In this paper, we develop an algebraic theory of modular forms, for connected, reductive groupsG overQ with the property that every arithmetic subgroup Γ ofG(Q) is finite. This theory includes our previous work  on semi-simple groupsG withG(R) compact, as well as the theory of algebraic Hecke characters for Serre tori . The theory of algebraic modular forms leads to a workable theory of modular forms (modp), which we hope can be used to parameterize odd modular Galois representations.
The theory developed here was inspired by a letter of Serre to Tate in 1987, which has appeared recently . I want to thank Serre for sending me a copy of this letter, and for many helpful discussions on the topic.
KeywordsModular Form Division Algebra Open Compact Subgroup Arithmetic Subgroup Central Involution
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