Twisting overconvergent p-adic eigenforms

(Still very preliminary)

Robert F. Coleman  William A. Stein

October 11, 1999

1 Twists can be approximated

Theorem 1.1. Let F be a form of tame level N and let ω^{r} be a power of the Teichmuller character ω at p, with (N, p) = 1. Then the twist F ⊗ ω^{r} can be approximated arbitrarily closely by eigenforms of level N.

Proof. First we lose nothing by starting with an old form F on X_{1}(Np), since if we approximate its twist by a form whose slope is large, but small with respect to its weight, it is congruent to the form it came from on X_{1}(N). Suppose the weight of F is k and its slope is α. Let F (q) = ∑ a_{n}q^{n} be the q-expansion of F. Let A be a positive integer. Then if m is an integer sufficiently small p-adically there exists an overconvergent eigenform G of weight k−m and slope α such that G(q) ≡ F (q) (mod p^{A}). If k − m ≤ 0, then F_{1} = θ^{m−k+1}G is an overconvergent eigenform form of weight k_{1} := m − k + 2 and slope α_{1} := α + k_{1} − 1. If F_{1}(q) = ∑ b_{n}q^{n}, then

b_{n} ≡ n^{k_{1}−1}a_{n} (mod p^{A}).

It follows that if B is large there exists an eigenform G_{1} of weight k_{1} − p^{B}r and slope α_{1} such that G_{1}(q) ≡ F_{1}(q) (mod p^{A}). Now if k_{1} − p^{B}r < 0, then F_{2} := θ^{p^{B}r−k_{1}+1}G_{1} is an overconvergent eigenform of weight k_{2} := p^{B}r − k_{1} + 2 and slope α_{2} := α_{1} + k_{2} − 1. If F_{2}(q) = ∑ c_{n}q^{n}, then

c_{n} ≡ n^{k_{2}−1}b_{n} ≡ n^{p^{B}r}a_{n} (mod p^{A}).

It follows that if v is sufficently small p-adically, then there exists a classical eigenform F_{3} of weight k_{2} +v = k +v +p^{B}r −m and slope α_{2} such that F_{3}(q) ≡ F_{2}(q) (mod p^{A}). It follows from the above that if B > A then F_{3} is congruent to the twist of F by the r-th power Teichmuller character modulo p^{A}.

2 Some numerical experiments

We have experimentally investigated the question of approximation of twists of finite slope forms.

One way to make “approximate” precise and falsifiable, is to ask the following: let n ≥ 1 be an integer; does there exist a newform g ∈ S_{k+φ(p^{n})}(Γ_{0}(Np); Z_{p}) such that g ≡ f^{ε} (mod p^{n})? (By this congruence, we mean that ord_{p}(g − f^{ε}) = min_{i≥1}{ord_{p}(a_{i}(g) − a_{i}(f^{ε}))} ≥ n.)

2.1 A false conjecture

In this section we state a very general, and very false, conjecture. Let p be a prime and let N be a positive integer not divisible by p.

Conjecture 2.1 (False conjecture). Let f ∈ S_{k}(Γ_{0}(N), Z_{p}) be a finite slope newform, and ε a character mod p such that ε^{2} = 1. Then for all n ≥ 1 there exists an eigenform g ∈ S_{k+φ(p^{n})}(Γ_{0}(Np); Z_{p}) such that g ≡ f^{ε} (mod p^{n}).

2.2 Numerical experiments

In the remainder of this paper we report on numerical experiments designed to gather evidence about Conjecture 2.1. These computations were carried out in Magma (cf. [Can]).

We perform the following experiments:

— Section 2.3. Let f = Δ be the unique cusp form of weight 12. For each p = 2, 3, 5, 7, 11 consider the twist f^{ε} by either the trivial or quadratic mod p character. We compute ord_{p}(g − f^{ε}) for each rational newform g ∈ S_{k+φ(p^{n})}(Γ_{0}(Np); Z_{p}), for the first few values of n. In fact, ord_{p}(g−f^{ε}) was computed using only 17 terms of the q-expansion, which is probably enough, but we have not proved this.

— Section 2.3.1. Let f be the weight-2 newform corresponding to the elliptic curve X_{0}(11). For p = 2, 3, 5 we perform the same computation as for Δ.

Since, in this experiment, we only twist f by the trivial or quadratic character, we adopt the notation f^{(1)} and f^{(−1)} for these twists, respectively. Thus f^{(1)} is the p-deprivation of f, and f^{(−1)} is the quadratic twist.

Remark 2.2. The newforms at level Np all have slope (k − 2)/2.

2.3 Twisting Δ

Consider the weight-12 cuspform Δ; the q-expansion of Δ is

Δ = q ∏(1 − q^{n})^{24}
= q − 24q^{2} + 252q^{3} − 1472q^{4} + 4830q^{5} − 6048q^{6} − 16744q^{7} + · · · .

Let p be a prime and let ε : (Z/pZ)^{∗} → {±1} be a character; then ε^{2} = 1, so Δ^{ε} ∈ S_{12}(Γ_{0}(9)). As outlined in Section 2.2, we compute each of the rational newforms g ∈ S_{12+φ(p^{n})}(Γ_{0}(p), Z_{p}), and then ord_{p}(Δ^{ε} − g).

