Date: Wednesday, February 27, 2013
Location: 3866 East Hall (3:00 PM to 4:00 PM)
Title: The Herbrand-Ribet Theorem
Abstract: Kummer proved that Fermat's Last Theorem holds for so-called regular primes and introduced "Kummer's Criterion": a prime $p$ is irregular if and only if $p$ divides the numerator of some Bernoulli number $B_{2k}$ with $2 < 2k < p - 1$. In 1932, Herbrand showed that if the Galois group of $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ acts on a subgroup of order $p$ in the ideal class group in a prescribed fashion, then $p$ divides a specific Bernoulli number. Nearly 45 years later, Ribet was able to prove the converse to Herbrand's theorem, using Eisenstein series, modular forms, the Eichler-Shimura relation, and Galois representations to explicitly construct an unramified cyclic extension of degree $p$ over $\mathbb{Q}(\zeta_p)$ with the desired behavior.
In this talk we will discuss the relationships established between these tools and sketch Ribet's proof, with some difficulties black-boxed. As time permits, we will discuss connections to Vandiver's conjecture.
Speaker: Brandon Carter
Institution: UM
Event Organizer:
|
