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. Topics are sorted by the date of the last update (most recent updates coming first).

Effective bounds for relative class number one (last update: 1 Feb 24)

The strongest form of the relative class number one problem for number fields is the following: find all L/K where K is a totally real number field, L is a totally imaginary quadratic extension of K, and the ratio h_L/h_K (which in this situation is guaranteed to be an integer) is equal to 1. This problem is completely solved when K = Q (Gauss's class number one problem), [K:Q] = 2 (by various results), or L/Q is Galois (by Hoffman-Sircana, building on much previous work).

It is expected that there are only finitely many such extensions in all, even if we allow [K:Q] to be arbitrary. This finiteness is known conditionally (under GRH by Lee-Kwon, or under analytic continuation of all Artin L-functions for CM fields by Peng-Jie Wong). It is known unconditionally for fields with solvable Galois closure (by Wong), or more generally for fields of the form L/L_0/Q where L_0/Q is subnormal and L/L_0 has solvable Galois closure. It would be natural to try to determine the precise list of relative class number one fields under these conditions; a few special Galois groups have been settled already.

Censuses of varieties over small finite fields (last update: 1 Feb 24)

For various reasons, it would be useful to have exhaustive tables of certain types of algebraic varieties over small finite fields. For example, with Sutherland we made a census of the smooth quartic K3 surfaces over F_2 and their zeta functions; Auel, Kulkarni, Petok, and Weinbaum have made a census of cubic fourfolds over F_2.

One natural example is curves of low genus. Since the moduli space of curves of genus g is unirational for g up to at least 10, and of dimension 3g-3, it is conceivable to tabulate at least this far. However, one must come up with a fairly efficient mechanism for enumerating the curves to avoid generating too much overhead. Then one needs to optimize computation of the zeta functions. Over F_2 this has been recently completed in genus 6 and is ongoing in genus 7; further progress in this direction would require making explicit the "flowchart" for describing a canonical curve in terms of special divisors (and doing this over a finite field rather than an algebraically closed field).

Point counts on moduli spaces of curves (last update: 1 Feb 24)

While the stable cohomology of moduli spaces of curves is well-understood, our knowledge of the complete cohomology of moduli spaces of curves is limited even in low genus. For example, it was only recently shown that the (stacky) number of genus-6 curves over F_q is a polynomial in q (and even more recently the polynomial was computed).

One can ask similar questions for other loci in the moduli spaces, and with a few exceptions (such as the hyperelliptic locus) even less is known. For instance, the stacky number of trigonal curves over F_q is only known up to genus 5.

Drinfeld's lemma for (phi, Gamma)-modules (last update: 3 Oct 23)

In geometric Langlands, it is helpful for functoriality to be able to manipulate representations of finite powers of the relevant Galois group; various statements about such representations are grouped together under the umbrella name of "Drinfeld's lemma".

For the Galois group of a p-adic local field, one has a form of Drinfeld's lemma for etale multivariate (phi, Gamma)-modules as described by Carter-Kedlaya-Zábrádi; however, at the moment this is not known for general multivariate (phi, Gamma)-modules. On one hand, in the de Rham case we can deduce such a result from the multivariate p-adic local monodromy theorem (which implies a form of "de Rham -> potentially semistable" even in the non-étale case). On the other hand, there is no corresponding result for vector bundles on a power of a Fargues-Fontaine curve, which makes it less clear how to adapt ideas from the univariate case.

Relative class number problems for function fields (last update: 13 Jul 23)

We recently resolved the relative class number one problem for arbitrary extensions of function fields. However, this does not currently yield an effective lower bound on relative class numbers, which would be a welcome improvement. Adam Logan has further suggested trying to prove a finiteness result for extensions where the relative class group has fixed exponent (say exponent 2).

Strictly totally disconnected adic spaces (last update: 30 Jun 23)

While Scholze's theory of diamonds was originally built in terms of sheaves on the category of perfectoid spaces in characteristic p, it was subsequently noted (e.g., in "Etale cohomology of diamonds") that one can get by using a much more restricted family of test objects, namely the "strictly totally disconnected" adic spaces (terminology suggested by Bogdan Zavyalov). We say that a space is strictly totally disconnected if it is qcqs and each connected component is isomorphic to Spa(C, C^+) for some algebraically closed valued field C (the trivial valuation being allowed).

Can one use this setup to reproduce Huber's theory of etale cohomology of adic spaces in full generality by adapting Scholze's approach? By including the condition that C be algebraically closed in the definition of strictly totally disconnected, this forces the analogue of almost purity to hold for trivial reasons.

