Brief introduction to cyclotomic theory over Q using adeles. Discussion of the definitions of modular forms and automorphic forms. Introducing the adelic automorphic forms via strong approximation theorem. Discussion of the connected components of Shimura varieties (modular curves). Smooth/admissible representations of locally finite groups. Definition and admissibility of (cuspidal) automorphi...

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...

In 1952, Gelfand and Fomin noticed that classical modular forms were related to representations of SL2(R). As a result of this realization, Gelfand later defined GLr automorphic forms via representation theory. A metaplectic form is just an automorphic form defined on a cover of GLr, called a metaplectic group. In this talk, we will carefully construct the metaplectic covers of GL2(F) where F i...

Automorphic forms play an important rôle in physics, especially in the realm of string and M-theory dualities. Notably, the hidden symmetries of 11-dimensional supergravity compactifications, discovered by Cremmer and Julia, motivate the study of automorphic forms for exceptional arithmetic groups En(Z) (including their n ≤ 5 classical A andD instances). These Notes are a pedestrian introductio...

That interesting new L functions with Euler products arise in the classical theory of modular forms is in some sense an accident, and even a bit deceptive. For algebraic number fields other than Q the relationship between classical forms and L functions is more complicated. It ought to be no surprise to anyone familiar with John Tate’s thesis that the correct groups with which to do automorphic...

1 Orbital integrals and the Harish-Chandra transform. This section is devoted to a rapid review of some of the basic analysis that is necessary in representation theory and the basic theory of automorphic forms. Even though the material below looks complicated it is just the tip of the iceberg. 1.1 Left invariant measures. Let X be a locally compact topological space with a countable basis for ...

In this paper, we study the period map of a certain one-parameter family of quartic K3 surfaces with an S5-action. We construct automorphic forms on the period domain as the pull-backs of theta constants of genus 2 by a modular embedding. Using these automorphic forms, we give an explicit presentation of the inverse period map.

This integral trick assumes greater significance when the function f is known to have strong decay properties both at 0 and at ∞, since then the Mellin transform is entire in s. One way to ensure such rapid decay is via eigenfunction properties in the context of automorphic forms. [2] • The archetype Mellin transform: zeta from theta • Abstracting to holomorphic modular forms • Variation: wavef...