Ranks of rational points of the Jacobian varieties of hyperelliptic curves

TL;DR

This paper establishes bounds for the Mordell-Weil ranks of Jacobian varieties of hyperelliptic curves over cyclotomic extensions, extending previous Iwasawa theory results to more general abelian varieties with non-ordinary reduction.

Contribution

It generalizes Iwasawa theory bounds for Mordell-Weil ranks to hyperelliptic Jacobians with arbitrary dimension and non-ordinary reduction over ramified cyclotomic fields.

Findings

Bounds for ranks over cyclotomic extensions of hyperelliptic Jacobians.

Extension of Perrin-Riou's Iwasawa theory to higher-dimensional abelian varieties.

Application to ramified hyperelliptic curves with specific equations.

Abstract

In this paper, we obtain bounds for the Mordell-Weil ranks over cyclotomic extensions of a wide range of abelian varieties defined over a number field $F$ whose primes above $p$ are totally ramified over $F/\mathbb{Q}$. We assume that the abelian varieties may have good non-ordinary reduction at those primes. Our work is a generalization of \cite{Kim}, in which the second author generalized Perrin-Riou's Iwasawa theory for elliptic curves over $\mathbb{Q}$ with supersingular reduction (\cite{Perrin-Riou}) to elliptic curves defined over the above-mentioned number field $F$. On top of non-ordinary reduction and the ramification of the field $F$, we deal with the additional difficulty that the dimensions of the abelian varieties can be any number bigger than 1 which causes a variety of issues. As a result, we obtain bounds for the ranks over cyclotomic extensions $\mathbb{Q}(\mu_{p^{\max(M,N)+n}})$ of the Jacobian varieties of {\it ramified} hyperelliptic curves $y^{2p^M}=x^{3p^N}+ax^{p^N}+b$ among others.