Forthcoming events in this series
Let $E/k$ be an elliptic curve over a number field and $K/k$ a Galois extension with group $G$. What can we say about $E(K)$ as a Galois module? Not just what complex representations appear, but its structure as a $\mathbb{Z}[G]$-module. We will look at some examples with small $G$.
Guided by the analogy with certain moments of the Bessel function that appear as Feynman integrals, Broadhurst and Roberts recently studied a family of L-functions built up by assembling symmetric power moments of Kloosterman sums over finite fields. I will prove that these L-functions arise from potentially automorphic motives over the field of rational numbers, and hence admit a meromorphic continuation to the complex plane that satisfies the expected functional equation. If time permits, I will identify the periods of the corresponding motives with the Bessel moments and make a few comments about the special values of the L-functions. This is a joint work with Claude Sabbah and Jeng-Daw Yu.
I will discuss work in progress with Sary Drappeau and Maksym Radziwill on low-lying zeros of Dirichlet L-functions. By way of motivation I will discuss some results on the spacings of zeros of the Riemann zeta function, and the conjectures of Katz and Sarnak relating the distribution of low-lying zeros of L-functions to eigenvalues of random matrices. I will then describe some ideas behind the proof of our theorem.
We report on a line of work initiated by Dan-Cohen and Wewers and continued by Dan-Cohen and the speaker to explicitly compute the zero loci arising in Kim's non-abelian Chabauty's method. We explain how this works, an important step of which is to compute bases of a certain motivic Hopf algebra in low degrees. We will summarize recent work by Dan-Cohen and the speaker, extending previous computations to $\mathbb{Z}[1/3]$ and proposing a general algorithm for solving the unit equation. Many of the methods in the more recent work are inspired by recent ideas of Francis Brown. Finally, we indicate future work, in which we hope to use elliptic motivic periods to explicitly compute points on punctured elliptic curves and beyond.
We compute the asymptotics of Arakelov functions if smooth curves degenerate to semistable singular curves. The motivation was to determine whether the delta function defines a metric on the boundary of moduli space. In fact things are slightly more complicated. The main result states that the asymptotics is mostly governed by the graph associated to the degeneration, with some subleties. The topic has been also treated by R. deJong and my student R. Wilms.
For a family of proper hyperbolic complex curves $f: X \longrightarrow \Delta^*$ over a puntured disc $\Delta^*$ with semistable reduction at the center, Oda proved, with transcendental methods, that the outer monodromy action of $\pi_1(\Delta^*) \cong \mathbb{Z}$ on the classical unipotent fundamental group of the generic fiber of $f$ is trivial if and only if $f$ has good reduction at the center. In this talk I explain a joint work with B. Chiarellotto and A. Shiho in which we give a purely algebraic proof of Oda's result.
I will discuss the arithmetic significance of Fourier expansions of modular forms at cusps. I will talk about joint work with F. Brunault, where we determine the number field generated by Fourier coefficients of newforms at a cusp. Then I will discuss joint work with A. Saha and K. Česnavičius where we find denominator bounds for Fourier expansions at cusps and apply these bounds to a conjecture on the Manin constants of elliptic curves.
In 1983, Erdos and Szemerédi conjectured that for any finite subset of the integers, either the sumset or the product set has nearly quadratic growth. Applications include incidence geometry, exponential sums, compressed image sensing, computer science, and elsewhere. We discuss recent progress towards the main conjecture and related questions.
We discuss a nonlinear variant of Roth’s theorem on the existence of three-term progressions in dense sets of integers, focusing on an effective version of such a result. This is joint work with Sarah Peluse.
We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve". In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.
Berry's random wave conjecture is a heuristic that the eigenfunctions of a classically ergodic system ought to display Gaussian random behaviour, as though they were random waves, in the large eigenvalue limit. We discuss two manifestations of this conjecture for eigenfunctions of the Laplacian on the modular surface: Planck scale mass equidistribution, and an asymptotic for the fourth moment. We will highlight how the resolution of these two problems in this number-theoretic setting involves a delicate understanding of the behaviour of certain families of L-functions.
After a brief introduction to the theory of transcendental numbers, I will discuss Nesterenko's 1996 celebrated theorem on the algebraic independence of values of Eisenstein series, and some related open problems. This motivates the second part of the talk, in which I will report on a recent geometric generalization of Nesterenko's method.
I will talk about recent work with Jack Thorne in which we find the average size of the Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type E_8. I will focus on the role representation theory plays in our proofs.
When S is a finite set of natural numbers, a GCD-sum is a particular kind of double-sum over the elements of S, and they arise naturally in several settings. In particular, these sums play a role when one studies the local statistics of point sequences on the unit circle. There are known upper bounds for the size of a GCD-sum in terms of the size of the set S, most recently due to de la Bretèche and Tenenbaum, and these bounds are sharp. Yet the known examples of sets S for which the GCD-sum over S provides a matching lower bound all possess strong multiplicative structure, whereas in applications the set S often comes with additive structure. In this talk I will describe recent joint work with Thomas Bloom in which we apply an estimate from sum-product theory to prove a much stronger upper bound on a GCD-sum over an additively structured set. I will also describe an application of this improvement to the study of the distribution of points on the unit circle, with a further application to arbitrary infinite subsets of squares.
