  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.
