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Algebraic modular forms. (English) Zbl 0965.11020

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 \(p\).
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.

MSC:

11F55 Other groups and their modular and automorphic forms (several variables)
11F80 Galois representations
20G30 Linear algebraic groups over global fields and their integers
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Citations:

Zbl 0870.11030
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Full Text: DOI

References:

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