Lorentz Center

September 17-21, 2001

L-functions from algebraic geometry

Organizers: H. W. Lenstra, Jr (Leiden/Berkeley), P. Stevenhagen (Leiden)

Invited speakers:
Gebhard Boeckle (Zurich)
Jan Denef (Leuven)
Bas Edixhoven(Rennes)
Günter Harder (Bonn)
Alan Lauder (Oxford)
Bernadette Perrin-Riou (Paris)
Chad Schoen (Bonn)
Rene Schoof (Rome)
Jasper Scholten (Nijmegen)
Daqing Wan (Irvine)

The schedule of the workshop is now available.

Please register on-line if you want to attend. There is no registration fee.

About the topic
In 2000, the Clay Mathematics Institute published a list of seven one-million dollar Millennium Prize Problems. These problems, selected by a panel of leading mathematicians, are widely regarded as the most important open problems in mathematics. One problem on the list is the Riemann Hypothesis, which concerns the location of the zeroes of the so-called Riemann zeta function. A proof of the Riemann Hypothesis would have major implications for many questions in number theory, notably on the distribution of prime numbers. Another Clay problem is the conjecture of Birch and Swinnerton-Dyer. It asserts that one can obtain very precise information about the arithmetic of an elliptic curve---or, in plain terms, about the rational solutions to cubic equations in two variables---by analytic means; namely, by examining a special value of the Hasse-Weil L-function associated to the curve.

Both the Riemann zeta function and the Hasse-Weil L-functions are examples of L-functions, and their occurrence on the Clay list illustrates the central role that L-functions play in number theory in general and in arithmetic algebraic geometry in particular. `Arithmetic algebraic geometry' is the modern name for the age-old theory of diophantine equations, with an emphasis on the use of tools from algebraic geometry.

To many objects in arithmetic algebraic geometry one can attach an L-function. L-functions are analytic in nature, but they encode important arithmetic information about the object in question; for example, they may assist in counting the number of rational solutions to systems of equations and estimating their size. It is hoped that they provide the key to a deeper understanding of arithmetic properties of algebraic varieties, but they are still largely shrouded in mystery.

The aim of the workshop is to explore general arithmetic and analytic properties of L-functions that arise from algebraic varieties. It is organized on the occasion of a visit that Daqing Wan will pay to the Mathematisch Instituut of the Universiteit Leiden during the month of September, 2001. Daqing Wan recently achieved a breakthrough in the area of p-adic zeta functions of varieties, by viewing them from an entirely new perspective. It is often felt that new perspectives may also lead to breakthroughs for other types of L-functions, and it is accordingly expected that the workshop will have a speculative component as well. This workshop is partially financed by Lenstra's 1999 Spinoza Award.

Maintained by Peter Stevenhagen (psh@math.leidenuniv.nl)
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