Iwasawa theory for elliptic curves at supersingular primes: A pair of main conjectures

TL;DR

This paper extends Iwasawa theory for elliptic curves at supersingular primes to cases where the trace of Frobenius is non-zero, constructing new p-adic L-functions and formulating related main conjectures.

Contribution

It introduces algebraic constructions of new p-adic L-functions and generalizes existing Selmer group frameworks for supersingular primes with non-zero Frobenius trace.

Findings

Constructed p-adic L-functions $L_p^{lat}$ and $L_p^{lat}$ matching classical cases.

Formulated main conjectures relating Selmer groups and p-adic L-functions.

Proved divisibility results using Kato's theorems.

Abstract

We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case $a_p \neq 0$, where $a_p$ is the trace of Frobenius. To do this, we algebraically construct $p$-adic $L$-functions $L_p^{\sharp}$ and $L_p^{\flat}$ with the good growth properties of the classical Pollack $p$-adic $L$-functions that in fact match them exactly when $a_p=0$ and $p$ is odd. We then generalize Kobayashi's methods to define two Selmer groups $\Sel^{\sharp}$ and $\Sel^{\flat}$ and formulate a main conjecture, stating that each characteristic ideal of the duals of these Selmer groups is generated by our $p$-adic $L$-functions $L_p^{\sharp}$ and $L_p^{\flat}$. We then use results by Kato to prove a divisibility statement.