[1] Despite contrary assertions in the literature, rewriting Eisenstein series, as opposed to more general automorphic forms, on adele groups does not use Strong Approximation. Strong Approximation does make precise the relation between general automorphic forms on adele groups and automorphic forms on SL2 and even on SLn, but rewriting these Eisenstein series does not need this comparison. Ind...

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.

The modern theory of automorphic forms is a response to many different impulses and influences, above all the work of Hecke, but also class-field theory and the study of quadratic forms, the theory of representations of reductive groups, and of complexmultiplication, but so far many of the most powerful techniques are the issue, direct or indirect, of the introduction by Maass and then Selberg ...

In this paper, we study modular forms on two simply connected groups of type D4 over Q. One group, Gs, is a globally split group of type D4, viewed as the group of isotopies of the split rational octonions. The other, Gc, is the isotopy group of the rational (non-split) octonions. We study automorphic forms on Gs in analogy to the work of Gross, Gan, and Savin on G2; namely we study automorphic...

We present a level raising result for families of p-adic automorphic forms for a definite quaternion algebra D over Q. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and Taylor. One of the ingredients in the proof of Diamond and Taylor’s theorem (which also played a role in earlier work of Taylor) is the definition of a suitable pairing on the space ...

In this paper a theory of Hecke operators for higher order modular forms is established. The definition of higher order forms is extended beyond the realm of parabolic invariants. A canonical inner product is introduced. The role of representation theoretic methods is clarified and, motivated by higher order forms, new convolution products of L-functions are introduced.