Ranks of rational points of the Jacobian varieties of hyperelliptic curves

Abstract

In this paper, we obtain bounds for the Mordell Weil ranks over certain Z(p)-extensions (including cyclotomic Z(p)-extensions) of a wide range of abelian varieties defined over a number field F whose primes above p are totally ramified over F/Q. We assume that the abelian varieties may have good non-ordinary reduction at primes above p. Our work is a generalization of [5], in which the second author generalized Perrin-Riou's Iwasawa theory for elliptic curves over Q with supersingular reduction ([14]) to elliptic curves defined over the above-mentioned number field F. As a result, we obtain bounds of the Mordell Weil ranks over cyclotomic extensions of the Jacobian varieties of y(2) = x(3pN) + ax(pN) + b and y(2pM) = X-3pN + ax(pN) + b. (C) 2018 Elsevier Inc. All rights reserved.