Exact formulas for traces of singular moduli of higher level modular functions

Zagier proved that the traces of singular values of the classical j-invariant are the Fourier coefficients of a weight 3/2 modular form and Duke provided a new proof of the result by establishing an exact formula for the traces using Niebur's work on a certain class of non-holomorphic modular forms. In this short note, by utilizing Niebur's work again, we generalize Duke's result to exact formulas for traces of singular moduli of higher level modular functions.Comment: 8 page

Modification of fluorometric assay for thiamin in chicken muscle

Call number: LD2668 .T4 1986 K55Master of ScienceHuman Nutritio

Hecke equivariance of generalized Borcherds products of type O(2,1)O(2,1)

Recently, a weak converse theorem for Borcherds' lifting operator of type $O(2,1)$ for \G_0(N) is proved and the logarithmic derivative of a modular form for \G_0(N) is explicitly described in terms of the values of Niebur-Poincar\'e series at its divisors in the complex upper half-plane. In this paper, we prove that the generalized Borcherds' lifting operator of type $O(2,1)$ is Hecke equivariant under the extension of Guerzhoy's multiplicative Hecke operator on the integral weight meromorphic modular forms and the Hecke operator on half-integral weight vector-valued harmonic weak Maass forms. Additionally, we show that the logarithmic differential operator is also Hecke equivariant under the multiplicative Hecke operator and the Hecke operator on integral weight meromorphic modular forms. As applications of Hecke equivariance of the two operators, we obtain relations for twisted traces of singular moduli modulo prime powers and congruences for twisted class numbers modulo primes, including those associated to genus $1$ modular curves.Comment: 18 page