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

Abstract

We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case $a_p\ne 0$, where $a_p$ is the trace of Frobenius. To do this, we algebraically construct $p$-adic $L$-functions $L_p^{♯}$ and $L_p^{♭}$ 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 $L_p^{♯}$ and $L_p^{♭}$. We then use results by Kato to prove a divisibility statement.

arXiv comment26 pages. Basically a new article. Has been expanded and revised incorporating comments made by the referee and many others

Cited by in corpus (1)

• The Shafarevich-Tate group in cyclotomic Z_p-extensions at supersingular primes