Eisenstein Series and Applications

August 15 to August 19, 2005

at the

American Institute of Mathematics, Palo Alto, California

organized by

Wee Teck Gan, Stephen Kudla, and Yuri Tschinkel

This workshop, sponsored by AIM and the NSF, will consider some recent applications of Eisenstein series to problems in arithmetic geometry and number theory.

Eisenstein series are an essential ingredient in the spectral theory of automorphic forms and an important tool in the theory of L-functions. They have also been exploited extensively by number theorists for many arithmetic purposes. But the Eisenstein series themselves are often treated as auxillary objects. For example, in the classic work of Gross and Zagier leading to the Gross--Zagier formula, the critical role played by the derivative of a certain Eisenstein series is seldom emphasized. This workshop will focus on the Eisenstein series which arise in several recent developments in arithmetic:

  1. Arakelov intersection theory on Shimura varieties (such as the work of Kudla-Rapoport-Yang).
  2. Special values of L-functions and Iwasawa theory (such as the work of Skinner-Urban).
  3. Equidistribution of rational/integer points on homogeneous varieties (such as the work of Shalika--Takloo-Bighash--Tschinkel)
A central goal of the workshop will be to try to understand the common structural properties of the Eisenstein series occuring in these and related applications. For example, is it possible to identify a class of Eisenstein series whose Fourier coefficients (resp. special values) encode a significant arithmetic information? Do such series fit into p-adic families? Are the Eisenstein series which arise in counting problems of this type? We hope that expository lectures emphasizing the role of Eisenstein series in the three areas above will serve as a basis for stimulating discussion on these and related questions.

Different users of Eisenstein series often focus on different aspects of the theory. By bringing together users from rather different areas who do not usually interact with each other, the workshop will provide participants with an exposure to unfamiliar ways in which Eisenstein series have been used. This juxtaposition of perspectives should provide deeper insight into the arithmetic of Eisenstein series. and foster fruitful new collaborations.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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