American Institute of Mathematics, Palo Alto, California
Manjul Bhargava, Juergen Klueners, Joe Buhler, and Benedict Gross
This workshop, sponsored by AIM and the NSF, will be devoted to questions surrounding some of the various exciting developments on the Cohen-Lenstra heuristics. A number of these recent developments have appeared in the last few years, while several other recent advances have not yet been published. This is thus a very ripe time for the various different people who have been working in the area to get together, share ideas, and, hopefully, prove some theorems!
Besides engaging in discussions about the various recent results and evidence towards the original Cohen-Lenstra heuristics for class groups (and their extensions by Cohen-Martinet), the workshop will also consider some of the further distribution conjectures now formulated for other types of groups such as the Tate-Shafarevich groups of elliptic curves (see the reference due to Delaunay), which has also attracted much recent attention. We will also discuss the extensions of the Cohen-Lenstra heuristics for quadratic fields to the prime 2 by Frank Gerth III.
For function fields, there has been much more evidence for the Cohen-Lenstra conjectures. More recent works have shown that the conjectures in the function field case in large part are related to certain very ''believable'' topological conjectures, which may soon be proved. These works help in understanding what other things might also be true in the number field case.
In addition to the work for number and function fields, there has also been much recent work on class groups of orders, and in this situation one of the conjectures of Cohen and Lenstra has recently been disproved (though it is certainly not any major hole in the conjectures, and can be easily fixed). We will discuss how the conjecture should be revised in this and other related cases, and we hope to discuss further cases in which one might obtain proofs of these revised conjectures.
Similarly, some recent computations have shown that the Cohen-Martinet heuristics, in cases where there was earlier doubt, can be extended/corrected through the appropriate consideration of roots of unity in the ground field. These revised/extended heuristics will also be discussed.
In a related direction, there have been recently discovered families of extensions (such as certain higher degree number fields that have a power integral basis) where the Cohen-Lenstra heuristics do not seem to hold at all (provably!). Ideas such as these will help fuel discussion on what assumptions on a set of fields is necessary to expect random distribution of class group behavior as the Cohen-Lenstra conjectures predict!
At the meeting, we hope to have presentations on the various aspects of the Cohen-Lenstra conjectures described above, including the recent re-thinking of the conjectures by Lenstra himself. With the numerous experts in the area present, we will have theoretical discussions, computations, and problem sessions, which we hope will help everyone reach a higher understanding of the area, and perhaps begin new collaborations which will help unify the various results that have been achieved in the area in recent years.
It is remarkable that after nearly 20 years of very little progress on the conjectures, in only a few years a number of different results have been achieved in different directions by many different authors. This meeting hopes to bring these disparate results together in one place, where they can be discussed in unison and thus spawn further progress!
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 parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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