Past Number Theory Seminar

In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension (the genus of the Siegel forms) but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidality" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidality we want to consider: the more non-cuspidal the forms, the smaller the dimension. This is a joint work with Riccardo Brasca.

The talk will be based on some of the material in the joint survey with Etienne Ghys

"Signatures in algebra, topology and dynamics"

http://arxiv.org/abs/1512.092582

In the 19th century Sturm's theorem on the number of roots of a real polynomial motivated Sylvester to define the signature of a quadratic form. In the 20th century the classification of quadratic forms over algebraic number fields motivated Witt to introduce the "Witt groups" of stable isomorphism classes of quadratic forms over arbitrary fields. Still in the 20th century the study of high-dimensional topological manifolds with nontrivial fundamental group motivated Wall to introduce the "Wall groups" of stable isomorphism classes of quadratic forms over arbitrary rings with involution. In our survey we interpreted Sturm's theorem in terms of the Witt-Wall groups of function fields. The talk will emphasize the common thread running through this developments, namely the notion of the localization of a ring inverting elements. More recently, the Cohn localization of inverting matrices over a noncommutative ring has been applied to topology in the 21st century, in the context of the speaker's algebraic theory of surgery.

We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.

I will describe how the moduli of various congruences between Hecke eigenvalues of automorphic forms ought to show up in ratios of critical values of $\text{GSP}_2 \times \text{GL}_2$ L-functions. To test this experimentally requires the full force of Farmer and Ryan's technique for approximating L-values given few coefficients in the Dirichlet series.

I will discuss the notion of badly approximable points and recent progress and problems in this area, including Schmidt's conjecture, badly approximable points on manifolds and real numbers badly approximable by algebraic numbers.

I will describe some diophantine problems and results motivated by the analogy between powers of the modular curve and powers of the multiplicative group in the context of the Zilber-Pink conjecture.

TBA

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. 
Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on  Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

Iwasawa theory is a powerful technique for relating the behaviour of arithmetic objects to the special values of L-functions. Iwasawa originally developed this theory in order to study the class groups of number fields, but it has since been generalised to many other settings. In this talk, I will discuss some new results in the Iwasawa theory of the symmetric square of a modular form. This is a joint project with Sarah Zerbes, and the main tool in this work is the Euler system of Beilinson-Flach elements, constructed in our earlier works with Kings and Lei.

Given a family $F$ of elliptic curves defined over $Q$, we are interested in the set $H(Y)$ of curves $E$ in $F$, of positive rank, and for which the minimum of the canonical heights of non-torsion rational points on $E$ is bounded by some parameter $Y$. When one can show that this set is finite, it is natural to investigate statistical properties of arithmetic objects attached to elliptic curves in the set $H(Y)$. We will describe some problems related to this, and will state some results in the case of families of quadratic twists of a fixed elliptic curve.

We discuss moments of $L$-functions in function fields, in the hyperelliptic ensemble, focusing on the fourth moment of quadratic Dirichlet $L$-functions at the critical point. We explain how to obtain an asymptotic formula with some of the secondary main terms.

A period is a certain type of number obtained by integrating algebraic differential forms over algebraic domains. Examples include pi, algebraic numbers, values of the Riemann zeta function at integers, and other classical constants.
Difficult transcendence conjectures due to Grothendieck suggest that there should be a Galois theory of periods.
I will explain these notions in very introductory terms and show how to set up such a Galois theory in certain situations.
I will then discuss some applications, in particular to Kim's method for bounding $S$-integral solutions to the equation $u+v=1$, and possibly to high-energy physics.

The number of steps required by the Euclidean algorithm to find the greatest common divisor of a pair of integers $u,v$ with $1<u<v<n$ has been investigated since at least the 16th century, with an asymptotic for the mean number of steps being found independently by H. Heilbronn and J.D. Dixon in around 1970. It was subsequently shown by D. Hensley in 1994 that the number of steps asymptotically follows a normal distribution about this mean. Existing proofs of this fact rely on extensive effective estimates on the Gauss-Kuzman-Wirsing operator which run to many dozens of pages. I will describe how this central limit theorem can be obtained instead by a much shorter Tauberian argument. If time permits, I will discuss some related work on the number of steps for the binary Euclidean algorithm.

The prime number race is the competition between different coprime residue classes mod $q$ to contain the most primes, up to a point $x$ . Rubinstein and Sarnak showed, assuming two $L$-function conjectures, that as $x$ varies the problem is equivalent to a problem about orderings of certain random variables, having weak correlations coming from number theory. In particular, as $q \rightarrow \infty$ the number of primes in any fixed set of $r$ coprime classes will achieve any given ordering for $\sim 1/r!$ values of $x$. In this talk I will try to explain what happens when $r$ is allowed to grow as a function of $q$. It turns out that one still sees uniformity of orderings in many situations, but not always. The proofs involve various probabilistic ideas, and also some harmonic analysis related to the circle method. This is joint work with Youness Lamzouri.

I will discuss some new results on the Iwasawa theory for the $3$-dimensional symmetric square Galois representation of a modular form, using the Euler system of Beilinson-Flach elements I constructed in joint work with Kings, Lei and Loeffler.

The Möbius function plays a central role in number theory; both the prime number theorem and the Riemann Hypothesis are naturally formulated in terms of the amount of cancellations one gets when summing the Möbius function. In a recent joint work with Maksym Radziwill we have shown that the sum of the Möbius function exhibits cancellation in "almost all intervals" of arbitrarily slowly increasing length. This goes beyond what was previously known conditionally on the Riemann Hypothesis. Our result holds in fact in much greater generality, and has several further applications, some of which I will discuss in the talk. For instance the general result implies that between a fixed number of consecutive squares there is always an integer composed of only "small" prime factors. This settles a conjecture on "smooth" or "friable" numbers and is related to the running time of Lenstra's factoring algorithm.

In contrast to the Artin-Schreier Theorem, its $p$-adic analog(s) involve infinite Galois theory, e.g., the absolute Galois group of $p$-adic fields.  We plan to give a characterization of $p$-adic $p$-Henselian valuations in an essentially finite way. This relates to the $Z/p$ metabelian form of the birational $p$-adic Grothendieck section conjecture.

I will explain the formulation and proof of Arthur's multiplicity formula for automorphic representations of special orthogonal groups and certain inner forms of symplectic groups $G$ over a number field $F$. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set $S$ of real places of $F$ such that $G$ has discrete series at places in $S$ and is quasi-split at places outside $S$, and restricting to automorphic representations of $G(A_{F})$ which have algebraic regular infinitesimal character at the places in $S$. In particular, this proves the general multiplicity formula for groups $G$ such that $F$ is totally real, $G$ is compact at all real places of $F$ and quasi-split at all finite places of $F$. Crucially, the formulation of Arthur's multiplicity formula is made possible by Kaletha's recent work on local and global Galois
gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors.

Solving systems of linear equations $Ax=b$ is easy, but how can we solve such a system when given a "noisy" version of $b$? Over the reals one can use the least squares method, but the problem is harder when working over a finite field. Recently this subject has become very important in cryptography, due to the introduction of new cryptosystems with interesting properties.

The talk will survey work in this area. I will discuss connections with coding theory and cryptography. I will also explain how Fourier analysis in finite groups can be used to solve variants of this problem, and will briefly describe some other applications of Fourier analysis in cryptography. The talk will be accessible to a general mathematical audience.

In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramaré, Elkies and many others, we complete the proof of Jacobi's observation.

The talk will discuss the mean value theorem and Wooley's breakthrough with his "efficent congruencing" method.

I will discuss joint work with Ana Caraiani, Matthew Emerton and David Savitt, in which we construct moduli stacks of two-dimensional potentially Barsotti-Tate Galois representations, and study the relationship of their geometry to the weight part of Serre's conjecture.

I describe joint work with Alastair Irving in which we improve a result of
D.H.J. Polymath on the length of intervals in $[N,2N]$ that can be shown to
contain $m$ primes. Here $m$ should be thought of as large but fixed, while $N$
tends to infinity.
The Harman sieve is the key to the improvement. The preprint will be
available on the Math ArXiv before the date of the talk.

