# 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).

After a long hiatus, I made a major update to this list in late 2017. I plan to add some additional topics later, including the following.

- Geometrization of
*p*-adic local Langlands after Fargues, Scholze. - Computational aspects of modular forms.

### 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.)

### Crystalline companions (last update: 28 Dec 17)

For varieties over a finite field of characteristic p, one has an l-adic Weil cohomology theory for each prime l, namely etale cohomology for l different from p and Berthelot's rigid cohomology (generalizing rational crystalline cohomology in the smooth proper case and Monsky-Washnitzer formal cohomology in the smooth affine case) for l equal to p. In each theory, one has a category of "smooth" (i.e., of locally constant rank) coefficient objects; these are the lisse Weil sheaves in etale cohomology and the overconvergent F-isocrystals in crystalline cohomology.

It is a major piece of unfinished business from the resolution of the Weil conjectures to establish that an individual coefficient object cannot exist in isolation, but must always come embedded in a "compatible system" of coefficients, one for each place of an algebraic closure of Q, which all have matching Frobenius traces at closed points. On curves, this is resolved by the existence of motivic origins of such coefficients arising from the global Langlands correspondence for function fields (for the group GL(n)), established in etale cohomology by Laurent Lafforgue and extended to rigid cohomology by Tomoyuki Abe. For higher-dimensional varieties, results of Deligne, Drinfeld, Abe-Esnault, and myself have led to some serious progress; for instance, any p-adic coefficient object gives rise to l-adic companions, but the converse is not yet known.

### 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).

### Sato-Tate groups of motives (last update: 27 Dec 17)

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.

*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?