ is obtained by the composition

Gal(Q_{ℓ}^{nr}(ℓ^{ℓ−1})/Q_{ℓ}^{nr}) → μ_{ℓ−1}(\overline{Q}_{ℓ}) → μ_{ℓ−1}(\overline{F}_{ℓ}^{∗}) = F_{ℓ}^{∗}.

Let π be such that π^{ℓ−1} = ℓ. Then Ψ(σ) = \frac{σ(π)}{π} (mod π). Let ζ ∈ \overline{Q}_{ℓ} be a primitive ℓth root of unity. Now

∏_{a=1}^{ℓ−1}(ζ^{a} − 1) = ℓ,

so

(ζ − 1)^{ℓ−1} ∏_{a=1}^{ℓ−1} \frac{ζ^{a} − 1}{ζ − 1} = ℓ

and (this is where Wilson’s theorem is used),

∏_{a=1}^{ℓ−1} \frac{ζ_{ℓ}^{a} − 1}{ζ − 1} ≡ 1 (mod ζ − 1).

Since the polynomial x^{ℓ−1} − 1 has roots over F_{ℓ}, by Hensel’s lemma there is a unit u ∈ Q_{ℓ}(ℓ^{ℓ−1}) such that

u^{ℓ−1} = ∏_{a=1}^{ℓ−1} \frac{ζ^{a} − 1}{ζ − 1}.

We can take π = (ζ − 1)u. Then

\frac{σ(π)}{π} = \frac{(ζ^{χ(σ)} − 1)σ(u)}{(ζ − 1)u}
= \frac{(ζ − 1)(ζ^{χ(σ)−1} + ⋯ + 1)σ(u)}{(ζ − 1)u}
= (ζ^{χ(σ)−1} + ⋯ + 1)σ(u)/u
≡ χ(σ) (mod ζ − 1)

Solution 27. We write N = N(ρ) and k = k(ρ) to save space. The essential tool is Theorem 5.

1. ℓ = 5: N = 6, k = 6, ℓ > 5, N = 30, k = 2.

2. ℓ = 5: N = 2 · 3 · 7, k = 6; ℓ = 7: N = 2 · 3 · 5, k = 8; ℓ > 7: N = 2 · 3 · 5 · 7,

k = 2.

3. ℓ = 3: N = 2·5·11, k = 4; ℓ = 5: N = 2·3·11, k = 6; ℓ = 7: N = 2·3·5·11,

k = 2; ℓ = 11: N = 2 · 3 · 5, k = 12; ℓ > 11: N = 2 · 3 · 5 · 11, k = 2.

4. ℓ = 3: N = 7 · 13, k = 2; ℓ = 5: N = 7 · 13, k = 6; ℓ = 7: N = 5 · 13, k = 8;

ℓ = 13: N = 5 · 7, k = 14; ℓ = 11, ℓ > 13: N = 5 · 7 · 13, k = 2.

5. ℓ = 3: N = 2·11·19, k = 2; ℓ = 7: N = 2·11·19, k = 8; ℓ = 11: N = 2·7·19,

k = 12; ℓ = 19: N = 2 · 7 · 11, k = 20; ℓ other: N = 2 · 7 · 11 · 19, k = 2.

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