Comparison of semi-simplifications of Galois representations

Abstract

Let Γ be the absolute Galois group of a global field. Let ρ 1 and ρ 2 be two p-adic, finite dimensional representations of Γ . Then there exists a finite number of primes q such that if the characteristic polynomials of ρ 1 (Frob q ) and ρ 2 (Frob q ) are equal, then ρ 1 and ρ 2 have isomorphic semi-simplifications and so the same L-functions. We give a method to compute a sufficient list of primes, based on the ramification and the dimension of the representations. We use then the result to prove a conjecture by B. van Geemen and J. Top.

Galois representations associated with a non-selfdual automorphic representation of GL(3)

“…When it can be applied, it sometimes gives an efficient criterion to test the equivalence of ℓ-adic representations. The following result is proved in [15,Section 4].…”

“…In [15], Grenié gave a method to obtain a set T n,S,E satisfying the above condition. His result can be applied to ℓ-adic representations satisfying some restrictive conditions only.…”

“…His result can be applied to ℓ-adic representations satisfying some restrictive conditions only. (For the precise statement, see [15,Theorem 3]. ) When it can be applied, it sometimes gives an efficient criterion to test the equivalence of ℓ-adic representations.…”

“…We shall show the equivalence of ρ vGT,2 and ρ π,2 using Grenié's results [15]. To use his results, we need to show the image of ρ π,2 is contained in a subgroup conjugate to GL…”

“…(1),(2) are proved in[12, Section 3.7] except for the absolute irreducibility. The absolute irreducibility of ρ vGT,ℓ is proved in[12, Remark 3.12] and the last paragraph of[15, Section 4.1]. (3) As explained in [12, Section 3.8], we haveb p = ι ℓ (Tr ρ vGT,ℓ (Frob p )) = 1 2 ι ℓ (Tr(Frob p ; T Q ℓ )) − √ −1 2 ι ℓ (Tr(Frob p •φ * ; T Q ℓ )).It is enough to show that Tr(Frob p ; T Q ℓ ), Tr(Frob p •φ * ; T Q ℓ ) are rational numbers which are independent of ℓ = p. This follows from [12, Proposition 3.6] and [12, Proposition 3.9].…”