Hilbert Poincaré series and kernels for products of $L$-functions
Mingkuan Zhang, Yichao Zhang
TL;DR
The paper extends kernel methods for L-values to totally real fields by developing Hilbert Poincaré series with seed functions, enabling explicit Rankin-Cohen bracket formulas and kernel representations. It constructs Cohen's kernel and double Eisenstein series as convergent series of Poincaré series and shows how their values yield special values of products of Hilbert L-functions, generalizing Zagier's kernel formula. The results connect Petersson inner products with Rankin-Selberg L-values and recover Shimura-type algebraicity results for Hilbert L-values in the parallel weight case, while providing a framework that unifies adélic Hilbert modular forms with Poincaré-series techniques. Overall, the work provides concrete kernel tools for analyzing Hilbert L-functions and their algebraicity properties over totally real fields.
Abstract
We study Hilbert Poincaré series associated to general seed functions and construct Cohen's kernels and double Eisenstein series as series of Hilbert Poincaré series. Then we calculate the Rankin-Cohen brackets of Hilbert Poincaré series and Hilbert modular forms and extend Zagier's kernel formula to totally real number fields. Finally, we show that the Rankin-Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant.