Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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)?

share|improve this question
    
Are you assuming trivial central character? The given answers are only for $Psl(2, \mathbb{R})$, and which is fine, depending upon your definition of $\Gamma_0(N)$. –  plusepsilon.de Apr 15 '12 at 9:09
add comment

2 Answers 2

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".

share|improve this answer
1  
Stein's book is a good reference, but beware of a typo in Theorem 5.9 which means that some formulae for Eisenstein series with character are not quite correct (some $\psi$'s should be $\bar{\psi}$'s). See trac.sagemath.org/sage_trac/ticket/4062 (a bug in the Sage computer algebra system which was traced to an incorrect formula copied from Stein's book). –  David Loeffler Apr 15 '12 at 20:33
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.