 Let $X$ be a smooth cubic hypersurface of dimension $m$ defined over a global field $K$. A conjecture of Colliot-Thelene(02) states that $X$ satisfies the Hasse Principle and Weak approximation as long as $m\geq 3$. We use a global version of Hardy-Littlewood circle method along with the theory of global $L$-functions to establish this for $m\geq 6$, in the case $K=\mathbb{F}_q(t)$, where $\text{char}(\mathbb{F}_{q})> 3$.

Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt.

**Joint seminar with Logic. 

I will discuss some recent results on the distribution of the real-analytic Eisenstein series on thin sets, such as a geodesic segment. These investigations are related to mean values of the Riemann zeta function, and have connections to quantum chaos.

The Cohen-Lenstra heuristics, postulated in the early 80s, conceptually explained numerous phenomena in the behaviour of ideal class groups of number fields that had puzzled mathematicians for decades, by proposing a probabilistic model: the probability that the class group of an imaginary quadratic field is isomorphic to a given group $A$ is inverse proportional to $\#\text{Aut}(A)$. This is a very natural model for random algebraic objects. But the probability weights for more general number fields, while agreeing well with experiments, look rather mysterious. I will explain how to recover the original heuristic in a very conceptual way by phrasing it in terms of Arakelov class groups instead. The main difficulty that one needs to overcome is that Arakelov class groups typically have infinitely many automorphisms. We build up a theory of commensurability of modules, of groups, and of rings, in order to remove this obstacle. This is joint work with Hendrik Lenstra.

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution. Joint work with Francesco Cellarosi.

I will review some classical problems in number theory concerning the statistical distribution of the primes, square-free numbers and values of the divisor function; for example, fluctuations in the number of primes in short intervals and in arithmetic progressions.  I will then explain how analogues of these problems in the function field setting can be resolved by expressing them in terms of matrix integrals. 

I will describe joint work with Manjul Bhargava (Princeton) and Tom Fisher (Cambridge) in which we determine the probability that random equation from certain families  has a solution either locally (over the reals or the p-adics), everywhere locally,  or globally. Three kinds of equation will be considered: quadratics in any number of variables, ternary cubics and hyperelliptic quartics.

Thanks to the p-adic local Langlands correspondence for GL_2(Q_p), one "knows" all admissible unitary topologically irreducible representations of GL_2(Z_p). In this talk I will focus on some elementary properties of their restriction to GL_2(Z_p): for instance, to what extent does the restriction to GL_2(Z_p) allow one to recover the original representation, when is the restriction of finite length, etc.

In sieve theory, one is concerned with estimating the size of a sifted set, which avoids certain residue classes modulo many primes. For example, the problem of counting primes corresponds to the situation when the residue class 0 is removed for each prime in a suitable range. This talk will be concerned about what happens when a positive proportion of residue classes is removed for each prime, and especially when this proporition is more than a half. In doing so we will come across an algebraic question: given a polynomial f(x) in Z[x], what is the average size of the value set of f reduced modulo primes?

Given an abelian variety A over a number field k, its Kummer variety X is the quotient of A by the automorphism that sends each point P to -P. We study p-adic density and weak approximation on X by relating its rational points to rational points of quadratic twists of A. This leads to many examples of K3 surfaces over Q whose rational points lie dense in the p-adic topology, or in product topologies arising from p-adic topologies. Finally, the same method is used to prove that if the Brauer--Manin obstruction controls the failure of weak approximation on all Kummer varieties, then ranks of quadratic twists of (non-trivial) abelian varieties are unbounded. This last fact arises from joint work with David Holmes.

In 1989, Selberg defined what came to be known as the "Selberg class" of $L$-functions, giving rise to a new subfield of analytic number theory in the intervening quarter century. Despite its popularity, a few things have always bugged me about the definition of the Selberg class. I will discuss these nitpicks and describe some modest attempts at overcoming them, with new applications.

Fix a prime $p$. In this talk, we will discuss the $p$-adic properties of the *coefficients* of the characteristic power series of $U_{p}$ acting on spaces of overconvergent $p$-adic modular forms. These coefficients are, by a theorem of Coleman, power series in the weight variable over $Z_{p}$.  Our first goal will be to show that in tame level one, the simplest case, every coefficient is non-zero mod $p$ and then to give some idea of the (finitely many) roots of each coefficient. The second goal will be to explain how it the previous result fails in higher levels, along with possible salvages. This will include revisiting the tame level one case. The progress we've made has applications, and lends understanding, to recent work being made elsewhere on the geometric structure of the eigencurve "near its boundary". This is joint work with Rob Pollack.

Rational points on Kummer varieties can be studied through the variation of Selmer groups of quadratic twists of the underlying abelian variety, using an idea of Swinnerton-Dyer. We consider the case when the Galois action on 2-torsion has a large image. Under a mild additional assumption we prove the Hasse principle assuming the finiteness of relevant Shafarevich-Tate groups. This approach is inspired by the work of Mazur and Rubin.

The classical conjecture of Serre (proved by Khare-Winterberger) states that a continuous, absolutely irreducible, odd representation of the absolute Galois group of Q on two-dimensional F_p-vector space is modular. We show how one can formulate its analogue in characteristic 0. In particular we discuss the weight part of the conjecture. This is a joint work with John Bergdall.

In 1953 Roth proved that any positive density subset of the integers contains a non-trivial three term arithmetic progression. I will present a recent quantitative improvement for this theorem, give an overview of the main ideas of the proof, and discuss its relation to other recent work in the area. I will also discuss some closely related problems. 

We discuss a new method to bound the number of primes in certain very thin sets. The sets $S$ under consideration have the property that if $p\in S$ and $q$ is prime with $q|(p-1)$, then $q\in S$. For each prime $p$, only 1 or 2 residue classes modulo $p$ are omitted, and thus the traditional small sieve furnishes only the bound $O(x/\log^2 x)$ (at best) for the counting function of $S$. Using a different strategy, one related to the theory of prime chains and Pratt trees, we prove that either $S$
contains all primes or $\# \{p\in S : p\le x \} = O(x^{1-c})$ for some positive $c$. Such sets arise, for example, in work on Carmichael's conjecture for Euler's function.

Let f be an elliptic modular newform of weight at least 2. The 
problem of the automorphy of the symmetric power L-functions of f is a 
key example of Langlands' functoriality conjectures. Recently, the 
potential automorphy of these L-functions has been established, using 
automorphy lifting techniques, and leading to a proof of the Sato-Tate 
conjecture. I will discuss a new approach to the automorphy of these 
L-functions that shows the existence of Sym^m f for m = 1,...,8.

I'll discuss work (part with Savitt, part with Dembele and Roberts) on two related questions: describing local factors at primes over p in mod p automorphic representations, and describing reductions of local crystalline Galois representations with prescribed Hodge-Tate weights.

In this seminar I will discuss a function field analogue of classical problems in analytic number theory, concerning the auto-correlations of divisor functions, in the limit of a large finite field.

In this talk, we will discuss the dimension of $J_0(N)[m]$ at an Eisenstein prime m for
square-free level N. We will also study the structure of $J_0(N)[m]$ as a Galois module.
This work generalizes Mazur’s work on Eisenstein ideals of prime level to the case of
arbitrary square-free level up to small exceptional cases.

If X/F is a smooth and proper variety over a global function field of

characteristic p, then for all l different from p the co-ordinate ring of the l-adic

unipotent fundamental group is a Galois representation, which is unramified at all

places of good reduction. In this talk, I will ask the question of what the correct

p-adic analogue of this is, by spreading out over a smooth model for C and proving a

version of the homotopy exact sequence associated to a fibration. There is also a

version for path torsors, which enables me to define an function field analogue of

the global period map used by Minhyong Kim to study rational points.

For the last few years Soundararajan and I have been developing an alternative "pretentious" approach to analytic number theory. Recently Harper established a more intuitive proof of Halasz's Theorem, the key result in the area, which has allowed the three of us to provide new (and somewhat simpler) proofs to several difficult theorems (like Linnik's Theorem), as well as to suggest some new directions. We shall review these developments in this talk.