Arc-descent and isocrystals over (semi)perfect schemes (last update: 22 Oct 22)

It is known that isocrystals over perfect schemes of characteristic p obey descent for the arc-topology (Bhatt-Scholze, Ivanov). Can this be used to recover expected properties, such as semicontinuity for the Newton polygon and slope filtrations?

Better yet, can this be used to recover deep properties of isocrystals over smooth schemes of finite type such as full faithfulness of restriction to an open dense subscheme? This might require generalizing from perfect schemes to semiperfect schemes.

Arithmetic statistics of modular forms (last update: 11 Dec 20)

Fix a finite field F. Can one make a reasonable prediction about the distribution of modular forms with associated Galois representations mapping into various subgroups of GL_2(F)? (To make this a well-formed question, one should consider forms of level up to some bound, then take the limit as the bound goes to infinity.) The case of dihedral image can be related to Cohen-Lenstra heuristics; we examined the characteristic-2 case with Medvedovsky.

It should be relatively easy to collect numerical data about these questions by computing Hecke algebras using modular symbols (as implemented in several computer algebra systems).

Sato-Tate groups of motives (last update: 11 Dec 20)

Given a motive (i.e., a "piece" of the cohomology of an algebraic variety) over a number field, one gets an associated L-function by computing the trace of Frobenius at all places (including places of bad reduction and archimedean places). For example, this construction gives rise to Dirichlet L-functions, Dedekind zeta functions associated to number fields, Artin L-functions, L-functions associated to elliptic curves, L-functions associated to modular forms, and so on.

While it is rare for the Euler factors comprising these L-functions to have an "explicit" description, one can conjecturally give a good description of their average behavior. A basic example of this is the Chebotarev density theorem, which describes the average behavior of Euler factors of an Artin L-function in terms of an associated Galois group. An analogous statement, but whose proof lies much deeper, is the (now proved) Sato-Tate conjecture, which describes the average behavior of Euler factors of an elliptic curve L-function in terms of the group SU(2).

A general conjecture of Serre (which would follow from sufficient analytic continuation of certain L-functions) implies that the Euler factors of an arbitrary motivic L-function are controlled in aggregate by a certain compact Lie group. It is an interesting problem to classify the groups arising from various classes of motives. For example, with Fité, Rotger, and Sutherland we have done this for abelian surfaces; there are 52 distinct possibilities. With Fité and Sutherland, we have done likewise for abelian threefolds; there are 410 distinct possibilities.

Motives associated to A-hypergeometric systems (last update: 27 May 20)

Of late there has been much attention paid to families of motives over P^1 for which the associated Picard-Fuchs differential equation is hypergeometric. In particular, one can compute the L-functions of these motives at large scale, making them ripe for arithmetic investigations.

Hypergeometric equations are known to be a special case of A-hypergeometric systems in the sense of Gelfand-Kapranov-Zelevinsky. Are there similar families of motives associated to such systems (over projective spaces of suitable dimension)? If so, can their L-functions again be computed efficiently?

Rational points on modular curves (last update: 12 Oct 19)

In his paper "Modular curves and the Eisenstein ideal", Mazur exploits the fact that the modular curve X_0(N) of prime level N has Jacobian which projects onto a large abelian variety with finite Mordell-Weil group, in order to make a uniform statement about the Q-points of such curves. Subsequent work of Kamienny and Merel extends this to number fields, but only for the larger modular curve X_1(N); that is, Merel's theorem controls the presence of rational points of large prime order on elliptic curves over a number field (in terms of the degree of the number field), but not rational torsion subgroups of large prime order. Is it possible to prove such a bound if one excludes elliptic curves with complex multiplication, using either a Mazur-type method or the quadratic Chabauty method (to gain information from projection to an abelian variety with Mordell-Weil rank equal to its dimension)?

On a related note, for N prime, consider the modular curve X_{ns}^+(N) corresponding to a nonsplit Cartan subgroup of GL_2(Z/NZ). Given a CM point x and a prime number p, can one use quadratic Chabauty to show that x is the unique rational point in its mod-p residue disc?

Counting l-adic local systems (last update: 20 Aug 18)

Based on his proof of the Langlands correspondence for GL(2) over a function field with finite base field, Drinfeld gave a formula for the number of irreducible rank 2 l-adic local systems on a smooth proper curve over a finite field. Deligne has conjectured a vast generalization of this formula, essentially stating that local systems of arbitrary rank admit counts which behave like the counts of rational points on some scheme (or stack). Many results are known in this direction (by Deligne-Flicker, Flicker, Yu) using trace formula methods in the style of Drinfeld's original work (but availing of L. Lafforgue's extension of Drinfeld's work to GL(n) for arbitrary n).

