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 ψ. By generalizing a construction of [20], we construct a p-adic L-function interpolating the special values of the L-function L(s, Sym 2 (f ) ⊗ ξ), where ξ is a Dirichlet character. When s = k − 1 and ξ = ψ −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 −

Derivative of symmetric square p-adic L-functions via pull-back formula

“…That's why in [Ros13a,Ros13b] and in this article we can deal only with forms which are Steinberg at p.…”

“…This method has been used successfully many other times [Mok09,Ros13a,Ros13b]. It is very robust and easily adaptable to many situations in which the expected order of the trivial zero is one.…”

“…For example, in [Gre94,Mok09] we have s(κ) = 1 2 Log p (π(κ)) and in [Ros13a,Ros13b] we have s(κ) = Log p (π(κ)) − 1. In the first case the line corresponds to the vanishing on the central critical line which is a consequence of the fact that the ε-factor is constant in the family.…”

“…where C is a constant independent of κ, explicitly determined by the comparison of C κ here and in [Ros13b,Theorem 4.14].…”

Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form

“…In particular, thanks to the recent work of Urban [Urb] on families of nearly-overconvergent modular forms, this method can now be successfully applied in the case F = Q. This is the main subject of [Ros13b]. We point out that understanding the order of this zero and an exact formula for the derivative of the p-adic L-function has recently become more important after the work of Urban on the main conjecture for the symmetric square representation [Urb06] of an elliptic modular form.…”