**Title:** Geometric deformations of modular Galois representations

**Speaker:** Professor Mark Kisin

**Speaker Info:** University of Munster

**Brief Description:**

**Special Note**:

**Abstract:**

A p-adic representation of the absolute Galois group of Q is called "geometric" if it is unramified outside finitely many places, and if its restriction to a decomposition group at p satisfies a certain subtle condition (being "de Rham"). It is known that representations which come from geometry are geometric, and the converse has been conjectured by Fontaine and Mazur.In this talk I will explain a result which says that a Galois representation attached to a modular form (such representations are geometric) has no non-trivial deformations which are geometric. This result is consistent with the conjecture of Fontaine-Mazur, which implies that there should not exist one parameter families of geometric deformations. It also gives support to a philosophy of Deligne and Beilinson relating special values of L-functions and extensions of motives. A more down to earth consequence of this result is that the universal deformation space of a modular residual representation is smooth at modular points.

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