Any reference on Eisenstein Series for \Gamma_o(N) in GL(2)

2012-04-14T15:40:17Z

What's the best reference on Eisenstein Series for $\Gamma_o(N)$ in GL(2,R)?

For fixed $\Gamma_o(N)$, should there be several Eisenstein series(corresponding to each cusp)?

Answer by GH for Any reference on Eisenstein Series for \Gamma_o(N) in GL(2)

2012-04-14T16:59:58Z

For detailed information I recommend "Chapter 7. Eisenstein series" in Miyake: Modular Forms (Springer Verlag, 2006).

Answer by Joël for Any reference on Eisenstein Series for \Gamma_o(N) in GL(2)

2012-04-14T20:07:38Z

Miyake, already recommended by GH, is a very complete reference, which is perhaps the only place to contain complete proofs about the subject. However, for that reason, and also because he works with weird modular groups $\Gamma_1(a,b)$ conjugate to but not equal to the more familiar $\Gamma_1(ab)$, it is difficult to read.

So I propose another reference which contains a clear complete basis of Eisenstein series, which are eigenforms for $\Gamma_1(N)$, together with their nebentypus, and their Hecke eigenvalues : W. Stein, Modular Forms, A Computational Approach, available at http://modular.math.washington.edu/books/modform/modform/index.html . More precisely, see here.

To get Eisenstein series for $\Gamma_0(N)$ from those for $\Gamma_1(N)$, just restrict to those with trivial nebentypus, that is $\psi \chi=1$ in the notation of the reference given.

And about your question: in weight $>2$, there are as many independent Eisenstein series as cusps; in weight $2$, the dimension of the Eisenstein subspace is the number of cusp minus 1, because "$E_2$ is missing".

Any reference on Eisenstein Series for \Gamma_o(N) in GL(2) - MathOverflow

Any reference on Eisenstein Series for \Gamma_o(N) in GL(2)

Answer by GH for Any reference on Eisenstein Series for \Gamma_o(N) in GL(2)

Answer by Joël for Any reference on Eisenstein Series for \Gamma_o(N) in GL(2)