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Hecke stability and weight \(1\) modular forms

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Abstract

The Galois representations associated to weight 1 eigenforms over \(\bar{\mathbb {F}}_{p}\) are remarkable in that they are unramified at \(p\), but the effective computation of these modular forms presents challenges. One complication in this setting is that a weight 1 cusp form over \(\bar{\mathbb {F}}_{p}\) need not arise from reducing a weight 1 cusp form over \(\bar{\mathbb {Q}}\). 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 that are motivated by the refined inverse Galois problem.

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Acknowledgments

The author would like to thank Akshay Venkatesh, John Voight, Kevin Buzzard, Frank Calegari, Dave Roberts, Chandrashekhar Khare, and Haruzo Hida for their input and support. The author would also like to thank the anonymous referee for their helpful comments and suggestions.

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  1. Department of Mathematics, UCLA, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA, 90095, USA

    George J. Schaeffer

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  1. George J. Schaeffer
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Correspondence to George J. Schaeffer.

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The author was partially supported by the NSF under Grants DMS-0838697 and DMS-0854949.

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Schaeffer, G.J. Hecke stability and weight \(1\) modular forms. Math. Z. 281, 159–191 (2015). https://doi.org/10.1007/s00209-015-1477-9

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