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∞/F be a Zp-extension, and for simplicity, assume that there is only one prime p of F∞ above p, and F∞,p/Qp is totally ramified and abelian. (For example, we can take F=Q(ζpN) for some N, and F∞=Q(ζp∞).) As Perrin-Riou did in [11], we use Fontaine's theory ([3]) of group schemes to construct series of points over each Fn,p which satisfy norm relations associated to the Dieudonné module of A∨ (in the case of elliptic curves, simply the Euler factor at p), and use these points to construct characteristic power series Lα∈Qp[[X]] analogous to Mazur's characteristic polynomials in the case of good ordinary reduction. By studying Lα, we obtain a weak bound for rankE(Fn).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∞ are the same as above. Also as above, we assume E has supersingular reduction at 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/Fp. Then, we apply Sprung's ([14]) and Perrin-Riou's insights to construct integral characteristic polynomials Lalg♯ and Lalg♭. One of the consequences of this construction is that if Lalg♯ and Lalg♭ are not divisible by a certain power of p, then E(F∞) has a finite rank modulo torsions.