This is a short list of books to get you started on learning automorphic representations. Before I talk about them, I will first define automorphic representation, which will take a few paragraphs.

To start, we need an affine algebraic $F$-group scheme $G$ where $F$ is a number field or function field. We let $\A_F$ be the adeles of $F$. The idelic norm is defined as

\[|-| = \prod_v |-|_v:F^\times\backslash\A_F^\times\to \R.\] That is, the idelic norm is the product of all the norms where the product runs over all the places of $F$. We define

\[ G(\A_F)^1 = \cap_{\chi\in X^*(G)}\ker(|-|\circ\chi). \] That is $G(\A_F)^1$ is a subgroup of $G(\A_F)$ consisting of all elements $g$ such that $|\chi(g) = 1|$ for all characters $\chi\in X^*(G)$. The reason for introducing this subgroup rather than working with the full adelic group $G(\A_F)^1$ is representation-theoretic: the group $G(\A_F)^1$ is unimodular and under the unique-up-to-scale Haar measure, the quotient $G(F)\backslash G(\A_F)^1$ has finite volume, and therefore we can do a lot of representation theory compared to working with $G(\A_F)$.

So far, we are talking about pretty concrete objects. However, if you are a little shaky with adeles and places, a good place to start is the book

Ramakrishnan, Dinakar; Valenza, Robert J. Fourier analysis on number fields. Graduate Texts in Mathematics, 186. Springer-Verlag, New York, 1999. xxii+350 pp. ISBN: 0-387-98436-4

You should know the first five chapters of this book pretty well. It will take you through some basic representation theory, local fields and global fields, and adeles. Anyways, let's continue with our definition of an automorphic representation. We have the group $G(\A_F)^1$, and I mentioned that it is unimodular. We fix some Haar measure, and consider the quotient $G(F)\backslash G(\A_F)^1$, which has finite volume under the induced measure. It is natural therefore to consider the space $L^2(G(F)\backslash G(\A_F)^1)$. This space has a natural pairing defined by an integral:

\[ (f_1,f_2) = \int_{G(F)\backslash G(\A_F)^1} f_1(x)\overline{f_2(x)}{\rm d}x.\]

## Hecke algebras

In order to define automorphic representation, we need to introduce the Hecke algebra. The Hecke algebra is actually a pretty concrete object, especially for nonarchimedean fields. In fact, if you've never seen these types of constructions before, it would be a good idea to consult Fiona Murnaghan's course notes on the representation theory of locally compact groups and reductive groups.

There are two definitions of the Hecke algebra, one for number fields and one for function fields. The one for function fields is easier, because we don't have to worry about the infinite places. If $F$ is a function field, the Hecke algebra of $G$ is just the space of locally constant (a.k.a. smooth) compactly supported functions on $G(\A_F)$. We denote this space by $\Hcl$.

If $F$ is a number field, then we define $\Hcl^\infty$ as the locally constant, compactly supported functions on $G(\A_F^\infty)$, where $\A_F^\infty$ is the subgroup of the adeles that is the restricted direct product of all the $F_v$ over the nonarchimedean places $v$ of $F$. So, it's just like the function field case, because in both cases we are just considering a restricted direct product over nonarchimedean local fields.

However, we still want to define the Hecke algebra $\Hcl$ for number fields, and we only have $\H^\infty$, which is just one part of $\Hcl$. The other part is defined as follows. Let $K$ be a maximal compact subgroup of the real Lie group $G(R\otimes_\Q F)$. Then we define $\Hcl_\infty$ as the convolution algebra of distributions of $G(R\otimes_Q F)$ supported on $K$. Then the Hecke algebra when $F$ is a number field is defined as $\Hcl = \Hcl_\infty\otimes\Hcl^\infty.$ Just for understanding the definition, it might be easier to just think of function fields, but it is the number field case that is the most interesting from a number-theoretic perspective.

## The definition

There is a natural action of the Hecke algebra $\Hcl$ on the Hilbert space $L^2 = L^2(G(F)\backslash G(\A_F)^1)$ defined by

\[\begin{align*}R:\Hcl\times L^2&\longrightarrow L^2\\

(f,\phi)&\longmapsto \left(g\mapsto \int_{G(\A_F)}\phi(gh)f(h){\rm d}h\right).

\end{align*}\] Note that because $f$ is locally constant and compactly supported, this integral makes sense. This gives a representation of $\Hcl$. An *automorphic representation* is defined to be an admissible representation of $\Hcl$ isomorphic to a subquotient of the representation of $\Hcl$ on $L^2$. Here, a representation of $\Hcl$ on $V$ is admissible if the fixed point set $V^K$ is finite dimensional for every open compact $K$ and $V$ is nondegenerate. Nondegenerate means that every element of $V$ can be written as $\sum h_iv_i$ for $h_i\in H$ and $v\in V$; this is not a trivial condition since Hecke algebras for noncompact groups do not have an identity.

## Books on automorphic representations

People studying automorphic representations are really lucky to have a few good books on the topic, some of which have come out in the last ten years. An obvious addition is the two volume series:

Goldfeld, Dorian; Hundley, Joseph. Automorphic representations and $L$-functions for the general linear group. Volume I. With exercises and a preface by Xander Faber. Cambridge Studies in Advanced Mathematics, 129. Cambridge University Press, Cambridge, 2011. xx+550 pp. ISBN: 978-0-521-47423-8

Goldfeld, Dorian; Hundley, Joseph. Automorphic representations and $L$-functions for the general linear group. Volume II. With exercises and a preface by Xander Faber. Cambridge Studies in Advanced Mathematics, 130. Cambridge University Press, Cambridge, 2011. xx+188 pp. ISBN: 978-1-107-00799-4

These books are great because prerequisites are kept to a minimum and everything is done for the general linear group ${\rm GL}_n$. This special case has a lot of simplifications compared to a general reductive group. Also in this book is the connection to automorphic forms and other classical number theory topics.

Students may face some difficulties with Goldfeld and Hundley's books, partially because they are so long and contain a lot of different but important details. For readers looking for a more compact source but still very readable, the book

Bump, Daniel. Automorphic forms and representations. Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge, 1997. {\rm xiv}+574 pp. ISBN: 0-521-55098-X

should be useful. It's especially good for someone who already has some familiarity with modular forms, although it can be read by someone with little knowledge of them as well. Gelbart's book

Gelbart, Stephen S. Automorphic forms on adÃ¨le groups. Annals of Mathematics Studies, No. 83. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. {\rm x}+267 pp.

is also good, as it focuses on GL2 and uses adeles.

Also exciting and even more modern treatment is the Springer GTM An Introduction to Automorphic Representations with a view toward Trace Formulae by Jayce Getz and Heekyoung Hahn. It is currently being written and should come out this year. However, at the time of this post, the first sixteen chapters are available for download at that link. This book is quite different than the books by Goldfeld and Hundley. As the title suggests, this book is much more focused on the mathematics behind Arthur-Selberg-like trace formula, orbital integrals, and Langlands functoriality. Once completed, this book will certainly stand as the best entry into trace formula.

There is also a two volume series edited by Borel and Casselman

Borel, Armand, and William Casselman, eds. Automorphic Forms, Representations and L-Functions. Vol. 1/2. American Mathematical Soc., 1979.

which contains one of the best summaries of the mathematics surrounding automorphic representations.

## 1 Comment

Shout out to Jayce and Heekyoung! Whoot whoot! :D