Asymptotic bounds for special values of shifted convolution Dirichlet series

Abstract

In [15], Hoffstein and Hulse defined the shifted convolution series of two cusp forms by "shifting" the usual Rankin-Selberg convolution L-series by a parameter h. We use the theory of harmonic Maass forms to study the behavior in h-aspect of certain values of these series and prove a polynomial bound as h → ∞. Our method relies on a result of Mertens and Ono [22], who showed that these values are Fourier coefficients of mixed mock modular forms.

Shifted convolution L-series values for elliptic curves

“…They are generalizations of the classical Rankin-Selberg convolutions [17], [20] which were used to bound the growth of the Fourier coefficients of cusp forms. Properties of these shifted convolution Dirichlet series have been investigated recently; it was first shown that these shifted convolution values are essentially coefficients of mixed mock modular forms (see [15]) and later the p-adic properties of these series (see [4]) and their asymptotic behavior (see [2]) were investigated. Initially motivated by a desire to compute these L-values and understand the explicit construction of the Weierstrass mock modular form (in terms of other known elliptic invariants), we offer a closed formula for these generating functions L f E (z).…”

“…They are generalizations of the classical Rankin-Selberg convolutions [17], [20] which were used to bound the growth of the Fourier coefficients of cusp forms. Properties of these shifted convolution Dirichlet series have been investigated recently; it was first shown that these shifted convolution values are essentially coefficients of mixed mock modular forms (see [15]) and later the p-adic properties of these series (see [4]) and their asymptotic behavior (see [2]) were investigated.…”