Derivative of the standard 𝑝-adic 𝐿-function associated with a Siegel form

Abstract

In this paper we first construct a two-variable p p -adic L L -function for the standard representation associated with a Hida family of parallel weight genus g g Siegel forms, using a method developed by Böcherer–Schmidt in one variable. When a form f f has weight g + 1 g+1 a non-crystalline trivial zero could appear. In this case, using the two-variable p p -adic L L -function we have constructed, we can apply the method of Greenberg–Stevens to calculate the first derivative of the p p -adic L L -function for f f and show that it has the form predicted by a conjecture of Greenberg on trivial zeros.