The rank of Abelian varieties over infinite Galois extensions

G, Frey and M. Jarden (1974, Proc, London Math. Soc, 28, 112 128) asked if every Abelian variety A defined over a number field k with dim A > 0 has infinite rank over the maximal Abelian extension k(ab) of k. We verify this for the Jacobians of cyclic covers of P-1, with no hypothesis on the Weierstrass points or on the base field, We also derive an infinite rank criterion by analyzing the ramification of division points of an Abelian variety. As an application, we show that any d-dimensional Abelian variety A over k with a degree n projective embedding over k has infinite rank over the compositum of all extensions of k of degree <n(4d+2). (C) 2001 Elsevier Science (USA).