Winfried Kohnen

The arithmetic of the values of modular functions and the divisors of modular forms

We investigate the arithmetic and combinatorial significance of the values of the polynomials j n ( x ) defined by the q -expansion \[\sum_{n=0}^{\infty}j_n(x)q^n:=\frac{E_4(z)^2E_6(z)}{\Delta(z)}\cdot\frac{1}{j(z)-x}.\] They allow us to provide an explicit description of the action of the Ramanujan Theta-operator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p -adic class number formulas.

On the Signs of Fourier Coefficients of Cusp Forms

Let Γ be a discrete subgroup of SL(2, ℝ) with a fundamental region of finite hyperbolic volume. (Then, Γ is a finitely generated Fuchsian group of the first kind.) Let

LINEAR RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES

f\left( z \right) = \sum\limits_{n + \kappa > 0} {a{{\left( n \right)}^{e2\pi i\left( {n + \kappa } \right)z/\lambda }}} ,z \in H.

A SHORT NOTE ON FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS

be a nontrivial cusp form, with multiplier system, with respect to Γ. Responding to a question of Geoffrey Mason, the authors present simple proofs of the following two results, under natural restrictions upon Γ.

The first sign change of Fourier coefficients of cusp forms

We show that for arbitrary even genus 2 n with

ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS

n\equiv {0,1}

Some numerical computations concerning Spinor Zeta functions in genus 2 at the central point

(mod 4) the subspace of Siegel cusp forms of weight

Nonvanishing of symmetric square -functions of cusp forms inside the critical strip

k+n