Arithmeticity in the theory of automorphic forms, by Goro Shimura, Mathematical

Introduction to Shimura Varieties

This is an introduction to the theory of Shimura varieties, or, in other words, to the arithmetic theory of automorphic functions and holomorphic automorphic forms.

Automorphic forms and cohomology theories on Shimura curves of small discriminant

We apply Lurie’s theorem to produce spectra associated to 1dimensional formal group laws on the Shimura curves of discriminants 6, 10, and 14. We compute rings of automorphic forms on these curves and the homotopy of the associated spectra. At p = 3, we find that the curve of discriminant 10 recovers much the same as the topological modular forms spectrum, and the curve of discriminant 14 gives...

Research Statement Modular Forms and Cycles in Locally Symmetric Spaces and Shimura Varieties: Interplay between Automorphic Forms, Representation Theory, and Arithmetic and Differential Geometry

The Shimura curve of discriminant 15 and topological automorphic forms

We find defining equations for the Shimura curve of discriminant 15 over Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of “topological automorphic forms” associated to this curve, as well as one associated to a quotient by an Atkin-Lehner involution.

Modular Forms and - adic Representations ∗

the circumstance that L-functions can be introduced not only in the context of automorphic forms, with which he has had some experience, but also in the context of diophantine geometry. That this circumstance can be the source of deep problems was, I believe, first perceived by E. Artin. He was, to be sure, concerned with forms on GL(1) and with varieties of dimension 0. This remains the only c...