Determinants of subquotients of Galois representations associated with abelian varieties

Abstract

Given an abelian variety A of dimension g over a number field K, and a prime ℓ, the ℓ n -torsion points of A give rise to a representation ρ A,ℓ n : Gal(K/K) → GL 2g (Z/ℓ n Z). In particular, we get a mod-ℓ representation ρ A,ℓ : Gal(K/K) → GL 2g (F ℓ ) and an ℓ-adic representation ρ A,ℓ ∞ : Gal(K/K) → GL 2g (Z ℓ ). In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.Applying our results in dimension g = 1, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur's bound on rational isogenies of prime degree to number fields.The study of associated characters of these kinds goes back to Serre's foundational work on the Open Image Theorem, which states that for an elliptic curve E without complex multiplication, the action of the absolute Galois group of a number field on the adèlic Tate module. This is proved in two principal steps. First, in [16], Serre shows that the ℓ-adic image ρ E,ℓ ∞ : G K → GL 2 (Z ℓ ) has finite index for all ℓ. Second, in [17], Serre shows that for sufficiently large primes, the mod-ℓ image ρ E,ℓ : G K → GL 2 (Z ℓ ) is surjective. In each case, the proof consists of reducing the problem to the study of the ℓ-adic and mod-ℓ associated characters of E respectively, i.e. studying the Galois action on the 1-dimensional subquotients.A major conjecture, which is still open, is whether the index of the image of G K in GL 2 ( Z) in Serre's theorem is bounded uniformly in E. The first step towards this result, for K = Q, is Mazur's seminal theorem on isogenies [9]. This theorem is equivalent to the statement that, for an elliptic curve E over Q and for a prime ℓ > 163, the ℓ-torsion module E[ℓ] is irreducible (equivalently, no isogenies E → E ′ defined over Q have kernel of order ℓ). An essential step of Mazur's proof is to analyze the possible associated characters (up to torsion of small degree) of subquotients of E[ℓ], and show that for ℓ > 163 the list of possible associated characters is empty.Momose in [13] gives an exhaustion (i.e. a list containing all possibilities, perhaps with excess) for the mod-ℓ associated characters of elliptic curves over number fields K attached to subquotients of E[ℓ], for ℓ sufficiently large depending on K. When K is quadratic, either real, or imaginary of class number greater than one (i.e. as long as K = Q[ √ D] for D ∈ {−1, −2, −3, −7, −11, −19, −43, −67, −163}), the list of possible associated characters is empty. In particular, any elliptic curve E over such a quadratic field K has irreducible torsion module E[ℓ] (equivalently, admits no ℓ-isogenies) as long as ℓ > C K for some constant C K that depends only on K.The main theorem of our paper gives an analogous exhaustion for abelian varieties of dimension g over K. When applied to elliptic curves, we obtain slightly stronger versions of the above results of Mom...

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