University of Georgia
Mathematics Department Colloquium
Chris Skinner
Institute for Advanced Study
January 21 1999
Recent results on modularity of two-dimensional Galois
representations
Abstract:
Two-dimensional $p$-adic representations of $G_Q=Gal(\overline Q/Q)$, the
absolute Galois group of the field of rational numbers $Q$,
(i.e., homomorphisms $G_Q \rightarrow GL_2(R)$, $R$ the ring of integers
of a finite extension of the $p$-adics $Q_p$) occur often in number theory.
For example, the Tate-module at
a prime $p$ of an elliptic curve over $Q$ gives rise to such a representation,
and in a similar manner such representations appear in the Tate-modules of
Abelian varieties with "lots of real multiplications."
Similarly, a holomorphic
modular form that is an eigenform for the Hecke operators has associated to it
such a two-dimensional representation for each prime $p$.
In all of these cases
the representations encode much of the arithmetic information about the objects
(e.g., the number of points over finite fields, the $L$-series, the Hecke
eigenvalues).
The celebrated conjecture of Shimura and Taniyama asserts that the
representations in the first two examples also belong to the third example.
This has been subsumed in a conjecture of Mazur and Fontaine which purports to
give criteria for any two-dimensional $p$-adic representation of $G_Q$ to be a
representation associated to an eigenform. Such a representation is said to be
``modular''. As is well-known, Wiles' proof of Fermat's Last Theorem
included the first significant results in the direction of these conjectures.
In this talk, I will recall the above notions, discuss the conjectures, and
talk
about some recent work extending/complementing the earlier work of Wiles. In
particular I will discuss some recent joint work of Wiles and myself on the
modularity of representations that are ordinary at $p$. This work entails not
just establishing modularity of two-dimensional representations of $G_Q$ but
also suitable generalizations to totally real fields.