Higher coherent cohomology and \(p\)-adic modular forms of singular weights
type: Item
instance of: scholarly article
title: Higher coherent cohomology and \(p\)-adic modular forms of singular weights
zbMATH Open document ID: 1481.11051
DOI: 10.1215/00127094-2019-0075
published in: Duke Mathematical Journal
publication date: 2020-08-03T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1588903316
review text: ``In this article, we investigate the theory of \(p\)-adic families of automorphic forms for the group \(\text{GSp}_4/\mathbb Q\) whose component at infinity has singular Harish-Chandra parameter and is a nondegenerate limit of discrete series. The automorphic forms we consider can be realized in the coherent cohomology of an appropriate automorphic vector bundle over a Siegel threefold (see [\textit{M. Harris}, Proc. Lond. Math. Soc. (3) 59, No. 1, 1--22 (1989; Zbl 0711.14011)]). The Siegel threefolds are finite unions of arithmetic quotients of the \(3\)-dimensional Siegel upper half-space. They have a modular interpretation as moduli spaces of abelian surfaces with polarization and level structure and they have canonical models over number fields. Using this coherent realization, one can prove that the Hecke parameters of these automorphic forms are defined over number fields, and one can construct, using congruences, compatible systems of \(4\)-dimensional Galois representations (see, e.g., [the author and \textit{B. Stroh}, Ann. Math. Qué. 40, No. 1, 167--202 (2016; Zbl 1357.14033); \textit{R. Taylor}, Duke Math. J. 63, No. 2, 281--332 (1991; Zbl 0810.11033)]).'' The author constructs \(p\)-adic families of (cuspidal) cohomology classes for the sheaves \(\{\Omega^{(k,2)}\}_{k\geq 0}\) (attached to the representation \(\text{Sym}^k\text{St} \otimes \det^2 \text{St}\) of the group \(\text{GL}_2\)) in degree \(0\) and \(1\) (Theorems 1.1 and 1.2). ``It is conjectured that for every simple abelian surface \(A\) over \(\mathbb Q\) there should exist a cuspidal automorphic form \(\pi\) on \(\text{GSp}_4\mathbb Q\) such that the spin \(L\)-function of \(\pi\) and the \(L\)-function of \(H^1(A)\) coincide. When \(\text{End}(A) \not=\mathbb Z\), this is known (see, e.g., [\textit{M. Kisin}, Invent. Math. 178, No. 3, 587--634 (2009; Zbl 1304.11043); \textit{H. Yoshida}, Invent. Math. 60, 193--248 (1980; Zbl 0453.10022); \textit{A. Brumer} and \textit{K. Kramer}, Trans. A…
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11G18
zbMATH DE Number: 7226649
zbMATH Keywords: coherent cohomology
zbMATH Keywords: \(p\)-adic modular forms
zbMATH Keywords: automorphic vector bundles
zbMATH Keywords: Siegel threefolds
zbMATH Keywords: Shimura varieties
zbMATH Keywords: Galois representations
author: Vincent Pilloni
MaRDI profile type: Publication
OpenAlex ID: W3021335334
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