Hecke stability and weight $$1$$ 1 modular forms

Abstract

The Galois representations associated to weight 1 newforms overF p are remarkable in that they are unramified at p, but the computation of weight 1 modular forms has proven to be difficult. One complication in this setting is that a weight 1 cusp form overF p need not arise from reducing a weight 1 cusp form overQ.In this article we propose a unified Hecke stability method for computing spaces of weight 1 modular forms of a given level in all characteristics simultaneously. Our main theorems outline conditions under which a finite-dimensional Hecke module of ratios of modular forms must consist of genuine modular forms.We conclude with some applications of the Hecke stability method motivated by the refined inverse Galois problem.

Pseudorepresentations of weight one are unramified

“…For general m, one knows that the map: H 0 (X 1 (N ), ω) → H 0 (X 1 (N ), ω/̟ m ) need not be surjective. This was first observed by Mestre for N = 1429 and p = 2, (see [Edi06, Appendix A]), and many examples for larger p have been subsequently computed by Buzzard and Schaeffer [Buz14,Sch15]. In particular, if T denotes the subring of…”

Computation of weight 1 modular forms with exotic representations

“…While a trace formula for all automorphic forms of weight 1 can be generated, it is not possible to extract the holomorphic contribution (see [15,Chapter 10,Section 4]). As such, computation of the a p has been performed a number of indirect ways (see [11,Section 13.6.1] for a review of three methods), with the most efficient being the Hecke stability method presented in [23]. This is performed as follows:…”

“…Practically, 'computing a space' means computing a matrix of basis coefficients up to the Sturm bound. It is also shown in [23] that this method works (under additional constraints) for computing weight 1 forms over characteristic p. Computation over such fields is considerably faster than computation over Q, and given that newforms over Q project onto F p as newforms, one might hope that only the characteristic p computation is needed. Unfortunately, there exist so-called 'ethereal forms' (see [14, Appendix A]) over characteristic p which are not the projection of forms over Q.…”

“…Proof. In [23] it is proven that, for any Eisenstein series E ∈ E 1 (N, χ) and p ∤ N , the maximal T p -stable subspace of S 2 (N, ½)/E is a subspace of M 1 (N, χ) containing S 1 (N, χ). Thus:…”

“…See [11,Section 13.6.1] for a review of three such methods. The most efficient in general is the Hecke stability method, presented in [23].…”