The question here is to obtain similar results using a different approach: the fact that every coefficient object, in either l-adic (etale) or p-adic (crystalline) cohomology, extends to a compatible system (again a consequence of Lafforgue's work, as augmented by T. Abe to cover the p-adic case). This transforms the problem at hand into counting rational points on certain moduli spaces of vector bundles with connection on algebraic curves; this may shed some additional light on Deligne's conjectures.

The Riemann extension theorem for adic and perfectoid spaces (last update: 24 Apr 18)

The original Riemann extension theorem states that a holomorphic function on a punctured disc extends across the puncture if and only if it is bounded in some neighborhood of the puncture. A similar statement holds for a normal complex analytic variety minus a closed analytic subvariety.

In rigid analytic geometry, a corresponding statement for normal rigid spaces was proved by Bartenwerfer. Recently, a corresponding statement for perfectoid spaces has appeared in several important contexts, notably Scholze's study of perfectoid Siegel modular varieties (as applied to the construction of automorphic Galois representations) and the work of André and Bhatt on the direct summand conjecture in commutative algebra. Notably, at the cost of passing from ordinary to "almost" commutative algebra, one obtains a far more robust (under derived functors) statement with simpler hypotheses and a more streamlined proof.

Can this statement be used to recover, sharpen, and/or generalize Bartenwerfer's result?

Revisiting the Ax-Sen-Tate theorem (last update: 05 Feb 18)

The Ax-Sen-Tate theorem states that if K is a nonarchimedean field of mixed characteristics (0,p) with Galois group G and L is a completed algebraic closure of K, then the fixed subfield of L under G is equal to K; moreover, every element of K which is "nearly invariant", in the sense that it differs from each of its G-images by an element of norm at most t, is "near an invariant", in the sense that it differs from some element of K by an element of norm c*t for some absolute constant c (depending only on p, not on the element or even on K).

The bound given by Ax is almost surely not optimal; for K a discretely valued field with finite residue field, the optimal constant was found by Le Borgne. Can this be extended to general K?

Another formulation of the theorem is that H^0(G, O_L) = 0 and H^1(G, O_L) is annihilated by every element of O_K of norm at most c (for the same constant c as above). Can one say something about higher cohomology? One might be able to do this by proving that the groups H^i(G, O_L/O_K) are killed by every element of O_K of norm at most c_i, where c_i depends only on i and p.

It is likely that recent developments on perfectoid spaces, particularly the almost acyclicity of the integral structure sheaf, will be crucial here. For one thing, such results show that one can replace L with any perfectoid completion of a Galois extension of K and G with the corresponding Galois group.

Reproducibility in arithmetic geometry (last update: 28 Dec 17)

Often, important work in arithmetic geometry includes some computational component. Unfortunately, such computational components often are not easily reproduced by subsequent researchers, putting such work on shaky footing. One basic issue is that computations are often performed in an opaque manner, either because the user-level code is not available, or because the user-level code ultimately depends on a closed-source underlying system. The first issue is easily addressed; the second requires a deeper investment in open-source software for mathematics. There is also the third issue of durability of code: computer hardware and software both evolve rapidly and today's software may not run on tomorrow's hardware.

Mochizuki's approach to diophantine approximation (last update: 28 Dec 17)

At the heart of number theory lies a network of interrelated (and ultimately equivalent) problems in diophantine approximation, including the Masser-Oesterlé ABC conjecture, Szpiro's conjecture on conductors of elliptic curves, and (some special cases of) Vojta's conjectures motivated by Nevanlinna theory. In 2012, Shinichi Mochizuki announced a proof of these conjectures using a collection of rather exotic ideas derived very loosely from Grothendieck's principle of anabelian geometry. While (as of late 2017) Mochizuki's four papers on "inter-universal Teichmüller theory" (IUT) appear to have been accepted for publication, there has been little independent exposition of the central concepts, and most specialists (including myself) are not currently treating this proof as fully verified.

Resolving this impasse is obviously a question of pressing importance, albeit one which may not be advisable for a young researcher who needs to establish a research portfolio in order to secure a permanent academic position. One potential starting point would be to clarify the foundations of the subject: the nature of the argument is to derive diophantine inequalities by quantifying over certain "degrees of indeterminacy" in anabelian reconstruction, and it is possible that modern tools like homotopy type theory can be used to better pin down what "reconstruction" means in practice.

