Very simple 2-adic representations and hyperelliptic jacobians

2001/09/03 by Zarhin, Yuri G.
#11G10 #11G30 #14H40 #14K05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0109014

Abstract

Let K be a number field, n>4 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group is either the full symmetric group S_n or the alternating group A_n. Suppose C:y²=f(x) is the corresponding hyperelliptic curve and J its jacobian defined over K. For each prime ℓ we write V_ℓ(J) for the Q_ℓ-Tate module of J and e_ℓ for the Riemann form on V_ℓ(J) attached to the theta divisor. (Here Q_ℓ is the field of ℓ-adic numbers.) We write sp(V_ℓ(J)) for the Q_ℓ-Lie algebra of the symplectic group of e_ℓ. We write g_ℓ for the Lie algebra of the image of the Galois group Gal(K) of K in Aut(V_ℓ(J)). We prove that g_ℓ coincides with the direct sum Q_ℓI ⊕ sp(V_ℓ(J)) where I is the identity operator.

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