There is a Shimura lifting which sends cusp forms of a half-integral weight to holomorphic modular forms of an even integral weight. Niwa and Cipra studied this lifting using the theta series attached to an indefinite quadratic form; later, Borcherds and Bruinier extended this lifting to weakly holomorphic modular forms and harmonic weak Maass forms of weight 1/2, respectively. We apply Niwa's theta kernel to weak Maass forms by using a regularized integral. We show that the lifted function satisfies...
Suppose that f is an elliptic modular form with integral coefficients. Sturm obtained bounds for a nonnegative integer n such that every Fourier coefficient of f vanishes modulo a prime p if the first n Fourier coefficients of f are zero modulo p. In the present note, we study analogues of Sturm's bounds for Siegel modular forms of genus 2. As an application, we study congruences involving an analogue of Atkin's U(p)-operator for the Fourier coefficients of Siegel modular forms of genus 2.
Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality is satisfied for , where denotes the Euler(-Mascheroni) constant. We show by elementary methods that if does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that must be divisible by a fifth power . As consequence we obtain that RH holds true iff every natural number divisible by a fifth power...
We employ recent results on Jacobi forms to investigate congruences and filtrations of Siegel modular forms of degree . In particular, we determine when an analog of Atkin’s -operator applied to a Siegel modular form of degree is nonzero modulo a prime . Furthermore, we discuss explicit examples to illustrate our results.
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