Decomposition and parity of p-adic representations attached to algebraic automorphic forms on GL(4)

Abstract

Let F be a number field with adele ring A_F, and \pi an isobaric, algebraic automorphic representation of GL_4(A_F) of a fixed archimedean weight, which is quasi-regular, meaning that at every archimedean place v of F, the 4-dimensional representation \sigma_v of the Weil group W_{F_v} attached to \pi_v is multiplicity free. Suppose there is an associated 4-dimensional, Hodge-Tate p-adic representation \rho of the absolute Galois group G_F, whose local L-factors agree with those of \pi (up to a shift) at almost all primes P of F. Then our first result is that the semisimplification of \rho does not contain any irreducible 2-dimensional Galois representation which is even. The second result is that if \pi is regular and \rho crystalline, then for sufficiently large p (see the article for a precise statement), the decomposition type of \rho is the same as the isobaric type of \pi. A consequence is that \rho is irreducible when \pi is cuspidal (and regular algebraic), which has also been proved by F. Calegari and T. Gee, in fact with no hypothesis on p. The third and final result, using Taylor's potential modularity theorem, is that given a pair (\sigma, \sigma') of odd, 2-dimensional p-adic representations of the same weight and distinct Hodge-Tate types, such that their direct sum is automorphic, the dimension of the G_F-invariants of the tensor product \eta of the dual of \sigma with \sigma' equals, for large p, the order of pole at s=1 of the L-function of \eta (with the bad factors removed). This is as predicted by the Tate conjecture when \sigma, \sigma' occur in p-adic etale cohmology of smooth projective varieties over F, and it also provides a useful link to a small piece of the work of C. Skinner and E. Urban.