The Iwasawa theory for elliptic curves (and modular forms) relates two distant objects: On the analytic side, we have p-adic L-functions, whereas on the algebraic side, we have Selmer groups. This theory has traditionally been in better shape at good ordinary primes, whereas in the good supersingular case, the analogous objects don´t behave as well. In this talk, we introduce a *pair* of convenient objects in the supersingular case and derive some arithmetic consequences whose true origins still are a mystery.