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

Text: We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case a_p≠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^# 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 L_p^# and L_p^≠. We then use results by Kato to prove a divisibility statement. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Y7gPQsBZo6s.