Cycles on modular varieties (11w5125)
In a highly influential 2001 paper, Henri Darmon proposed a systematic, conjectural "modular" construction of algebraic points on elliptic curves. Using p-adic analysis, he constructed local points on elliptic curves, conjectured them to be global points, and gave precise predictions governing their field of definition. This construction is genuinely novel in that it lies outside the scope of the theory of complex multiplication. Since 2001, Darmon's construction has grown into a theory that has yielded a host of generalizations. All of these generalizations have certain features in common:
(1) They produce interesting (conjecturally) global objects, such as algebraic points on elliptic curves or units in rings of algebraic integers, by evaluating certain Abel-Jacobi maps on cycles.
(2) These objects are, at least conjecturally, intimately related to special values of L-functions or their derivatives. In particular, their existence is compatible with the conjecture of Birch and Swinnerton-Dyer.
In many cases, the rationality of the objects constructed in this way can be related to fundamental conjectures in arithmetic and geometry, namely, the Tate conjecture, the Hodge conjecture and the Shafarevich-Tate conjecture. In other cases, the mechanism by which algebraic quantities are produced remains highly mysterious. Ten years have passed since the publication of Darmon's original construction, and we feel that the time is ripe to run a focused workshop, taking stock of the field's rapid growth, presenting state-of-the-art results, and planning future attacks on the host of fascinating and fundamental conjectures which have been developed.
The last years have also witnessed the emergence of profound connections between Darmon's theory of modular points and other branches of mathematics related to cycles on modular varieties which are experiencing equally rapid growth and becoming central themes in arithmetic geometry - p-adic Hodge theory, Kudla's program, and the theory of theta liftings and harmonic weak Maas forms, for instance. A main objective of the proposed workshop is to bring together mathematicians studying Darmon's theory of modular points with researchers in these related fields in order to study the connections in detail and expand the scope of the theory.
For these reasons, this is a particularly appropriate time to hold a workshop devoted to the theory of modular points. With its importance being recognized within the mathematical community, interest in the field is extremely high. In addition, the diverse collection of constructions that has constituted the theory for the past ten years is beginning to coalesce into a beautifully structured theoretical framework. Thus, a one-week BIRS workshop will spur the advancement of this science by bringing researchers together and allowing for a pooling of effort and knowledge at this crucial point in the theory's development. Also, it will serve as an access point for researchers in related areas wishing to enter the field. The expected flow of ideas from other areas into the theory of modular points will lead to increased understanding and new results.