Overconvergent Families of Siegel–Hilbert Modular Forms

Mok, Chung Pang; Tan, Fucheng

doi:10.4153/cjm-2014-017-9

Cited by 3 publications

(1 citation statement)

References 14 publications

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“…Until recently, further coherent-cohomological constructions of M λ for general weights λ, e.g. [KL05,MT12], have relied (as did Coleman and Mazur) on tricks involving Eisenstein series or lifts of Hasse invariants, and have only given rise to one-dimensional families. However, canonical constructions of overconvergent modular forms of arbitrary p-adic weight on Shimura varieties have recently been discovered [AIS13, AIP13, Bra14, CHJ14] which do give rise to universal coherent-cohomological eigenvarieties.…”

Section: Following Ideas Of Stevens Andmentioning

confidence: 99%

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

Hansen

Newton

2015

Journal Für Die Reine Und Angewandte Mathematik

146

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Using the overconvergent cohomology modules introduced by Ash and Stevens, we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces, building on work of Ash-Stevens, Urban, and others. We also formulate a precise modularity conjecture linking trianguline Galois representations with overconvergent cohomology classes. In the course of giving evidence for this conjecture, we establish several new instances of p-adic Langlands functoriality. Our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of Chenevier's interpolation theorem.

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Section: Following Ideas Of Stevens Andmentioning

confidence: 99%

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

Hansen

Newton

2015

Journal Für Die Reine Und Angewandte Mathematik

146

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Abelian surfaces over totally real fields are potentially modular

et al. 2021

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We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces $A$ A over ${\mathbf {Q}}$ Q with $\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}$ End C A = Z . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

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Eigenvarieties for Cuspforms Over PEL Type Shimura Varieties With Dense Ordinary Locus

Brasca¹

2016

Can. j. math.

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Abstract. Let p > be a prime and let X be a compacti ed PEL Shimura variety of type (A) or (C) such that p is an unrami ed prime for the PEL datum and such that the ordinary locus is dense in the reduction of X. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens we de ne the notion of families of overconvergent locally analytic p-adic modular forms of Iwahoric level for X. We show that the system of eigenvalues of any nite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected dimension that parameterize nite slope systems of eigenvalues appearing in the space of families of cuspidal forms.

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Overconvergent Families of Siegel–Hilbert Modular Forms

Abstract: Abstract.We construct one-parameter families of overconvergent Siegel-Hilbert modular forms. This result has applications to construction of Galois representations for automorphic forms of non-cohomological weights.

Cited by 3 publications

References 14 publications

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

Abelian surfaces over totally real fields are potentially modular

Eigenvarieties for Cuspforms Over PEL Type Shimura Varieties With Dense Ordinary Locus

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