Two-dimensional mod p Galois representations attached to modular forms

Classical newforms are cusp forms on congruence subgroups of SL(2,Z) that are eigenvectors for the Hecke operators. These modular forms give rise to two-dimensional representations of the absolute Galois group of the rational field. Conversely, if one starts with a semisimple two-dimensional representation of this Galois group over a finite field, the representation should arise from a newform if a mild necessary condition (involving complex
conjugation) is satisfied. When the representation is irreducible, Serre’s conjecture (which is essentially a theorem) predicts the modularity and specifies that possible weights and levels of newforms that give rise to the representation. When the representation is reducible, the set of weights and levels that give rise to it is apparently more complicated to describe. I will discuss the simplest possible examples of this phenomenon, where the representation is about as uncomplicated as possible, the weight is 2 and the level is either a prime number or the product of two distinct primes.

Speaker Details

Kenneth Ribet studied at Brown University and Harvard University. He received his PhD in 1973 from Harvard, where his advisor was John Tate. After three years of teaching in Princeton and two years of research in Paris, Ribet joined the Berkeley faculty in 1978. He received his department’s Distinguished Teaching Award in 1985.Ribet is known for his work in number theory and algebraic geometry. He played a prominent role in the proof of Fermat’s Last Theorem by showing that this statement was a logical consequence of a conjecture about elliptic curves. (Andrew Wiles proved this conjecture in 1995, thereby obtaining Fermat’s Last Theorem as a corollary.) Ribet is a member of the scientific advisory board of the Institute for Pure & Applied Mathematics at UCLA and a member of the editorial boards of the following three Springer book series: Graduate Texts in Mathematics, Universitext, Undergraduate Texts in Mathematics. He serves also on the editorial boards of Mathematische Annalen, the Annales de l’Institut Fourier, the Journal of Number Theory and Mathematics Research Letters.Ribet was elected to the American Academy of Arts and Sciences in 1997 and the National Academy of Sciences in 2000. He was awarded the Fermat Prize in 1989 and received an honorary PhD from Brown University in 1998. Ribet was inducted as a Vigneron d’honneur by the Jurade de Saint Emilion in 1988.

Date:
Speakers:
Kenneth A. Ribet
Affiliation:
UC Berkeley