Arithmetic statistics (last update: 28 Dec 17)

The asymptotic counting of arithmetic structures has its origins in the work of Gauss, motivated by questions about the behavior of class groups of number fields. Aside from the results of Gauss (including the theory of genera of quadratic forms), the first significant result in this field is the theorem of Davenport-Heilbronn giving the asymptotic count of cubic number fields. Subsequent work of Cohen-Lenstra, Malle, Klueners, and especially Bhargava and his collaborators has established enough results for this branch of number theory to merit its own name (coined by Mazur on the occasion of an MSRI special program in 2011).

Within this topic, one theme of particular interest to me is the question of local-to-global principles for asymptotic formulae. One generally expects the count of a given class of number fields with some ramification-theoretic parameter (e.g., discriminant) bounded by X to have the asymptotic shape of a power of X times a power of log X (with exponents predictable by heuristics originally due to Malle, refined by Bhargava). In most cases, the leading coefficient in this asymptotic does not have any special form; however, there are a few cases where the leading coefficient is a product of local factors, which themselves can be computed by solving a (far easier) local counting problem. For instance, this is conjectured by Bhargava to hold for counting number fields of a fixed degree by discriminant (and proven by him for degrees up to 5); it is also known for abelian number fields by conductor (work of Wood) and for D_4-extensions counted by the conductor of the two-dimensional representation (work of Altug-Shankar-Varma-Wilson). One generally expects such behavior whenever the local counting formulas are sufficiently uniform; even figuring out when this occurs is a nontrivial problem in many cases. (Another class of problems that may be considered to be a form of arithmetic statistics is the classification of Sato-Tate groups; see separate topic.)

Foundations of nonarchimedean analytic geometry (last update: 28 Dec 17)

Recent advances in p-adic Hodge theory powered by the theory of perfectoid spaces (see separate topic) have prompted a closer look at the foundations of nonarchimedean analytic geometry. In prior formulations (work of Tate, Raynaud, Berkovich), these were limited to spaces locally of finite type over a field; however, it has become necessary to work with more general adic spaces (work of Huber, Fujiwara-Kato, Gabber-Ramero) for which certain foundational questions come to the fore. By analogy with the famous Scottish Book of problems in functional analysis, I am tracking questions on a web site called The Nonarchimedean Scottish Book.

The Chabauty-Kim approach to rational points on curves (last update: 27 Dec 17)

With the description of rational points on elliptic curves given by the Mordell-Weil theorem in mind, Mordell conjectured that a curve of genus at least 2 over a number field contains only finitely many rational points. This was first proved by Faltings, and then again by Vojta using different techniques. However, neither approach gives an effective procedure for determining the rational points on any given curve.

By contrast, it was shown by Chabauty that Mordell's conjecture holds whenever the Mordell-Weil rank of the Jacobian of the curve is less than the genus of the curve. Subsequent work of Coleman, Poonen, Stoll, et al. has turned this approach into a highly practical effective procedure.

Recently, inspired by ideas from gauge theory and from Grothendieck's concept of anabelian geometry, Kim has proposed a variant of Chabauty's method that allows for some weakening of the rank condition. This approach has scored some recent successes as a "second-line antibiotic" for determining rational points, notably on the modular curve X_s(13) by the work of Balakrishnan et al. It is an intriguing question whether the Chabauty-Kim method can be used to upgrade Mazur's method to complete the classification of rational points on modular curves (Serre's uniformity conjecture).

p-adic Hodge theory and perfectoid spaces (last update: 27 Dec 17)

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).

The past 30 years have seen several remarkable developments in p-adic Hodge theory. First, the pioneering work of Faltings fused the original ideas of Tate, the Fontaine-Wintenberger theory of norms, and additional geometric insight to make the crucial break into Fontaine's conjectures on comparison isomorphisms. Second, the theory of (phi, Gamma)-modules emerged out of the work of Fontaine, Colmez, Berger, Kisin, and others to provide more direct access to the "hidden structures" provided by Fontaine's theory of p-adic period rings. Third, the theory of perfectoid spaces emerged from the work of Kedlaya-Liu and Scholze to provide a drastically simplified foundation to the subject, leading to a torrent of new results. Fourth, the work of Bhatt-Morrow-Scholze extended our deep understanding of rational p-adic Hodge theory to the integral setting, further illuminating links with algebraic topology originally discovered by Hesselholt et al.

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).

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?