From: Boudewijn Moonen Subject: Re: Complex analysis problem Date: Thu, 18 May 2000 15:04:07 +0200 Newsgroups: sci.math,alt.algebra.help Summary: [missing] Piotr Zielinski wrote: > > Hello > > I have recently get a problem from complex analysis as well as (I > suppose simple number theory) and I don't know how to solve it. > > Let define G and F as follows: > > G(z) = (1-z)*(1-z^2)*(1-z^3)*(1-z^4)*... > F(z) = z * G(z)^2 * G(z^11)^2, where z is a complex number > > Using F we define f: > > f(z) = F(e^(2*pi*i*z)), where z is complex number such as Im z > 0 > > Now I can present the problem: > Assume that a,b,c,d are integers such that ad-bc=1 and c is divisible > by 11. How to show that: > > f((az+b)/(cz+d)) = (cz+d)^2 f(z) (for all z : Im z > 0) > > I'm interested also in partial results. > > Piotr The last equation means f is a modular form of weight 2 for the congruence subgroup of level 11. For this, look up any reasonable book on modular forms, e.g. G. Shimura, Introduction to the theory of automorphic forms (around 1972) or N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Math. No. 97, Springer-Verlag, New York, 1984. Second Edition 1993. Regards, -- Boudewijn Moonen Institut fuer Photogrammetrie der Universitaet Bonn Nussallee 15 D-53115 Bonn GERMANY e-mail: Boudewijn.Moonen@ipb.uni-bonn.de Tel.: GERMANY +49-228-732910 Fax.: GERMANY +49-228-732712 ============================================================================== From: Don Redmond Subject: Re: Complex analysis problem Date: Fri, 19 May 2000 21:35:04 -0500 Newsgroups: sci.math,alt.algebra.help In article <3923EA47.75D903C2@ipb.uni-bonn.de>, Boudewijn Moonen wrote: [quote of previous article deleted --djr] Another place to look might be M. Knopp, Modular Functions. In there he discusses Ramanujan's partition congruences, where some of these variations on the eta function occur. Don