Kato explicit reciprocity law for Siegel modular forms of weight $(3, 3)$

Abstract

We extend Kato explicit reciprocity law, in the version written by Scholl, for a modular curve to a product of two modular curves. By embedding the product of two modular curves in the Siegel threefold, we deduce an explicit reciprocity law for the unique critical twist of the $p$-adic Galois representation attached to cuspidal Siegel modular forms of weight $(3,3)$.