The computations are given in the following tables. For example, the 4th line of the first table below says that when n = 4 there are two Z_{2}-rational newforms g_{1} and g_{2}; and that Δ^{(1)} ≡ g_{1} (mod 2^{5}) and Δ^{(1)} ≡ g_{2} (mod 2^{5}). The non-Z_{p}-rational newforms in S_{12+φ(p^{n})}(Γ_{0}(p), \overline{Z}_{p}) are not listed.

Eigenforms on Γ_{0}(2) congruent mod 2 to the 2-deprivation of Δ.

n  ord_{2}(Δ^{(1)} − g_{i})
1  none
2  3, 3
3  4
4  5, 5
5  6, 6
6  7, 7, 7, 7

Eigenforms on Γ_{0}(3) congruent mod 3 to twists of Δ

n  ord_{3}(Δ^{(1)} − g_{i})  ord_{3}(Δ^{(−1)} − g_{i})
1  1, 1, 1  1, 1, 1
2  1, 2, 1  2, 1, 2
3  3, 1, 3, 3, 1  1, 3, 1, 1, 3
4  4, 4, 1, 1, 1, 3, 4, 4, 1, 1, 1 1, 1, 4, 4, 3, 1, 1, 1, 4, 4, 3

Eigenforms on Γ_{0}(5) congruent mod 5 to twists of Δ

n  ord_{5}(Δ^{(1)} − g_{i}), ord_{5}(Δ^{(−1)} − g_{i})
1  [1, 0, 1, 0, 0], [ 0, 1, 0, 1, 1]
2  [2, 1, 1, 0, 0, 2, 1, 1, 0, 0, 0], [0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 2]
3  [2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 3, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 2, 2, 2 ]

Eigenforms on Γ_{0}(7) congruent mod 7 to twists of Δ

n  ord_{7}(Δ^{(1)} − g_{i}), ord_{7}(Δ^{(−1)} − g_{i})
1  [1, 1, 0, 0, 1, 1, 0, 0, 0], [1, 1, 0, 0, 1, 1, 0, 0, 0]
2  [1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0],

[1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]

Eigenforms on Γ_{0}(11) congruent mod 11 to twists of Δ

n  ord_{11}(Δ^{(1)} − g_{i}), ord_{11}(Δ^{(−1)} − g_{i})
1  [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],

[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]

2.3.1 Twisting the weight-2 newform on X_{0}(11)

Let f be the form corresponding to the elliptic curve of conductor 11. Fix a prime p, and let ε be the trivial character modulo p. We list the congruences between f^{ε} and the Z_{2}-rational newforms in S_{2+φ(p^{n})}(Γ_{0}(p · 11)).

For example, the 2nd line of the table below says that when n = 2 there are three Z_{2}-rational newforms g_{1}, g_{2}, g_{3}; and that f^{(1)} ≡ g_{1} (mod 3), f^{(1)} ≢ g_{2} (mod 3), and f^{(1)} ≡ g_{3} (mod 3). The 3rd line says there are five Z_{2}-rational newforms in S_{2+φ(2^{3})}(Γ_{0}(22)), and the first, g, satisfies f^{(1)} ≡ g (mod 3^{2}).

Eigenforms on Γ_{0}(22) congruent mod 2 to the 2-deprivation of f

\begin{matrix}n & ord_{2}(f^{(1)} − g_{i}) \\ 1 & none \\ 2 & 1, 0, 1 \\ 3 & 2, 1, 0, 1, 2 \\ 4 & 1, 1, 3, 3, 0 \\ 5 & 4, 1, 0, 4, 0, 1, 0 \end{matrix}

Eigenforms on Γ_{0}(33) congruent mod 3 to twists of f

n  ord_{3}(f^{(1)} − g_{i})
1  0, 1, 1, 0  1, 0, 0, 0
2  1, 0, 0, 1, 0, 0  0, 2, 1, 0, 1, 0
3  0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1

Eigenforms on Γ_{0}(55) congruent mod 5 to twists of f

\begin{matrix}n & ord_{5}(f^{(−1)} − g_{i}) & ord_{5}(f^{(−1)} − g_{i}) \\ 1 & 1, 0 & 0, 0 \\ 2 & 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 & 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0 \end{matrix}

2.3.2 Twisting X_{0}(14)

Eigenforms on Γ_{0}(3 · 14) congruent mod 3 to twists of h ∈ S_{2}(Γ_{0}(14))

\begin{matrix}n & ord_{3}(h^{(1)} − g_{i}) & ord_{3}(h^{(−1)} − g_{i}) \\ 1 & 0, 1 & 0, 0 \\ 2 & 0, 1, 0, 0, 0, 0 & 0, 0, 2, 0, 2, 0 \end{matrix}

References

[Can] J. Cannon, The magma computational algebra system,

http://www.maths.usyd.edu.au:8000/u/magma/.

[CM98] R. Coleman and B. Mazur, The eigencurve, Galois representations in arithmetic algebraic geometry (Durham, 1996), Cambridge Univ. Press, Cambridge, 1998, pp. 1–113.

[Shi94] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, Princeton, NJ, 1994, Reprint of the 1971 original, Kan Memorial Lectures, 1.