40 research outputs found
Nonvanishing and Central Critical Values of Twisted -functions of Cusp Forms on Average
Let be a holomorphic cusp form of integral weight for
with nebentypus character . Generalising work of Kohnen
and Raghuram we construct a kernel function for the -function
of twisted by a primitive Dirichlet character and use it to show
that the average over an orthogonal basis
of does not vanish on certain line segments inside the critical
strip if the weight or the level is big enough. As another application
of the kernel function we prove an averaged version of Waldspurger's theorem
relating the central critical value of the -th twist ( a fundamental
discriminant) of the -function of a cusp form of even weight to the
square of the -th Fourier coefficient of a form of half-integral weight
associated to under the Shimura correspondence.Comment: 13 page
Eisenstein series for the Weil representation
We compute the Fourier expansion of vector valued Eisenstein series for the
Weil representation associated to an even lattice. To this end, we define
certain twists by Dirichlet characters of the usual Eisenstein series
associated to isotropic elements in the discriminant form of the underlying
lattice. These twisted functions still form a generating system for the space
of Eisenstein series but have better multiplicative properties than the
individual Eisenstein series. We adapt a method of Bruinier and Kuss to obtain
algebraic formulas for the Fourier coefficients of the twisted Eisenstein
series in terms of special values of Dirichlet -functions and representation
numbers modulo prime powers of the underlying lattice. In particular, we obtain
that the Fourier coefficients of the individual Eisenstein series are rational
numbers. Additionally, we show that the twisted Eisenstein series are
eigenforms of the Hecke operators on vector valued modular forms introduced by
Bruinier and Stein.Comment: 14 page