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

Abstract

We extend Kobayashiʼs formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case ap≠0, where ap is the trace of Frobenius. To do this, we algebraically construct p-adic L-functions Lp♯ and Lp♭ with the good growth properties of the classical Pollack p-adic L-functions that in fact match them exactly when ap=0 and p is odd. We then generalize Kobayashiʼs methods to define two Selmer groups Sel♯ and Sel♭ 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 Lp♯ and Lp♭. We then use results by Kato to prove a divisibility statement. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Y7gPQsBZo6s.