A refinement of Ogg’s conjecture

William A. Stein

September 14, 1999

§1. The conjecture

• p prime

• f = q + a_{2}q^{2} + ⋯ ∈ S_{2}(Γ_{0}(p); C) eigenform for T = Z[… T_{n} …]

• I_{f} := ker(T ⟶^{T_{p}↦a_{p}} K_{f} = Q(… a_{n} …))

• A_{f} := J_{0}(p)/I_{f}J_{0}(p), abelian variety, dimension d := [K_{f} : Q]

• A′_{f} := dual of A_{f}

• π : J_{0}(p) → A_{f}

Associated finite groups:

• A_{f}(Q)_{tor} rational torsion

• Φ_{A_{f}}(\overline{F}_{p}) component group of closed fiber of Néron model

Motivation: Both quantities are closely related to quantities that appear in the conjecture of Birch and Swinnerton-Dyer, so we must understand them completely.

Conjecture 1. (a) We have

#Φ_{A_{f}}(\overline{F}_{p}) = #A_{f}(Q)_{tor} = order(π(0 − ∞))

(b) (More tentative) We have

∏ #A_{f}(Q)_{tor} = numer (\frac{p − 1}{12}) ,

where the product is over the Galois conjugacy classes.

Why?

• It is the nicest possible answer.

• All of my computer computations suggest it.

Remark: Totally false when the level is composite! Counterexamples abound, e.g., E =33A, E(Q) = Z/2 × Z/2, Φ_{E}(\overline{F}_{3}) = Z/6Z.

§2. Computational evidence

• Conjecture 1 true for all 305 of the A_{f} of conductor p ≤ 631.

• For all 666 of the A_{f} of conductor p ≤ 1571 the conjecture is true, up to powers of 2. I haven’t checked the 2-power. Same statement for 880 of the abelian varieties up to level 2411 (some levels missed).

Example 1:

\begin{matrix}J_{0}(389) ∼ & A_{1} & ×A_{2} & ×A_{3} & ×A_{6} & ×A_{20} \\ #A_{f}(Q)_{tor} & 1 & 1 & 1 & 1 & 97 \\ #Φ_{f}(\overline{F}_{p}) & 1 & 1 & 1 & 1 & 97 \\ \frac{L(1)}{Ω} & 0 & 0 & 0 & 0 & \frac{2^{11}·5^{2}}{97} \end{matrix}

numer (\frac{389 − 1}{12}) = 97

Example 2:

\begin{matrix}J_{0}(113) ∼ & A_{1} & ×A_{2} & ×A_{3} & ×B_{3} \\ #A_{f}(Q)_{tor} & 2 & 2 & 1 & 7 \\ #Φ_{f}(\overline{F}_{p}) & 2 & 2 & 1 & 7 \\ \frac{L(1)}{Ω} & \frac{1}{2} & \frac{1}{2} & 0 & \frac{8}{7} \end{matrix}

numer (\frac{113 − 1}{12}) = 28

(Remark: Probably Sha(B_{3}) has order 8, and is explained by the Mordell-Weil group of A_{3}.)

§3. Theoretical evidence

Ribet and Mestre-Oesterlé (J. Reine Angew. Math., 1989): If A_{f} is an elliptic curve, then Conjecture 1(a) true.

Mazur:

• (Ogg’s conjecture)

#J_{0}(p)(Q)_{tor} = #Φ_{J_{0}(p)}(\overline{F}_{p}) = numer (\frac{p − 1}{12}) .

So my conjecture can be viewed as a refinement of Ogg’s.

• If ℓ | #A_{f}(Q) then ℓ | numer (\frac{p−1}{12})

(Mazur’s Modular curves …, page 141).

• Conjecture 1(b): Mazur’s comment on the conjecture is: What you are conjecturing is something I don’t think I ever thought of—at least in those terms—It has some implications which are interesting (to me). For example suppose that a prime q divides the numerator of (p − 1)/12 but q^{2} does not; let A be the q-Eisenstein quotient. Then only one optimal quotient of A would be allowed to have nontrivial q-torsion, and the other(s), if there are any, would, I guess, then have to settle for a μ_{q} subgroup. I suppose, though, that it is very often the case that (under the above hypotheses of q) the q-Eisenstein quotient is simple.

Ribet: If ℓ | #Φ_{A_{f}}(\overline{F}_{p}) then ρ_{f,𝔪} tends to be finite for some 𝔪 containing ℓ. If so, and ρ_{f,𝔪} is irreducible, then Ribet’s level lowering leads to a contradiction.

Remark: If Φ_{J} → Φ_{A_{f}} is surjective, then the primes on the left hand side of Conjecture 1b divide the right hand side. So we make the following weaker conjecture. Conjecture 2. The natural map Φ_{J} → Φ_{A_{f}} is surjective.

§4. Grothendieck’s monodromy pairing

• J := J_{0}(p).

• A := A_{f}.

• A := Néron model of A.

• T_{A} := (torus of closed fiber of A) = A_{F_{p}}^{o}, since A is purely toric.

