Infinite product exponents for modular forms

CLASSICAL AND p-ADIC MODULAR FORMS ARISING FROM THE BORCHERDS EXPONENTS OF OTHER MODULAR FORMS

Abstract. Let f(z) = q ∏∞ n=1(1−q) be a modular form on SL2(Z). Formal logarithmic differentiation of f yields a q-series g(z) := h −∞n=1 ∑ d|n c(d)dq n whose coefficients are uniquely determined by the exponents of the original form. We provide a formula, due to Bruinier, Kohnen, and Ono for g(z) in terms of the values of the classical j-function at the zeros and poles of f(z). Further, we giv...

Product Formulas for Modular Forms on O(2, N)

Zagier Duality for the Exponents of Borcherds Products for Hilbert Modular Forms

A certain sequence of weight 1/2 modular forms arises in the theory of Borcherds products for modular forms for SL2(Z). Zagier proved a family of identities between the coefficients of these weight 1/2 forms and a similar sequence of weight 3/2 modular forms, which interpolate traces of singular moduli. We obtain the analogous results for modular forms arising from Borcherds products for Hilber...

Generalized Exponents and Forms

We consider generalized exponents of a finite reflection group acting on a real or complex vector space V . These integers are the degrees in which an irreducible representation of the group occurs in the coinvariant algebra. A basis for each isotypic component arises in a natural way from a basis of invariant generalized forms. We investigate twisted reflection representations (V tensor a line...

Motives for modular forms

In [DeFM], Deligne constructs l-adic parabolic cohomology groups attached to holomorphic cusp forms of weight ≥ 2 on congruence subgroups of SL2(Z). These groups occur in the l-adic cohomology of certain smooth projective varieties over Q—the Kuga-Sato varieties— which are suitably compactified families of products of elliptic curves. In view of Grothendieck’s conjectural theory of motives it i...