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

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.