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

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