Any reference on Eisenstein Series for \Gamma_o(N) in GL(2) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:14:15Z http://mathoverflow.net/feeds/question/94037 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94037/any-reference-on-eisenstein-series-for-gamma-on-in-gl2 Any reference on Eisenstein Series for \Gamma_o(N) in GL(2) 7-adic 2012-04-14T15:40:17Z 2012-04-14T20:07:38Z <p>What's the best reference on Eisenstein Series for $\Gamma_o(N)$ in GL(2,R)?</p> <p>For fixed $\Gamma_o(N)$, should there be several Eisenstein series(corresponding to each cusp)?</p> http://mathoverflow.net/questions/94037/any-reference-on-eisenstein-series-for-gamma-on-in-gl2/94043#94043 Answer by GH for Any reference on Eisenstein Series for \Gamma_o(N) in GL(2) GH 2012-04-14T16:59:58Z 2012-04-14T16:59:58Z <p>For detailed information I recommend "Chapter 7. Eisenstein series" in Miyake: Modular Forms (Springer Verlag, 2006).</p> http://mathoverflow.net/questions/94037/any-reference-on-eisenstein-series-for-gamma-on-in-gl2/94058#94058 Answer by Joël for Any reference on Eisenstein Series for \Gamma_o(N) in GL(2) Joël 2012-04-14T20:07:38Z 2012-04-14T20:07:38Z <p>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.</p> <p>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 <a href="http://modular.math.washington.edu/books/modform/modform/index.html" rel="nofollow">http://modular.math.washington.edu/books/modform/modform/index.html</a> . More precisely, see <a href="http://modular.math.washington.edu/books/modform/modform/eisenstein.html#explicit-basis-for-the-eisenstein-subspace" rel="nofollow">here</a>.</p> <p>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.</p> <p>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". </p>