Algebraic modular forms
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.
scientific article; zbMATH DE number 1389704
scientific article