Traces of CM values of modular functions

Bruinier, J.; Funke, J.

Abstract

Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also study the theta lift for the weight 0 Eisenstein series for SL2() and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagier's Eisenstein series of weight 3/2. This recovers a result of Kudla, Rapoport and Yang.

Citation

Bruinier, J., & Funke, J. (2006). Traces of CM values of modular functions. Journal für die reine und angewandte Mathematik, 594, 1-33. https://doi.org/10.1515/crelle.2006.034

• Journal Article Type: Article
• Publication Date: May 1, 2006
• Deposit Date: Feb 16, 2009
• Publicly Available Date: Mar 18, 2014
• Journal: Journal für die reine und angewandte Mathematik
• Print ISSN: 0075-4102
• Electronic ISSN: 1435-5345
• Publisher: De Gruyter
• Peer Reviewed: Peer Reviewed
• Volume: 594
• Pages: 1-33
• DOI: https://doi.org/10.1515/crelle.2006.034
• Public URL: https://durham-repository.worktribe.com/output/1566395