Irreducibility of limits of Galois representations of Saito–Kurokawa type

We prove (under certain assumptions) the irreducibility of the limit [Formula: see text] of a sequence of irreducible essentially self-dual Galois representations [Formula: see text] (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to [Formula: see text] with [Formula: see text] irreducible, two-dimensional of determinant [Formula: see text] , where [Formula: see text] is the mod p cyclotomic character. More precisely, we assume that [Formula: see text] are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as [Formula: see text] appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to [Formula: see text] ) which we assume are non-zero.