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Vanishing Fourier coefficients of Hecke eigenforms

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Abstract

We prove that, for fixed level \((N,p) = 1\) and \(p > 2\), there are only finitely many Hecke eigenforms f of level \(\Gamma _1(N)\) and even weight with \(a_p(f) = 0\) which are not CM.

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Acknowledgements

This paper owes its genesis to two conversations between the authors during their respective number theory seminars, one in Wisconsin in February of 2018, and the second in Chicago in January of 2020. The first author would like to thank George Boxer from whom he learned the surprising fact that the characters \(z \rightarrow z^n\) are Zariski dense in \(\mathbf{Z }_p \llbracket {\mathcal {O}}^{\times }_K \rrbracket \), and both authors would like to thank Patrick Allen and Toby Gee for comments. The second author would like to thank Professors Nigel Boston, Jordan Ellenberg, Lue Pan and Richard Taylor for their comments on the earlier versions of this work and also he would like to thank the hospitality of the department of mathematics of University of Chicago.

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Correspondence to Frank Calegari.

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Communicated by Wei Zhang.

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Frank Calegari was supported in part by NSF Grants DMS-1701703 and DMS-2001097. Naser Talebizadeh Sardari was supported in part by NSF Grant DMS-2015305.

Appendix A. Finiteness of unramified deformation rings

Appendix A. Finiteness of unramified deformation rings

Let \({\overline{\rho }}: G_{\mathbf{Q }} \rightarrow {{\,\mathrm{GL}\,}}_2(k)\) be an absolutely irreducible odd Galois representation of the form \(\text {Ind}^{G_{\mathbf{Q }}}_{G_L} \chi \), where \(L/\mathbf{Q }\) is the quadratic subfield \(L \subset \mathbf{Q }(\zeta _p)\). Suppose that, up to unramified twist:

$$\begin{aligned} \left. {\overline{\rho }}\right| _{G_{\mathbf{Q }_p}} = \text {Ind}^{G_{\mathbf{Q }_p}}_{G_{K}} {\overline{\varepsilon }}^{n-1}_2, \quad n-1 \equiv \frac{p+1}{2} \bmod p+1. \end{aligned}$$
(8)

The main theorem of this section is the following.

Theorem 2.2

Assume that \(p \equiv 1 \mod 4\), so \(L/\mathbf{Q }\) is real. Let \((N,p) = 1\), and let \(R^{\text {split}}\) denote the universal deformation ring of \({\overline{\rho }}\) consisting of representations which are unramified outside N and totally split at p defined in Definition 2.4. Then \(R^{\text {split}}\) is finite over W(k).

Before proving this, we begin with a preliminary lemma:

Lemma 2.8

There exists a finite extension \(F/\mathbf{Q }\) with the following properties

  1. (1)

    F is totally real.

  2. (2)

    If v|N, then \({\overline{\rho }}|_{F_v}\) is trivial.

  3. (3)

    \({\overline{\rho }}|_{G_F}\) is absolutely irreducible.

  4. (4)

    If v|p, then \(F_v \simeq K\), the unique unramified quadratic extension of \(\mathbf{Q }_p\).

Proof

The existence of F follows immediately from [5, Proposition 3.2] (see also [14]). For example, one may can choose \(G={{\,\mathrm{GL}\,}}_2(k)\), and then make the following choices:

  1. (1)

    \(\phi _v\) for v|N is the map \({\overline{\rho }}|_{G_{v}} \rightarrow G\),

  2. (2)

    \(\phi _v\) for \(v = p\) is any injective map \(\text {Gal}(K/\mathbf{Q }_p) \rightarrow G\),

  3. (3)

    \(c_{\infty }\) is trivial.

The irreducibility of \({\overline{\rho }}|_{G_F}\) is then guaranteed by choosing \(F/\mathbf{Q }\) to be linearly disjoint from the fixed field M of \(\ker ({\overline{\rho }})\) by [5, Lemma 3.2] (2). \(\square \)

Proof of Theorem 2.2

We begin with some reductions. If F/E is any finite extension so that \({\overline{\rho }}|_{G_F}\) remains absolutely irreducible, and \(R_F\) and \(R_E\) denote the universal deformation rings of \({\overline{\rho }}|_{G_F}\) and \({\overline{\rho }}|_{G_E}\) respectively, then the map \(R_F \rightarrow R_E\) is always finite. As a consequence, to prove the finiteness of \(R^{\text {split}}\), it suffices to replace \(\mathbf{Q }\) by any totally real field in which \({\overline{\rho }}\) remains absolutely irreducible. We replace \(\mathbf{Q }\) by the field \(F/\mathbf{Q }\) constructed in Lemma 2.8, so that \({\overline{\rho }}|_{G_v}\) is trivial for each v|N. By Lemma 2.8, the field \(F_v\) for v|p is precisely the unramified extension \(K/\mathbf{Q }_p\) for all v|p. In particular,

$$\begin{aligned} {\overline{\rho }}|_{F_v} \simeq {\overline{\varepsilon }}^{n-1}_2 \oplus {\overline{\varepsilon }}^{p(n-1)}_2 \end{aligned}$$

is reducible and p-distinguished. Make an arbitrary choice of one of these characters for each v|p, which gives a distinguished choice \(U_k\) of the decomposition \(V_k = U_k \oplus U'_k\) as \(G_{F_v}\) modules for each v|p.

