Spectacle cycles with coefficients and modular forms of half-integral weight

Details Spectacle cycles with coefficients and modular forms of half-integral weight Spectacle cycles with coefficients and modular forms of half-integral weight

2011-08-26

arxiv.org/abs/1108.5309v1

In: Arithmetic Geometry and Automorphic forms, Volume in honor of the 60th birthday of Stephen S. Kudla, 91-154. Advanced Lect

Mathematics

Number Theory

Scientific paper

In this paper we present a geometric way to extend the Shintani lift from even weight cusp forms for congruence subgroups to arbitrary modular forms, in particular Eisenstein series. This is part of our efforts to extend in the noncompact situation the results of Kudla-Millson and Funke-Millson relating Fourier coefficients of (Siegel) modular forms with intersection numbers of cycles (with coefficients) on orthogonal locally symmetric spaces. In the present paper, the cycles in question are the classical modular symbols with nontrivial coefficients. We introduce "capped" modular symbols with coefficients which we call "spectacle cycles" and show that the generating series of cohomological periods of any modular form over the spectacle cycles is a modular form of half-integral weight. In the last section of the paper we develop a new simplicial homology theory with local coefficients (that are not locally constant) that allows us to extend the above results to orbifold quotients of the upper half plane.

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