Analytic continuation on Siegel varieties
type: Item
instance of: scholarly article
title: Analytic continuation on Siegel varieties
zbMATH Open document ID: 1315.11033
DOI: 10.1215/00127094-2011-004
author: Vincent Pilloni
published in: Duke Mathematical Journal
publication date: 2011-05-03T00:00:00Z
review text: The article studies the rigid-analytic continuation of overconvergent modular forms on Siegel modular threefolds, and gives a criterion for such forms to be classical. This is an important contribution to our understanding of congruences of Siegel modular forms; the analogous results were crucial in the study of forms on modular curves, and have interesting arithmetic consequences. Fix a prime \(p\) and an integer \(N\geq 3\) not divisible by \(p\). Let \(\mathcal{O}_K\) be the ring of integers of a finite extension \(K/\mathbb{Q}_p\). Let \(X\) denote the moduli space of principally polarized abelian surfaces over \(\mathcal{O}_K\) with principal level-\(N\) structure and Iwahori level structure at \(p\) (the latter means a choice of totally isotropic subgroups \(H_1\subset H_2\) of the \(p\)-torsion group). Given a weight \(\kappa=(k_1\geq k_2)\in\mathbb{Z}^2\), the Hodge bundle is a locally free sheaf \(\omega^\kappa\) on \(X\), constructed from the sheaf of relative differentials. The space of classical Siegel modular forms under consideration is the space of global sections \(M(\kappa, X)=H^0(X,\omega^\kappa)\). If \(\bar{X}\) denotes a toroidal compactification of \(X\), then \(\omega^\kappa\) has a canonical extension to \(\bar{X}\) and the Koecher principle says that \(H^0(\bar{X},\omega^\kappa)=H^0(X,\omega^\kappa)\). There are at least three natural ways of defining overconvergent forms in this setting, and it is not known whether they agree (they coincide when \(g=1\)). Let \(X_{\text{rig}}\) be the rigid-analytic fiber of the formal completion of \(X\) along its special fiber; similarly get \(\bar{X}_{\text{rig}}\) from \(\bar{X}\). The ordinary locus \(Y\subset X_{\text{rig}}\) is the open formal subscheme classifying abelian surfaces with ordinary reduction and \(H_2\cong\mu_p^2\). The analogous consideration gives the ordinary locus \(\bar{Y}\subset\bar{X}_{\text{rig}}\). The various spaces of overconvergent forms are built from \(Y\) or \(\bar{Y}\) a…
reviewed by: Alexandru Ghitza
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 5883141
zbMATH Keywords: \(p\)-adic Siegel modular forms
zbMATH Keywords: overconvergence
zbMATH Keywords: Iwahori level structure
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00127094-2011-004
OpenAlex ID: W2076094928
cites work: Canonical subgroups and p-adic vanishing cycles for abelian varieties
cites work: The canonical subgroup for families of abelian varieties
cites work: Q4149991
cites work: Analytic continuation of overconvergent eigenforms
cites work: Classical and overconvergent modular forms
cites work: Q4277784
cites work: Les Schémas de Modules de Courbes Elliptiques
cites work: La filtration de Harder-Narasimhan des schémas en groupes finis et plats
cites work: Q4521512
cites work: Iwasawa modules attached to congruences of cusp forms
cites work: CONTROL THEOREMS OF COHERENT SHEAVES ON SHIMURA VARIETIES OF PEL TYPE
cites work: A gluing lemma and overconvergent modular forms
cites work: Q3775667
cites work: Q3215315
cites work: Q3932928
cites work: The crystals associated to Barsotti-Tate groups: with applications to Abelian schemes
cites work: Alcoves and \(p\)-rank of abelian varieties
cites work: Partial canonical subgroups
cites work: Compactification de variétés de Siegel aux places de mauvaise réduction
cites work: Minimal compactification and bad reduction
cites work: Group schemes of prime order
cites work: Siegel varieties and \(p\)-adic Siegel modular forms
description: scientific article; zbMATH DE number 5883141