Irreducibility of limits of Galois representations of Saito-Kurokawa type

Abstract

We prove (under certain assumptions) the irreducibility of the limit ${\sigma }_2$ of a sequence of irreducible essentially self-dual Galois representations ${\sigma }_k:G_{\mathbf{Q}}\to {\mathrm{GL}}_4({\overline{\mathbf{Q}}}_p)$ (as $k$ approaches 2 in a $p$-adic sense) which mod $p$ reduce (after semi-simplifying) to $1\oplus \rho \oplus \chi$ with $\rho$ irreducible, two-dimensional of determinant $\chi$, where $\chi$ is the mod $p$ cyclotomic character. More precisely, we assume that ${\sigma }_k$ 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 $k\to 2$ 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 $\rho$) which we assume are non-zero.

arXiv comment21 pages