Shintani lifts and fractional derivatives for harmonic weak Maass forms

In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the L-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly holomorphic Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space of harmonic weak Maass forms which yield a “fractional derivative” from the space of half-integral weight harmonic weak Maass forms to half-integral weight weakly holomorphic modular forms. This fractional derivative complements the usual ξ-operator introduced by Bruinier and Funke.     

A remark on congruences for coefficients of Faber polynomials

Faber polynomials appear when weight zero Hecke operators act on the modular j-invariant. They are polynomials in j with rational integral coefficents. Using the theory of p-adic modular forms we establish some congruences and divisibilities for these coefficients. 2000 Mathematics Subject Classification Primary—11F33 Supported by NSF grant DMS-0501225.     

On Ramanujan congruences for modular forms of integral and half-integral weights

In 1916 Ramanujan observed a remarkable congruence: . The modern point of view is to interpret the Ramanujan congruence as a congruence between the Fourier coefficients of the unique normalized cusp form of weight and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number . In this paper we give a simple proof of the Ramanujan congruence and its generalizations to forms of higher integral and half-integral weights.

     

Some congruences for traces of singular moduli

We address a question posed by Ono [Ken Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Reg. Conf. Ser. Math., vol. 102, Amer. Math. Soc., Providence, RI, 2004, Problem 7.30], prove a general result for powers of an arbitrary prime, and provide an explanation for the appearance of higher congruence moduli for certain small primes. One of our results overlaps but does not coincide with a recent result of Jenkins [Paul Jenkins, p-Adic properties for traces of singular moduli, Int. J. Number Theory 1 (1) (2005) 103-107]. This result essentially coincides with a recent result of Edixhoven [Bas Edixhoven, On the p-adic geometry of traces of singular moduli, preprint, 2005, math.NT/0502213 v1], and we hope that the comparison of the methods, which are entirely different, may reveal a connection between the p-adic geometry and the arithmetic of half-integral weight Hecke operators.     

Farkas’ identities with quartic characters

The Ramanujan Journal - Farkas in (On an Arithmetical Function II. Complex Analysis and Dynamical Systems II. Contemporary Mathematics, American Mathematical Society, Providence, 2005) introduced...     

On the Hecke Equivariance of the Borcherds Isomorphism

The Borcherds isomorphism is proved to be Hecke equivariantif one considers multiplicative Hecke operators acting on theintegral weight meromorphic modular forms. This answers a partof a question of Borcherds (see ‘Automorphic forms onO_{s+2, 2}(R) and infinite products’, Invent. Math. 120 (1995)161–213, 17.10), using his suggestion to define the multiplicativeHecke operators. 2000 Mathematics Subject Classification 11F37.     

p-Adic liftings of the supersingular j-invariants and j-zeros of certain Eisenstein series

Let p>3 be a prime. We consider j-zeros of Eisenstein series Ek of weights k=p−1+Mpa(p²−1) with M,a?0 as elements of . If M=0, the j-zeros of Ep−1 belong to Qp(ζp²−1) by Hensel's lemma. Call these j-zeros p-adic liftings of supersingular j-invariants. We show that for every such lifting u there is a j-zero r of Ek such that ordp(r−u)>a. Applications of this result are considered. The proof is based on the techniques of formal groups.     

Irreducibility of some Faber polynomials

Apply weight 0 Hecke operators to the modular function j and express the result as a polynomial in j. These polynomials were considered long ago in analysis, and recently attracted the attention of number theorists primarily for their connection with Borcherds’ infinite products. In particular, Ken Ono conjectured that all of them are irreducible. We prove a partial result towards this conjecture by presenting infinite families of these polynomials which are proved to be irreducible. Supported by NSF grant DMS-0501225.     

Hecke operators for weakly holomorphic modular forms and supersingular congruences

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, we obtain congruences modulo supersingular primes, which connect Hecke eigenvalues and certain singular moduli.