• X_{A} := Hom(T_{A}, G_{m}); A ↦ X_{A} contravariant; X_{A} free abelian of rank X_{A} = dim A.

The key diagrams are as follows.

\xymatrix{A′ \ar@{^{(}->}[r]^{π′} \ar[dr]_{θ} & J \ar@{->>}[d]^{π} & X_{A} \ar@{^{(}->}[r]^{π^{∗}} \ar[dr]^{θ^{∗}} & X_{J} \ar@{->>}[d]^{π_{∗}} \\  & A &  & X_{A′} \ar[ul]^{θ_{∗}}}

Modular degree: m_{A} = \sqrt{deg θ}.

Monodromy:

⟨ , ⟩ : X_{J} × X_{J} → Z
⟨ , ⟩ : X_{A} × X_{A′} → Z

Component groups:

0 → X_{J} → Hom(X_{J}, Z) → Φ_{J} → 0
0 → X_{A′} → Hom(X_{A}, Z) → Φ_{A} → 0

W. A. Stein, A refinement of Ogg’s conjecture V

§5. Formulas for Φ_{A}

S := saturation of π^{∗}X_{A} For any L ⊂ S of finite index, define

m_{L} := #(X_{A′}/π_{∗}L)
Φ_{L} := coker(X_{J} → Hom(L, Z))

Theorem 3 (-). Let L ⊂ S be of finite index; then

1. Φ_{A} = Φ_{π^{∗}X_{A}}, i.e., X_{J} → Hom(π^{∗}X_{A}, Z) → Φ_{A} → 0 is exact;

2. \frac{#Φ_{A}}{m_{A}} = \frac{#Φ_{L}}{m_{L}};

3. image(Φ_{J} → Φ_{A}) ≅ Φ_{S}. Corollary 4.

# coker(Φ_{J} → Φ_{A}) = \frac{#Φ_{A}}{#Φ_{S}} = \frac{m_{A}}{m_{S}},

in particular, m_{S} | m_{A}. Conclusion:

[figure: Φ_{J} → → Φ_{A}; easy; hard; π^{∗}X_{A} is saturated; m_{A} = m_{S}]

Observation:

m_{S} = #(X_{A′}/π_{∗}(X_{J}[I_{f}]))
m_{A} = #(H_{1}(A, Z)^{+}/π_{∗}(H_{1}(J, Z)^{+}[I_{f}])) · (2 − power)

so if H_{1}(J, Z)^{+} ⊗ Z_{p} ≅ X_{J} ⊗ Z_{p} as T ⊗ Z_{p}-modules, then p ∤ # coker(Φ_{J} → Φ_{A}).

§6. Level lowering

The component group Φ_{A} of A “frequently forces” various A[𝔪] to arise from a finite flat group scheme. Ribet’s theorem implies such 𝔪 are reducible, and then Mazur’s theorem implies such 𝔪 Eisenstein.

Suppose ℓ | #Φ_{A}. Does ℓ | numer(\frac{p−1}{12})? X := X_{A}, Y := X_{A′}

\xymatrix{ & 0 \ar[r] \ar[d] & Y \ar[d] \ar[r]^{ℓ} & Y \ar[d] \ar[r] & Y/ℓY \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Hom(X/ℓX, μ_{ℓ}) \ar[d] \ar[r] & T \ar[d] \ar[r]^{·^{ℓ}} & T \ar[d] \ar[r] & 0 \ar[d] \\ 0 \ar[r] & A[ℓ] \ar[r] & A \ar[r]^{ℓ} & A \ar[r] & 0}

Monodromy: 0 → Y → Hom(X, Z) → Φ_{A} → 0, gives

0 → Φ_{A}[ℓ] → Y ⊗ Z/ℓZ → Hom(X, Z/ℓZ) → Φ_{A} ⊗ Z/ℓZ → 0.

Snake gives: 0 → Hom(X/ℓX, μ_{ℓ}) → A[ℓ] ⟶^{f} Y/ℓY → 0. So:

\xymatrix{ &  & f^{−1}(Φ_{A}[ℓ]) \ar@{^{(}->}[d] & Φ_{A}[ℓ] \ar@{^{(}->}[d] \\ 0 \ar[r] & Hom(X/ℓX, μ_{ℓ}) \ar[r] & A[ℓ] \ar[r] & Y/ℓY \ar[r] & 0.}

The maximal finite part of A[ℓ] is f^{−1}(Φ_{A}[ℓ]), which is bigger than Hom(X/ℓX, μ_{ℓ}). Is it possible to show that the f^{−1}(Φ_{A}[ℓ]) in A[ℓ] forces some A[𝔪] to be finite? Example: If dim A = 1 then Y/ℓY has dimension one and the sequence becomes

0 → Hom(X/ℓX, μ_{ℓ}) → A[ℓ] → Φ_{A}[ℓ] → 0

so A[ℓ] is finite, so ℓ ∤ numer (\frac{p−1}{12}).