We now recall the deformation ring \(R_{{\mathcal {D}}}\) defined in [23, §2] with respect to \({\mathcal {D}}= (W(k),\Sigma ,\emptyset )\) where \(\Sigma \) is the set of primes dividing N. Note that \({\overline{\rho }}|_{G_F}\) is absolutely irreducible and satisfies all the conditions of [23] by construction. The ring \(R_{{\mathcal {D}}}\) is global deformation ring of \({\overline{\rho }}\) unramified outside Np subject to the following condition: for all v|p, there exists a short exact sequence

$$\begin{aligned} 0 \rightarrow U_A \rightarrow V_A \rightarrow U'_A \rightarrow 0 \end{aligned}$$

of \(G_{F_v}\)-modules where \(U_A\) and \(U'_A\) are free over A of rank one and \(U_A/{\mathfrak {m}}= U_k\).

After extending F if necessary we may assume that \([F:\mathbf{Q }]\) is even and thus the hypothesis (\(\text {H}_{\text {even}}\)) of [23, §3] holds. By [2, Lemma 5.1.2], there exists a lift \(\rho _0\) of \({\overline{\rho }}\) giving rise to a point on \(R_{{\mathcal {D}}}\) relative to our choices both over F and any totally real extension of F in which \({\overline{\rho }}|_{G_F}\) remains irreducible. This implies that condition (\(\text {H}_{\text {def}}\)) of [23, §3] holds for any such F. In this situation we have a corresponding Hecke ring \(\mathbf{T }_{{\mathcal {D}}}\) as defined in [23, p.196], and a surjection \(R_{{\mathcal {D}}} \rightarrow \mathbf{T }_{{\mathcal {D}}}\) of \(\Lambda \)-modules where \(\Lambda \) is an Iwasawa algebra [23, p.191] (denoted \(\Lambda _{{\mathcal {O}}}\)). There is a surjection \(R_{{\mathcal {D}}} \rightarrow R^{\text {split}}\). Any deformation coming from \(R^{\text {split}}\) locally has the form \(U_A \oplus U'_A\) where the action of \(G_K\) on \(U_A\) and \(U'_A\) is via the Teichmüller lifts of \({\overline{\varepsilon }}^{n-1}_2\) and \({\overline{\varepsilon }}^{p(n-1)}_2\) respectively. In particular, the map

$$\begin{aligned} \Lambda \rightarrow R_{{\mathcal {D}}} \rightarrow R^{\text {split}}\end{aligned}$$

factors through a quotient of the form W(k), as can be seen from the formulas in [23] on the last line of p.191 and the first line of p.192 respectively. A more intrinsic way to see this is that the ring \(\Lambda \) represents weight space and all split representations lie in the same fixed unramified weight. Hence to prove that \(R^{\text {split}}\) is finite over W(k) it suffices to show that \(R_{{\mathcal {D}}}\) is finite over \(\Lambda \).

By taking the compositum of F with a suitably large totally subfield of a cyclotomic extension (exactly as in the first two lines at the top of page 204 of [23]) we may further ensure that the pair \((F,\rho _0)\) is good in the sense of [23, §4]. It follows from [23, Prop 4.1] and [23, Prop 8.2] that all primes of \(R_{{\mathcal {D}}}\) are pro-modular. If \(\mathbf{T }_{{\mathcal {D}}}\) is the Hecke ring defined in [23, p.196], it follows that there is an isomorphism \(R_{{\mathcal {D}}}/{\mathfrak {p}}\simeq \mathbf{T }_{{\mathcal {D}}}/{\mathfrak {p}}\) for every prime \({\mathfrak {p}}\) of \(R_{{\mathcal {D}}}\), which implies immediately that \((R_{{\mathcal {D}}})^{\text {red}} = (\mathbf{T }_{{\mathcal {D}}})^{\text {red}}\) and hence \((R_{{\mathcal {D}}})^{\text {red}}\) is finite over \(\Lambda \) (since \(\mathbf{T }_{{\mathcal {D}}}\) is finite over \(\Lambda \) by [23, Lemma 3.3]). Since \(R_{{\mathcal {D}}}\) is Noetherian, it follows that \(R_{{\mathcal {D}}}\) is also finite over \(\Lambda \). But \(R_{{\mathcal {D}}}\) surjects onto \(R^{\text {split}}/p\), and the kernel contains the image of the maximal ideal of \(\Lambda \). It follows that \(R^{\text {split}}/p\) is finite over k and hence \(R^{\text {split}}\) is also finite over W(k), as claimed. \(\square \)

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Calegari, F., Talebizadeh Sardari, N. Vanishing Fourier coefficients of Hecke eigenforms. Math. Ann. 381, 1197–1215 (2021). https://doi.org/10.1007/s00208-021-02178-7

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