Algebraic modular forms
type: Item
instance of: scholarly article
title: Algebraic modular forms
zbMATH Open document ID: 0965.11020
DOI: 10.1007/BF02780173
published in: Israel Journal of Mathematics
publication date: 2000-02-15T00:00:00Z
review text: The author develops an algebraic theory of modular forms for connected reductive groups \(G\) over \(\mathbb Q\) with the property that every arithmetic subgroup \(\Gamma\) of \(G({\mathbb Q})\) is finite. This algebraic theory enables the author to define ``modular forms mod \(p\)'' for such groups. The success of such a theory in this case results from the fact that modular forms in this context are combinatorial objects, i.e., essentially functions on finite sets of points. What makes the theory non-trivial and arithmetic is the Hecke action that these spaces of modular forms carry. The author makes conjectures about mod \(p\) Galois representations attached to mod \(p\) modular forms that are eigenforms for the Hecke action, even detailing the behaviour of such representations when restricted to the decomposition group at \(p\). \textit{J.-P. Serre} [Isr. J. Math. 95, 281-299 (1996; Zbl 0870.11030)] had earlier developed such a theory in the case when \(G\) is a definite quaternion algebra.
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 1389704
zbMATH Keywords: algebraic theory of modular forms
zbMATH Keywords: connected reductive groups
zbMATH Keywords: modular forms \(\text{mod }p\)
zbMATH Keywords: functions on finite sets of points
zbMATH Keywords: Hecke action
zbMATH Keywords: \(\text{mod }p\) Galois representations
author: Benedict H. Gross
reviewed by: Chandrashekhar Khare
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1389704