Ranks of the rational points of abelian varieties over ramified fields, and Iwasawa theory for primes with non-ordinary reduction

Let A be an abelian variety defined over a number field F. Suppose its dual abelian variety A(v) has good non-ordinary reduction at the primes above p. Let F-infinity/F be a Z(p)-extension, and for simplicity, assume that there is only one prime p of F-infinity above p, and F-infinity,F-p/Q(p) is totally ramified and abelian. (For example, we can take F = Q(zeta(pN)) for some N, and F-infinity = Q(& p infinity</INF>).) As Perrin-Riou did in [11], we use Fontaine's theory ([3]) of group schemes to construct series of points over each F-n,F-p which satisfy norm relations associated to the Dieudonne module of A(v) (in the case of elliptic curves, simply the Euler factor at p), and use these points to construct characteristic power series L-alpha is an element of Q(p)[[X]] analogous to Mazur's characteristic polynomials in the case of good ordinary reduction. By studying L-alpha, we obtain a weak bound for rank E(F-n).& para;& para;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-infinity 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/F-p. Then, we apply Sprung's ([14]) and Perrin-Riou's insights to construct integral characteristic polynomials L-alg(#) and L-alg(b). One of the consequences of this construction is that if L-alg(#) and L-alg(b) are not divisible by a certain power of p, then E(F-infinity) has a finite rank modulo torsions. (C) 2017 Elsevier Inc. All rights reserved.