Proceedings of the National Academy of Sciencesâ€¦

2010

A "mock modular form" is the holomorphic part of a harmonic Maass form f. The nonholomorphic part of f is a period integral of its "shadow," a cusp form g. A direct method for relating theâ€¦ (More)

Let M be the space of even integer weight meromorphic modular forms on SL2(Z) with integer coefficients, leading coefficient equal to one, and whose zeros and poles are supported at cusps andâ€¦ (More)

Abstract. Recently, a beautiful paper of Andrews and Sellers has established linear congruences for the Fishburn numbers modulo an infinite set of primes. Since then, a number of authors have provenâ€¦ (More)

The theory of p-adic modular forms was developed by J.-P. Serre [8] and N. Katz [5]. This theory is by now considered classical. Investigation of p-adic congruences for modular forms of half-integerâ€¦ (More)

The phenomenon of U(p)-congruences was recently studied by Ahlgren and Ono [1] and by Elkies, Ono and Yang [2]. We provide a necessary and sufficient condition which improves their general results.

Let j be the modular invariant. For the primes p â‰¤ 23 the q-expansion coefficients of U(jâˆ’ 744) are multiplicative as it was a Hecke eigenformmodulo a power of p which increases withm. This wasâ€¦ (More)

We consider the action of Hecke operators on weakly holomorphic modular forms and a Hecke-equivariant duality between the spaces of holomorphic and weakly holomorphic cusp forms. As an application,â€¦ (More)

The notion of mixed mock modular forms was recently introduced by Don Zagier. We show that certain solutions of Kaneko Zagier differential equation constitute simple yet non-trivial examples of thisâ€¦ (More)

Let p > 3 be a prime. We consider j-zeros of Eisenstein series Ek of weights k = pâˆ’1+Mp(pâˆ’1) with M,a â‰¥ 0 as elements of Qp. If M = 0, the j-zeros of Epâˆ’1 belong to Qp(Î¶p2âˆ’1) by Henselâ€™s Lemma. Callâ€¦ (More)

n>0 (1âˆ’ q) be Dedekindâ€™s eta-function. For a prime p, denote by U Atkinâ€™s Up-operator. We say that a function Ï† with a Fourier expansion Ï† = âˆ‘ u(n)q is congruent to zero modulo a power of a prime p,â€¦ (More)