Ranks of the Rational Points of Abelian Varieties over Ramified Fields, and Iwasawa Theory for Primes with Non-Ordinary Reduction

TL;DR

This paper develops new Iwasawa theoretic methods for abelian varieties and elliptic curves over ramified p-adic extensions, providing bounds on ranks of rational points using characteristic power series and norm relations.

Contribution

It introduces novel constructions of characteristic power series for abelian varieties with non-ordinary reduction and refines Iwasawa theory for supersingular elliptic curves over ramified extensions.

Findings

Established weak bounds for ranks of abelian varieties over ramified extensions.

Constructed integral characteristic polynomials for supersingular elliptic curves.

Proved conditions under which the Mordell-Weil group has finite rank modulo torsion.

Abstract

Let $A$ be an abelian variety defined over a number field $F$. Suppose its dual abelian variety $A'$ has good non-ordinary reduction at the primes above $p$. Let $F_{\infty}/F$ be a $\mathbb Z_p$-extension, and for simplicity, assume that there is only one prime $\mathfrak p$ of $F_{\infty}$ above $p$, and $F_{\infty, \mathfrak p}/\mathbb Q_p$ is totally ramified and abelian. (For example, we can take $F=\mathbb Q(\zeta_{p^N})$ for some $N$, and $F_{\infty}=\mathbb Q(\zeta_{p^{\infty}})$.) As Perrin-Riou did, we use Fontaine's theory of group schemes to construct series of points over each $F_{n, \mathfrak p}$ which satisfy norm relations associated to the Dieudonne module of $A'$ (in the case of elliptic curves, simply the Euler factor at $\mathfrak p$), and use these points to construct characteristic power series $\bf L_{\alpha} \in \mathbb Q_p[[X]]$ analogous to Mazur's characteristic polynomials in the case of good ordinary reduction. By studying $\bf L_{\alpha}$, we obtain a weak bound for $\text{rank} E(F_n)$. In the second part, we establish a more robust Iwasawa Theory for elliptic curves, and find a better bound for their ranks under the following conditions: Take an elliptic curve $E$ over a number field $F$. The conditions for $F$ and $F_{\infty}$ are the same as above. Also as above, we assume $E$ has supersingular reduction at $\mathfrak p$. We discover that we can construct series of local points which satisfy finer norm relations under some conditions related to the logarithm of $E/F_{\mathfrak p}$. Then, we apply Sprung's and Perrin-Riou's insights to construct \textit{integral} characteristic polynomials $\bf L_{alg}^{\sharp}$ and $\bf L_{alg}^{\flat}$. One of the consequences of this construction is that if $\bf L_{alg}^{\sharp}$ and $\bf L_{alg}^{\flat}$ are not divisible by a certain power of $p$, then $E(F_{\infty})$ has a finite rank modulo torsions.