Alex Torzewski

How common is formal complex multiplication?

An elliptic curve over a characteristic zero field is said to have complex multiplication when its endomorphism ring is larger than Z ("E has extra endomorphisms"). Generic elliptic curves don't have complex multiplication. Similarly, when the Tate module of E has extra endomorphisms we say E has complex multiplication of its Tate module. Over a number field, the Tate module of E has complex multiplication if and only if E does. Over a local field this need not be the case. Locally, in the case of supersingular reduction, complex multiplication of the Tate module coincides with the notion of formal complex multiplication. We investigate how often this happens via basic computations in p-adic Hodge theory.