Constructing plus/minus local points over Iwasawa Kummer extensions, and Iwasawa theory for supersingular reduction

Abstract

Suppose an elliptic curve E/Q has good supersingular reduction at a prime p, and satisfies 1+p−#E(Z/pZ)=0, which is always true if p is supersingular and p>3 by the Hasse inequality. Expanding Kobayashi's plus/minus-local points over the cyclotomic Zp-extension of Qp, where π is any element in Cp with positive p-adic valuation, we construct doubly indexed plus/minus local points over Qp(πpn−m,μpn) (which we will call local Iwasawa Kummer extensions) as n,m vary.Then, we explore various possible plus/minus Selmer groups over Zp-extensions of number fields over which p is ramified, or doubly indexed plus/minus Selmer groups over Iwasawa Kummer extensions, and study or speculate their properties using the above plus/minus local points.Finally, again using the above points, we study the following: Suppose k is a ramified quadratic extension of Qp. We find a bound of [E(mk(μpn)):E+(mk(μpn))+E−(mk(μpn))] as n varies.