ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

作者：Miller Alison^*; Pixton Aaron

来源：International Journal of Number Theory, 2010, 6(1): 69-87.

DOI：10.1142/S1793042110002818

• Singular moduli
• half-integral weight modular forms
• Poincare series
• Maass forms

We extend results of Bringmann and Ono that relate certain generalized traces of Maass-Poincare series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences Z(lambda, N, m): M(2-2 lambda)(#)(Gamma(0)(N)) -> M(lambda+1/2)(!) (Gamma(0)(4N)) and Z(lambda, N, n)': M(2-2 lambda)(#) (Gamma(0)(N)) -> M(3/2-lambda)(!) (Gamma(0)(4N)). We show that if f is a modular form of non-positive weight 2 - 2 lambda and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms Z(lambda, N, m)(f) and Z(lambda, N, n)'(f).

• 出版日期2010-2