A formula for the derivative of the p-adic L-function of the symmetric square of a finite slope modular form

Abstract

Let f be a modular form of weight k and Nebentypus $\psi$. By generalizing a construction of Dabrowski and Delbourgo, we construct a p-adic L-function interpolating the special values of the L-function $L(s,{\mathrm{Sym}}^2(f)\otimes \xi )$, where $\xi$ is a Dirichlet character. When s=k-1 and $\xi ={\psi }^{-1}$, this p-adic L-function vanishes due to the presence of a so-called trivial zero. We give a formula for the derivative at s=k-1 of this p-adic L-function when the form f is Steinberg at p. If the weight of f is even, the conductor is even and squarefree, and the Nebentypus is trivial this formula implies a conjecture of Benois.

References in corpus (2)

• Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form
• Derivative of symmetric square p-adic L-functions via pull-back formula

Cited by in corpus (4)

• Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form
• Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$-adic $L$-functions
• Iwasawa Invariants for Symmetric Square Representations
• P-adic L-functions for GL(3)