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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)
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)