What is Eisenstein series?

Question

There are several related questions here, the latter being especially interesting. We know the classical Eisenstein series.

1. What are the Eisenstein series on a group G and why they are interesting?
2. What's the geometric analogue of Eisenstein series? I think I saw the definition, but what's its importance?

Answer 1

1. The standard basic statement one would make about Eisenstein series is that they "make up the continuous part of the spectrum of L^2(G)". In other words, you can break up the regular representation of G into two parts, one will be a discrete direct sum of representations, the other part will be a "direct integral". For example, if you look at the unit circle S^1, the theory of Fourier series decompose the regular representation into a discrete sum of characters: every periodic L^2 function on the unit interval is a sum of exp(2πin) (n varies discretely over Z). Whereas for the reals, R, the theory of the Fourier transform decomposes the regular representation into an integral: every L^2 function on R can be written as f(x)=int f^v(x)exp(2πx)dx by Fourier inversion (the x here varies continuously, and the exp(2πix) are the "irreducible characters"). For a group like SL(2,R), there is both a discrete part and a continuous part.

Now the way you construct the Eisenstein series is as a sum of certain nice summands (1/(m+nτ)^2k). The way to think of this is that you are taking a summand that behaves pretty well and then averaging over all of its translates to get something automorphic. You can interpret the summand as a section of an induced representation from a representation of the Borel subgroup of upper triangular matrices given by a character of the diagonal subgroup. So for more general groups, you pick some parabolic subgroup, pick a character (i think) of its Levi to define a representation of the parabolic, induce from the parabolic to the full group, take a section and sum over all translates.

Why go to this trouble? Well, special values of L-functions occur as the "constant coefficient" of Eisenstein series. For example, take the character of the diagonal of SL(2,R) to be χ then, I believe, the constant coefficient of the q-expansion of the corresponding Eisenstein series is L(1-k,χ)/2 (where the k is related to what weight you are picking for the action of SL(2,R)). This general principle was proved by Langlands in the 70s, if I'm not mistaken, in his book "Euler products". If you then p-adically vary the Eisenstein series, you can construct p-adic L-functions using the varying constant term. This was first done by Serre in his Antwerp III paper. It is also the approach of Deligne-Ribet and Katz for constructing p-adic L-functions of characters of totally real fields (resp. CM fields). One can also study special values through taking Petersson inner products of Eisenstein series and other modular forms. You can use this to show algebraicity properties of the L-values.

There's also a more emerging principle that the Eisenstein series can be used to construct global Galois cohomology classes using congruences. This kinda started with Ribet's proof of the converse to Herbrand's theorem. See Harris-Li-Skinner's "The Rallis inner product formula and p-adic L-functions" for a more modern ongoing project.

There are probably other applications as well. I'm not really an expert on the subject. And sorry my statements are rather vague, I've never seriously looked into general Eisenstein series. You could check out Shimura's book "Euler products and Eisenstein series" though (and the Harris-Li-Skinner article probably has interesting references).

Answer 2

The geometric Eisenstein series is quite complicated, see the paper by Gaitsgory and Braverman http://arxiv.org/abs/math/9912097

As the authors point out, a naive guess for the geometric Eisenstein functor would be the functor from sheaves on T-Bundles to sheaves on G-Bundles (T is a maximal torus of G) given by pulling back to sheaves on B-Bundles and then pushing forward to sheaves on G-Bundles. However, they explain that the fact that the map from B-Bundles to G-Bundles is not proper introduces problems, making it so that the geometric Eisenstein series does not behave as expected. For this reason, they have to replace B-Bundles by their Drinfeld compactification and (for others reasons they explain) also tensor with the IC sheaf of the Drinfeld compactification to get the correct functor (note that the Drinfeld compactification is singular, so the IC sheaf is not just the constant sheaf).

Answer 3

I highly recommend Drinfeld's lectures on Eisenstein series, notes available at http://www.math.utexas.edu/users/benzvi/GRASP/lectures/drinfeldEisen.pdf