Uniform behavior of families of Galois representations on Siegel modular forms and the endoscopy conjecture. (English summary) Bol. Soc. Mat. Mexicana (3) 13 (2007), no. 2, 243–253. Let f be a genus two Siegel modular form of level N, weight k> 3, multiplicity one, with the family of four-dimensional symplectic Galois representations attached to it. We also assume that we are in a case in which this family is semistable. One of the consequences of Tate’s conjecture on the Siegel threefold is that reducibility for the Galois representations attached to f must be a uniform property: if it is verified at one prime, then all the representations in the family are reducible. In the present paper, the author proves this uniform principle: Theorem. Let f be a genus two Siegel cuspidal Hecke eigenform of weight k> 3 and level N, having multiplicity one, such that the attached Galois representations ρf,λ are semistable. Suppose that for some odd prime l0 ∤ N, λ0 | l0, the representation ρf,λ0 is reducible. Then the representations ρf,λ are reducible for every λ. Moreover, if this happens, either f is of Saito-Kurokawa type or f is endoscopic. After excluding the Saito-Kurokawa case, the author proves the result more generally for compatible families of geometric, pure and symplectic four-dimensional Galois representations which are semistable. The proof is given as follows: the author uses as the starting point Taylor’s recent results on the Fontaine-Mazur conjecture and the meromorphic continuation of L-functions for odd twodimensiona
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