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 $A^{\prime }$ 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 $A^{\prime }$ (in the case of elliptic curves, simply the Euler factor at $\mathfrak{p}$), and use these points to construct characteristic power series ${\mathbf{L}}_{\alpha }\in {\mathbb{Q}}_{\mathbf{p}}[[\mathbf{X}]]$ analogous to Mazur's characteristic polynomials in the case of good ordinary reduction. By studying ${\mathbf{L}}_{\alpha }$, 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 ${\mathbf{L}}_{\mathbf{alg}}^{♯}$ and ${\mathbf{L}}_{\mathbf{alg}}^{♭}$. One of the consequences of this construction is that if ${\mathbf{L}}_{\mathbf{alg}}^{♯}$ and ${\mathbf{L}}_{\mathbf{alg}}^{♭}$ are not divisible by a certain power of $p$, then $E(F_{\infty })$ has a finite rank modulo torsions.

References in corpus (1)

• Ranks of rational points of the Jacobian varieties of hyperelliptic curves

Cited by in corpus (2)

• On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions
• Explicit uniformizers for certain totally ramified extensions of the field of $p$-adic numbers