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}}} rightarrow {{,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 krightarrow 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.
Highlights
In [8] the authors studied the modularity of abelian surfaces with rational torsion
Assuming that ρ is irreducible, Serre’s conjecture (Theorem of Khare-Wintenberger) implies that the mod p representation looks like the reduction of that of a Saito–Kurakawa lift of an elliptic modular form f of weight 2
The Paramodular Conjecture predicts that this representation should be isomorphic to the Galois representation attached to a weight 2 Siegel modular form of paramodular level which is not in the space of Saito–Kurokawa lifts
Summary
In [8] the authors studied the modularity of abelian surfaces with rational torsion. Let A be an abelian surface over Q, let p be a prime and suppose that A has a rational point of order p, and a polarization of degree prime to p. Assuming that ρ is irreducible, Serre’s conjecture (Theorem of Khare-Wintenberger) implies that the mod p representation looks like the reduction of that of a Saito–Kurakawa lift of an elliptic modular form f of weight 2. Establishing the modularity of A by a Siegel modular form requires proving congruences between the Saito–Kurokawa lift SK (f ) and “non-lifted type (G)” Siegel modular forms. The latter are cuspforms staying cuspidal under the transfer to GL4, and are expected to be exactly the forms whose associated p-adic representation is irreducible
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