Abstract. We apply Zagier’s result for the traces of singular moduli to construct Borcherds products in higher level cases. 1.

this paper we suppose that k = m=2. (For later applications we will only need this case.) Then the condition 2k b + b + 0 (mod 4) is equivalent to requiring that b + is even. The condition k > 2 implies m 5. Under this assumption the formula of Theorem 4.6 can be considerably simplied

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In the present paper we find explicit formulas for the degrees of Heegner divisors on arithmetic quotients of the orthogonal group O(2, p) and for the integrals of certain automorphic Green’s functions associated with Heegner divisors. The latter quantities are important in the study of the arithmetic degrees of Heegner divisors in the context of Arakelov geometry. In particular, we obtain a different proof and a generalization of results of Kudla relating these quantities to the Fourier coefficients of certain non-holomorphic Eisenstein series of weight 1 + p/2 for the metaplectic group Mp 2(Z). 1

The title of this paper was chosen more in homage to Drinfeld’s famous note [D] than as a precise description of its contents. We will therefore describe the main results in the first three sections. 1. Embeddings into R All of the non-associative rings that we will study arise from Coxeter’s order R in the Q-algebra of Cayley’s octonions. Recall that the latter algebra has basis 〈1,e1,e2,...,e7 〉 over Q and multiplication rules: e 2 i = −1, ei(ei+1ei+3) = (eiei+1)ei+3 = −1, where the indices are taken modulo 7. The order R is generated over Z by the ei’s, and the additional elements 1 2 (1 + e1 + e2 + e4),

We consider the osp(1|2)-invariant bilinear operations on weighted densities on the supercircle S 1|1 called the supertransvectants. These operations are analogues of the famous Gordan transvectants (or Rankin-Cohen brackets). We prove that these operations coincide with the iterated Poisson and ghost Poisson brackets on R 2|1 and apply this result to construct star-products involving the supertransvectants. 1

Abstract. The Ramanujan τ-function satisfies well-known congruences modulo the so-called exceptional prime numbers 2,3,5, 7,23,691. In this paper we prove new congruences related to the irregular primes 131 and 593, involving generalized class numbers. As an application we obtain distribution results. We obtain a new proof of the famous 691 congruence and congruences of the related Rankin L-funtion.

this paper to her memory.

Abstract. Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let f(z) = P∞ n=0 a(n)qn be a holomorphic half integer weight modular form with integer coefficients. If ℓ is prime, then we shall be interested in congruences of the form a(ℓN) ≡ 0 mod ℓ where N is any quadratic residue (resp. non-residue) modulo ℓ. For every prime ℓ> 3 we exhibit a natural holomorphic weight ℓ + 1 modular form whose coefficients satisfy the 2 congruence a(ℓN) ≡ 0 mod ℓ for every N satisfying ` ´ −N = 1. This is proved by using ℓ the fact that the Fourier coefficients of these forms are essentially the special values of real Dirichlet L−series evaluated at s = 1−ℓ which are expressed as generalized Bernoulli 2 numbers whose numerators we show are multiples of ℓ. ¿From the works of Carlitz and Leopoldt, one can deduce that the Fourier coefficients of these forms are almost always a multiple of the denominator of a suitable Bernoulli number. Using these examples as a template, we establish sufficient conditions for which the Fourier coefficients of a half integer weight modular form are almost always divisible by a given positive integer M. We also present two examples of half-integer weight forms, whose coefficients are determined by the special values at the center of the critical strip for the quadratic twists of two modular L−functions, possess such congruence properties. These congruences are related to the non-triviality of the ℓ−primary parts of Shafarevich-Tate groups of certain infinite families of quadratic twists of modular elliptic curves with conductors 11 and 14. 1. Congruences for Fourier coefficients First we shall fix the following notation. If D ≡ 0, 1 mod 4 is the fundamental discriminant of the quadratic field Q ( √ D), then let χD denote the Kronecker character Key words and phrases. congruences, modular forms, special values of L−functions.

Abstract. It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth, fiberwise-polynomial functions on the cotangent bundle a oneparameter family of graded star products. For a particular value of the parameter (corresponding to half-densities) the star product is symmetric, and specializes in the projectively flat case to the one constructed previously by C. Duval, P. Lecomte and V. Ovsienko. A limiting form of this family of star products yields a commutative deformation of the symmetric tensor algebra of the manifold. A basic ingredient of the proofs is the construction of projectively invariant multilinear differential operators on bundles of weighted symmetric k-vectors. The construction works except for a discrete set of excluded weights and generalizes the Rankin-Cohen brackets of modular forms. 1.