Algebraic modular forms

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 [15] on semi-simple groupsG withG(R) compact, as well as the theory of algebraic Hecke characters for Serre tori [20]. 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 [21]. I want to thank Serre for sending me a copy of this letter, and for many helpful discussions on the topic.