Traces of CM values of modular functions

By Jan H. Bruinier, Jens Funke. 2004

112 citations recorded by the source corpus.

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(ℤ) 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7b3ffab0134fa7754c9a254090d93ca5aebbe592

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Borcherds forms and generalizations of singular moduli (2006, background): “it would be interesting to ﬁnd the analog of Corollary 1.2 when Z(U)K meets the divisor of Ψ(F). There exists lots of recent work on singular moduli, particularly traces of singular moduli (e.g. [1], [4] and [19]). By considering the case of signature (1,2), Theorem 1.3 of [9] can be recovered from ”
Unearthing the visions of a master: harmonic Maass forms and number theory (2008, unknown): “…with generalizations in a variety of directions in works by the author, Bringmann, Bruinier, D. Choi, W. Duke, A. Folsom, J. Funke, O. Imamo¯ glu, P. Jenkins, D. Jeon, S.-Y. Kang, C. Kim, R. Masri, A. Miller, A. Pixton, J. Rouse, and A. Toth [54] [58] [64] [65] [77] [78] [87] [88] [89] [95] [154].”
Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms (2011, background): “We begin by recalling some important facts about these theta functions in the setting of the present paper (see [26], [9]).”
Congruences for the coefficients of weakly holomorphic modular forms (2006, background): “J. H. Bruinier and J. Funke, ‘Traces of CM values of modular functions’, J. reine angew.”
Regularized theta liftings and periods of modular functions (2011, methodology): “Note that in [5] this quantity is denoted by trm,h(f).”
Modular functions and the uniform distribution of CM points (2006, methodology): “correspondences has been applied directly by Bruinier and Funke [ 2 ] to prove generalizations of such identities.”