## Number Theory

The research of the UCLA Number Theory Group is concerned with the arithmetic of modular forms, automorphic forms, Galois representations, Selmer groups, and classical and p-adic L-functions. The group has 5 permanent faculty: Don Blasius, William Duke, Haruzo Hida, Chandrashekhar Khare, and Romyar Sharifi. Each works broadly in number theory and has settled conjectures and introduced new topics of study. As a case in point, Hida’s theory of ordinary p-adic families, and the theories to which it gave rise constitute an area of research that is broadly studied by mathematicians around the world. His theory also provides a powerful tool for applications to Galois representations and p-adic L-functions. Likewise, Khare and his collaborator Wintenberger proved Serre’s Conjecture on modular forms, which was widely viewed as one of the most important conjectures in the field of arithmetic geometry. Since that proof, the extension of this conjecture to automorphic forms has taken hold and stimulated a great deal of research. Duke is a leading analytic number theorist, known for many basic contributions, including the settling, using modular forms, of a well-known equidistribution problem dating back to Gauss. Blasius has made basic contributions to the special values of L-functions, and pioneered, with J. Rogawski, the use of endoscopy in the study of Galois representations. Sharifi formulated a conjecture which provides a explicit refinement to the Iwasawa main conjecture. This has opened up an exciting new avenue of research concerned with the description of arithmetic objects in terms of higher-dimensional geometry.

At any given time, the Number Theory Group has two or more postdocs, and up to 10 graduate students. There is a weekly number theory seminar and typically several ongoing instructional seminars devoted to the study of current research papers or topics, and the presentation of research of group members at all levels.