We prove (under certain assumptions) the irreducibility of the limit σ 2 of a sequence of irreducible essentially self-dual Galois representations σ k : G Q → GL 4 ( Q ¯ p ) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to 1 ⊕ ρ ⊕ χ with ρ irreducible, two-dimensional of determinant χ , where χ is the mod p cyclotomic character. More precisely, we assume that σ k 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 k → 2 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 ρ ) which we assume are non-zero.
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