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In this paper we study the distribution of the coefficients a(n) of half integral weight modular forms modulo odd integers M. As a consequence we obtain improvements of indivisibility results for the central critical values of quadratic twists of L-functions associated with integral weight newforms established in [O-S]. Moreover, we find a simple criterion… (More)

- JAN H. BRUINIER, KEN ONO
- 2008

Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as " generating functions " for central values and derivatives of quadratic twists of weight 2 modular… (More)

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 holomor-phic) modular functions on… (More)

1. Introduction. Let k be an integer and N be a positive integer divisible by 4. If is a prime, denote by v a continuation of the usual-adic valuation on Q to a fixed algebraic closure. Let f be a modular form of weight k + 1/2 with respect to 0 (N) and Nebentypus character χ which has integral algebraic Fourier coefficients a(n), and put v (f) = inf n v… (More)

We study the Faltings height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura… (More)

We prove that the coefficients of certain weight −1/2 harmonic Maass forms are " traces " of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight −2 harmonic weak Maass forms to spaces of weight −1/2 vector-valued harmonic weak Maass forms on Mp 2 (Z), a result which is of independent interest. We then… (More)

We investigate the arithmetic and combinatorial significance of the values of the polynomials jn(x) defined by the q-expansion ∞ ∑ n=0 jn(x)q := E4(z)E6(z) ∆(z) · 1 j(z)− x They allow us to provide an explicit description of the action of the Ramanujan Thetaoperator on modular forms. There are a substantial number of consequences for this result. We obtain… (More)

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ 2−k (resp. D k−1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are… (More)

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a… (More)