# Reference for automorphic forms

I would like to know some reference to learn the theory of automorphic forms. Any (good) book or online lecture notes will be fine. I am particularly interested in the arithmetic point of view (e.g. galois representations associated).

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It would be easier to make more specific recommendations if you outlined your background (e.g. are you familiar with the modular form case of the theory, are you familiar with class field theory, are you familiar with Shimura varieties, or etale cohomology, or reductive groups, or ... ?). Regards, –  Matt E Jun 22 '11 at 20:36

The theory of automorphic forms and its relationship to Galois representations is not something you will learn in one sitting, so to speak.

For the very broadest outlines of the goals of the field, you might begin with Mark Kisin's article What is ... a Galois representation. For a discussion of one of the fundamental recent advances in the field (which is nevertheless rather out of date, since the field is moving very rapidly at the moment) you could read Barry Mazur's article on the Sato--Tate conjecture.

To actually learn the theory, I recommend a solid knowledge of the theory of modular forms, which is a part of the theory of automorphic forms for the particular group $GL_2$ over $\mathbb Q$. You can start with Serre's Course in Arithmetic; after this, Shimura's book mentioned in amWhy's answer is a possible subsequent text, or there is Bump's book mentioned by Amitesh. Both of these still only deal with the group $GL_2$ over $\mathbb Q$, though.

For automorphic forms on general groups, the standard way to learn the outlines of the field still seems to be to read the Corvalis volumes, available here. These are not easy reading, and I don't recommend studying them in isolation --- you will need a good advisor (or equivalent) to guide you through them and relate them to modern developments and concerns in the field.

Unfortunately there are no particular modern texts that deal with what you are asking about, although there have been many recent conferences since the proof of Sato--Tate which have online videos, e.g. at MSRI in 2006 and at the CIRM in Luminy in 2007. You could try watching some of these for more modern perspectives.

There is also the recent article The conjectural connections between automorphic representations and Galois representations by Kevin Buzzard and Toby Gee. This is written for experts, but the introduction and bibliography may give you some additional hints for reading.

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If you simply Google Automorphic Forms and Arithemetic, you'll land LOTS of hits...

It is a very broad area; perhaps you have something narrower in mind (additional limiting criteria...etc.?)

Books:

Here's a link to a text reviewed by the MAA: Introduction to the Arithmetic Theory of Automorphic Functions by Goro Shimura. At amazon, you can Look Inside

Also @amazon: Automorphic Forms and Representations (Cambridge Studies in Advanced Mathematics), by Daniel Bump.

Video Lectures

See *Automorphic Forms_Arithmetic Applications* for a video presentation that is one of the Institute for Advanced Study Video Lectures. The full title: Automorphic forms: Arithmetic applications of automorphic forms.

Lecture Notes

See Automorphic representations and Galois representations for lecture notes (pdf) on the topic from a series of lectures given by Michael Harris (Ordway Lectures). Also available for download from Paul Garrett's (Univ. Minnesota) website.

PDF paper

In pdf: H. Grobner and A. Raghuram, On some arithmetic properties of automorphic forms of $GL_m$ over a division algebra. (There is an extensive list of references in the paper which may be of interest.)

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+1 good references. –  Babak S. Feb 20 '13 at 10:33

A textbook is Automorphic Forms and Representations by Daniel Bump. Unfortunately, as I have not read this textbook, I cannot comment further on it. However, you may wish to look at the google preview for this textbook:

You're welcome. In fact, the part /books?id=... until (and without) the first & suffices. The rest is just noise encoding your search terms and the settings for your view. I left the last part to link to the preview page and not the book description page. –  t.b. Jun 21 '11 at 14:22