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

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.