Questions I'm thinking about

This page was split off from my notes for potential students to make it easier to update, since the list of questions I'm thinking about varies more than my general attitude towards advising.

Computing zeta functions of varieties over finite fields (last update: 12 Dec 09)

To each algebraic variety over a finite field is associated its zeta function, a rational function which records the number of points on this variety over all finite extensions of the base field. For a variety of reasons (including some applications outside of number theory), much interest has arisen recently in the general problem of computing the zeta function of an explicitly specified variety. (This is to be distinguished from the problem of computing the zeta function of an implicitly specified variety, such as a moduli space, which is more easily described as the solution of some universal problem than as the zero set of a particular collection of polynomial equations.)

There are a variety of techniques that can be applied to this problem. Some techniques I am interested in include the following.

• For varieties with a great deal of extra structure (e.g., a large group of automorphisms), one can use trace formula techniques to reduce the zeta function computation to a much smaller point enumeration than one might have expected. This approach seems to be severly underutilized.
• For curves, one can apply generic group algorithms to determine the order (and incidentally also the structure) of the class group. This has been pursued recently by Andrew Sutherland.
• For some relatively simple classes of varieties (e.g., hyperelliptic curves), methods of p-adic cohomology have proven quite successful. There has been a pronounced uptick in activity in this area in the last few years, but there are still some interesting questions yet to be explored.

Valuation theory and other algebra (last update: 12 Dec 09)

I've spent some time fiddling with constructions in valuation theory, such as Hahn's "generalized power series". My curiosity is more than my knowledge in this area, so my questions here reflect as much ignorance as anything else.

• Can the Ax-Kochen theorem (on decidability of the first-order theory of p-adic fields) be extended to equal characteristic? The state of the art on this question is due to Kuhlmann, who showed that one definitely needs extra axioms; but one can hope (as I do) that Kuhlmann's axiom set is complete.
• I have a method for computing in the algebraic closure of the rational function field over a finite field, using finite automata and generalized power series. Does it actually work in practice? I can't tell. (There has been a tiny bit of experimental work on this; contact me for details.)
• What's new in the theory of local uniformization, and should it give me any reason to be optimistic about progress on the resolution of singularities problem in positive characteristic? (Michael Temkin has recently made tremendous progress on this question, using ideas from Berkovich's theory of nonarchimedean analytic spaces.)
• Solve the polyhedral problem that Abramovich and Karu ran up against in their work on semistable reduction (in characteristic zero).
• The relationship between singular values and eigenvalues for a product of matrices over a nonarchimedean is governed by the Horn inequalities. These provide a necessary condition, which has been shown to be sufficient using a nontrivial detour through representation theory (work of Klyachko, Knutson-Tau-Woodward, Speyer). Is a direct proof of sufficiency possible?

p-adic transcendence theory (last update: 16 Aug 05)

Classical transcendence theory is very hard. For instance, Kontsevich and Zagier have a general philosophy that predicts that algebraic relations between "periods" (those numbers that occur as integrals of rational algebraic differentials on varieties over a number field) only exist when they can be explained as a relation among the defining integrals. This philosophy predicts all of the standard transcendence results, like the Gelfond-Schneider theorem, but also predicts many more statements which we have little hope of proving anytime soon.

On the other hand, there is an analogue of the notion of periods for "t-motives", which are a peculiar function-field analogue of the category of motives (which in turn form a sort of "universal cohomology" of algebraic varieties). A recent paper of Anderson, Brownawell, and Papanikolas (Determination of the algebraic relations among special Gamma-values in positive characteristic, Annals of Mathematics 160 (2004), 237--313) proves a theorem for these periods which fulfills the entire Kontsevich-Zagier philosophy! Somehow the idea is that the Frobenius action gives you a much better grip on these periods than their archimedean analogues.

I've only briefly looked at this work, but the algebra in this paper seems strikingly similar to the algebra of Frobenius actions on crystals on varieties. This raises the question: if one considers p-adic periods on varieties, rather than t-motives, can one reproduce the results of Anderson-Brownawell-Papanikolas?

p-adic Hodge theory (last update: 12 Dec 09)

Hodge theory is the study of the special structure of those vector spaces, and families of vector spaces, which occur as the cohomology of algebraic varieties over the complex numbers. p-adic Hodge theory is the analogous thing for varieties over p-adic fields. It plays a vital role in a lot of recent work in number theory, such as the modularity of Galois representations (i.e., the continuation of Wiles's work on the Fermat problem).

Primarily through the work of Berger (and more recently Kisin), new techniques have appeared in p-adic Hodge theory that parallel techniques used in p-adic cohomology, like the local monodromy theorem for p-adic differential equations ("Crew's conjecture") and the theory of slope filtrations for Frobenius modules. However, these constructions share a defect with earlier constructions in p-adic Hodge theory: they display a certain ad hoc character, possibly due to the fact that they come essentially from Galois theoretic considerations, whereas ordinary Hodge theory has a more analytic origin.

I am interested in finding a point of view from which p-adic Hodge theory becomes more like ordinary Hodge theory; I am in particular hoping that systematic use of Witt vectors and de Rham-Witt constructions may give such a point of view. For number-theoretic applications (including extending Kisin's modularity theorems beyond the potentially Barsotti-Tate case), it would be useful to also be able to sensibly do "integral p-adic Hodge theory", which so far has proved quite difficult. Kisin's theory of sigma-modules makes a tremendous advance in this area.

Very recently, notions of relative p-adic Hodge theory have begun to emerge. There are actually two distinct forms of relative p-adic Hodge theory, one concerned with p-adic families of Galois representations (considered by Berger, Colmez, Bellaïche, Chenevier, Liu) and another concerned with representations of arithmetic fundamental groups (considered by Faltings, Andreatta, Brinon, Iovita, Hartl).