The meaning of modularity

Authors

Dr. Scott Ahlgren, Department of Mathematics, Penn State University

Document Type

Presentation Abstract

Presentation Date

12-10-1998

Abstract

Andrew Wiles' proof of Fermat's Last Theorem relies in turn on his proof that a wide class of elliptic curves are "modular" (this is the so-called Taniyama-Shimura conjecture). I will begin by describing what this conjecture says in concrete, friendly terms.

But elliptic curves are not the only objects thought to be modular. In recent work, Ken Ono and I have devised a new method which proves the modularity of a certain "Calabi-Yau threefold". As a result we can prove Beukers' conjectured "supercongruence" for the Apery numbers (these are combinatorial sums introduced by Apery to establish the irrationality of certain values of the Riemann zeta function).

Additional Details

Thursday, 10 December 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Recommended Citation

Ahlgren, Dr. Scott, "The meaning of modularity" (1998). Colloquia of the Department of Mathematical Sciences. 30.
https://scholarworks.umt.edu/mathcolloquia/30