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

Abstract

In this paper we construct a two variables p-adic L-function for the standard representation associated with a Hida family of parallel weight genus g Siegel forms, using a method previously developed by B\"ocherer--Schmidt in one variable. When a form of weight g+1 is Steinberg at p, a trivial zero appears and, using the method of Greenberg--Stevens, we calculate the first derivative of this p-adic L-function and show that it has the form predicted by a conjecture of Greenberg on trivial zeros.