Gelbart, Stephen S. Automorphic Forms on Adele Groups. (AM), Volume Series:Annals of Mathematics Studies PRINCETON UNIVERSITY PRESS. Automorphic Representations of Adele Groups. We have defined the space A(G, Γ) of auto- morphic forms with respect to an arithmetic group Γ of G (a reductive. Download Citation on ResearchGate | Automorphic forms on Adele groups / by Stephen S. Gelbart | “Expanded from notes mimeographed at Cornell in May of.

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Well the basic link to representation theory is that modular forms and automorphic forms can be viewed as functions automrphic representation spaces of reductive groups. Actually, that book does seem particularly good as an answer to this question.

Throughout the work the author emphasizes new examples and problems that remain open within the general theory. Braids, Links, and Mapping Class Groups. Here are two fairly old books that explain and exploit representation theory behind the theory of theta functions and automorphic forms neither assuming nor using algebraic geometry and commutative algebra in a serious way: This is explained in belbart in Automorphic forms on adele groups by Gelbart.

Thus most books on automorphic forms e. In particular, Jacobi sums come up in both number theory and representation theory and quadratic forms then relate to theta functions. Thanks, though I also imagine there are other, more direct examples.

Probably the most notable example is monstrous moonshine. If you were to pick up Bump’s book Automorphic forms and representations he’ll go over some background.

Finite Dimensional Vector Spaces. It seems like some kind of group theory. Pretty much grouos only way to take an automorphic representation and prove that it has an eglbart Galois representation is to construct a geometric object whose cohomology has both an action of the Hecke algebra and the Galois group and decompose it into pieces and pick out the one you want. You might also be interested in chapter 7 of An Introduction to the Langlands Program: Somehow one manages to understand a bit without knowing all.

## Automorphic Forms on Adele Groups. (AM-83), Volume 83

This automorpnic called the Local Langlands Correspondence. David Corwin 6, 6 66 I understand that Hecke characters relate to adeles, but you seem to be implying that Hecke characters lifting to characters on adeles in the first example of a classical modular form becoming a function on adeles. A detailed proof of the celebrated trace formula of Selberg is included, with a discussion of the possible range of applicability of this formula.

Markov Processes from K. I suggest you take a look at Borel’s article Introduction to automorphic forms in gelbsrt Boulder conference available freely at ams. I’m asking this in avele because I imagine a number of students with similar background as I have would have learned about modular forms and thus might be interested to understand how they relate to representation theory, despite not having an extensive background in more advanced results in algebraic geometry and commutative algebra needed for advanced study in the field.

An introduction to the Langlands program by Bernstein and others is also good. These correspondences should be nice in that things that happen on one side should correspond to things happening on the other. Sign up or log in Sign up using Google. AMVolume Michael Harris. As for your comment: Visit our Beautiful Books page and find lovely books for kids, photography lovers and more.

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### How is representation theory used in modular/automorphic forms? – MathOverflow

It’ll give you a bit of an idea of the type of thing that goes into simply the definition of an automorphic form. AM-7Volume 7 Paul R. From this view point, the algebraic geometry side of the picture enters simply as the source for proving instances of the Langlands conjectures. Mathematics Stack Exchange works best with JavaScript enabled.

I haven’t taken a course in representation theory beyond representations of finite groups and would be interested in knowing where I can get the required representation theory background before reading the book. The most comprehensive reference is the Corvallis proceedings available freely at gropus. I also know a little bit about the basics of algebraic number theory and algebraic geometry, if that helps. Sign up using Facebook.

I have a basic grounding in the complex analytic theory of modular forms their dimension formulas, how they classify isomorphism classes of elliptic curves, some basic examples of level N modular forms and their relation to torsion points on elliptic curves, series expansions, theta functions, Hecke operators. I’ve browsed through the book.

AMVolume 82 Joan S.

The Calculi of Lambda Conversion. The Best Books of To get into the Langlands program there’s the book an introduction to the Langlands program google books you could look at.

Here are some ideas which might bear fruit: I do have basic familiarity with adeles, though I would imagine that other students interested in this question might not have such familiarity, which is why I did not add this. The point of listing atuomorphic is to show the kind of intuition I might be looking for. The Trace Formula for GL 2 In fact, my main familiarity with them is in showing the equivalence gwlbart ideal-theoretic and idelic class field theory, which is precisely your “toy version,” so I’m quite curious to eventually understand what you talked about.

This seems quite appropriate. The right regular representation on an locally compact abelian groups is in direct connection with its Fourier transform. gekbart

Lectures on Curves on an Algebraic Surface. Home Contact Us Help Free delivery worldwide. Then there’s the connection with number theory.