Looking for concise books on automorphic L-functions, Eisenstein series on adelic homogeneous spaces

Question

Indeed I am now trying to read a series of papers written by Einsiedler, Lindenstrauss, Michel and Venkatesh that study distribution of periodic torus orbits on homogeneous spaces. They make heavy use of automorphic L-functions, and Eisenstein series on adelic homogeneous spaces, but I almost know nothing about that. And it seems that many books on automorphic forms are quite involved and have too many pages for me. So I want to ask whether there are some concise books that contain things about automorphic L-functions, Eisenstein series on adelic homogeneous spaces. Thanks for your attention.

Answer 1

This is not exactly what you've asked for, but I'll address this article directly, because it is not related to automorphic L-functions "directly" but more to homogeneous dynamics. You can actually read it with minimal knowledge about those stuff, if you believe some "black boxes", or some technical gadgets.

First of all, I think that you might want to read the new article about Duke's theorem, although it uses different measure rigidity theorem (because it is rank 1 situation), it sheds some light over the whole situation and the more difficult papers (ELMV1-2). It is written in purpose as to be an exposition of that work (I was the guinea pig for this article).

Second, if you're reading the article carefully, you should notice that they use in the Eisenstein series for only two purposes:

1. Get positive entropy, which is a fundamental requirement in order to use Lindenstrauss' measure classification argument for diagonal actions. This has something to do with the improvement of the volume estimates for small balls to be three dimensional, from that you can get positive entropy via the "Bowen-balls" entropy estimates (it is also related to Brin-Katok), you can read about it in the Duke theorem article, this is roughly the same argument as there (given this volume estimate).

2. Control escape of mass at the cusp. This is done via the Siegel formula. (Basically, the positive entropy is achieved via similar Eisenstein series techniques, but it is different in their purpose from the first one, because these two obstacles can be treated separately in other related situations, escape of mass doesn't usually follow from measure classification).

I think the most complete ref. for the Eisenstein part is probably - "Spectral decomposition and Eisenstein series" by Moeglin and Waldspurger, although this book is very technical, and I don't think it is suitable for beginners.

The relation to the period integrals comes from the so-called "Waldspurger's Formula". You should try to read some of Venkatesh's works, probably - "Sparse equidistribution problems, period bounds, and subconvexity" and some of "The subconvexity problem for GL(2)", they discuss Venkatesh's geometrical approach to subconvexity, by which (using Waldspurger's formula) the equidistribution follows.

As for the use of the Waldspurger's formula, you can get the idea from looking at Harcos' article in the Clay proceedings - "Equidistribution on the modular surface and L-functions". He describes there a bit simpler situation than the one present in the paper, in particular - not adelic, so it is easier to read and understand this one before.

In my opinion, you should think about the subconvexity here as a-priori estimate (which follows from some number theoretical estimates), which will allow you to get estimates for some part of your spectrum, and from there you can get full equidistribution by ergodic methods. This is in somehow resembles the fact that bounds towards Ramanujan gives you Hecke equidistribution.