Special Values of Shifted Convolution Dirichlet Series

Abstract

In a recent important paper, Hoffstein and Hulse [14] generalized the notion of Rankin-Selberg convolution L-functions by defining shifted convolution L-functions. We investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and "mixed mock modular" forms.

Shifted convolution L-series values for elliptic curves

“…Since dim(S 2 (Γ 0 (N ))) = genus(X 0 (N )) = 1, the modular form f E is a scalar multiple of the Poincaré series f E = 1 β P (1, 2, N ; z), where P (m, k, N ; z) is the Poincaré series described in Proposition 2.11. Then the Petersson coefficient formula (see [15]) and (1.5) yields β = π vol(Λ E ) . Following the computation in Corollary 1.2 of [15], we obtain the holomorphic projection in terms of the Poincaré series…”

“…Then the Petersson coefficient formula (see [15]) and (1.5) yields β = π vol(Λ E ) . Following the computation in Corollary 1.2 of [15], we obtain the holomorphic projection in terms of the Poincaré series…”

“…The modular degree is the degree of the map φ E . In this paper, we only consider elliptic curves E of modular degree 1 with conductor N such that genus(X 0 (N )) = 1, which restricts N to the finite set {11, 14,15,17,19,21,27, 32, 36, 49}. Our expressions make use of the Weierstrass mock modular form associated to E (a more detailed exposition can be found in Section 2.2).…”

“…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).…”

“…These identities use the characterization of holomorphic projection given by Mertens and Ono in [15] to explicitly relate the Fourier coefficients of weight 2 newforms to elliptic curve invariants and to understand the behavior of the Weierstrass mock modular form. These identities can also be used to compute the shifted convolution L-values to arbitrary precision.…”