The Asai (or twisted tensor) L-function attached to a Bianchi modular form is the 'restriction to the rationals' of the standard L-function. Introduced by Asai in 1977, subsequent study has linked its special values to the arithmetic of the corresponding form. In this talk, I will discuss joint work with David Loeffler in which we construct a p-adic Asai L-function -- that is, a measure on Z_p* that interpolates the critical values L^As(f,chi,1) -- for ordinary weight 2 Bianchi modular forms. We use a new method for constructing p-adic L-functions, using Kato's system of Siegel units to build a 'Betti analogue' of an Euler system, building on algebraicity results of Ghate. I will start by giving a brief introduction to p-adic L-functions and Bianchi modular forms, and if time permits, I will briefly mention another case where the method should apply, that of non-self-dual automorphic representations for GL(3).
Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. I'll discuss some recent refinements of Gallagher's theorem, one of which is joint work with Niclas Technau. A key new ingredient is the correspondence between Bohr sets and generalised arithmetic progressions. It is hoped that these are the first steps towards a metric theory of multiplicative diophantine approximation on manifolds.
Given a "random" polynomial over the integers, it is expected that, with high probability, it is irreducible and has a big Galois group over the rationals. Such results have been long known when the degree is bounded and the coefficients are chosen uniformly at random from some interval, but the case of bounded coefficients and unbounded degree remained open. Very recently, Emmanuel Breuillard and Peter Varju settled the case of bounded coefficients conditionally on the Riemann Hypothesis for certain Dedekind zeta functions. In this talk, I will present unconditional progress towards this problem, joint with Lior Bary-Soroker and Gady Kozma.
Seminal work of Hida tells us that if a modular eigenform is ordinary at p then we can always find other eigenforms, of different weights, that are congruent to our given form. Even better, it says that we can find q-expansions with coefficients in p-adic analytic function of the weight variable k that when evaluated at positive integers give the q-expansion of classical eigenforms. His construction of these families uses mainly the geometry of the modular curve and its ordinary locus.
In a joint work with Marc-Hubert Nicole, we obtained similar results for Drinfeld modular forms over function fields. After an extensive introduction to Drinfeld modules, their moduli spaces, and Drinfeld modular forms, we shall explain how to construct Hida families for ordinary Drinfeld modular forms.
Multiple zeta values are generalizations of the special values of Riemann's zeta function at positive integers. They satisfy a large number of algebraic relations some of which were already known to Euler. More recently, the interpretation of multiple zeta values as periods of mixed Tate motives has led to important new results. However, this interpretation seems insufficient to explain the occurrence of several phenomena related to modular forms.
The aim of this talk is to describe an analog of multiple zeta values for complex elliptic curves introduced by Enriquez. We will see that these define holomorphic functions on the upper half-plane which degenerate to multiple zeta values at cusps. If time permits, we will explain how some of the rather mysterious modular phenomena pertaining to multiple zeta values can be interpreted directly via the algebraic structure of their elliptic analogs.
The celebrated Lang-Vojta Conjecture predicts degeneracy of S-integral points on varieties of log general type defined over number fields. It admits a geometric analogue over function fields, where stronger results have been obtained applying a method developed by Corvaja and Zannier. In this talk, we present a recent result for non-isotrivial surfaces over function fields dominating a two-dimensional torus. This extends Corvaja and Zannier’s result in the isotrivial case and it is based on a refinement of gcd estimates for polynomials evaluated at S-units. This is a joint work with A. Turchet.
In 1973, Serre observed that the Hecke eigenvalues of Eisenstein series can be p-adically interpolated. In other words, Eisenstein series can be viewed as specializations of a p-adic family parametrized by the weight. The notion of p-adic variations of modular forms was later generalized by Hida to include families of ordinary cuspforms. In 1998, Coleman and Mazur defined the eigencurve, a rigid analytic space classifying much more general p-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is well-understood at points corresponding to cuspforms of weight k ≥ 2, while the weight one case is far more intricate.
In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations. Via this Galois-theoretic method, we obtain the q-expansion of some non-classical overconvergent forms in terms of p-adic logarithms of p-units in certain number field. Finally, we will explain how these calculations suggest a different approach to the Gross-Stark conjecture.
I will give an overview of some recent progress on potential automorphy results over CM fields, that is joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne. I will focus on explaining an application to the generalized Ramanujan-Petersson conjecture.
For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down down to -X for which the class group has trivial (non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss recent refinements of these classic results in which we consider the imaginary quadratic fields whose class number is indivisible (divisible) by p such that a given finite set of primes factor in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups satisfying almost any given finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.
Let F be a totally real or CM number field. Scholze has constructed Galois representations associated with torsion classes in the cohomology of locally symmetric spaces for GL_n(F). We show that the nilpotent ideal appearing in Scholze's construction can be removed when F splits completely at the relevant prime. As a key component of the proof, we show that the compactly supported cohomology of certain unitary and symplectic Shimura varieties with level Gamma_1(p^{\infty}) vanishes above the middle degree. This is joint work with Ana Caraiani, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih.
Understanding how shifts of multiplicative functions correlate with each other is a central question in multiplicative number theory. A well-known conjecture of Elliott predicts that there should be no correlation between shifted multiplicative functions unless the functions involved are ‘pretentious functions’ in a certain precise sense. The Elliott conjecture implies as a special case the famous Chowla conjecture on shifted products of the Möbius function.
In the last few years, there has been a lot of exciting progress on the Chowla and Elliott conjectures, and we give an overview of this. Nearly all of the previously obtained results have concerned correlations that are weighted logarithmically, and it is an interesting question whether one can remove these logarithmic weights. We show that one can indeed remove logarithmic averaging from the known results on the Chowla and Elliott conjectures, provided that one restricts to almost all scales in a suitable sense.
This is joint work with Terry Tao.
We will see some results and conjectures on the zeta and multizeta values in the function field context, and see how they relate to homological-homotopical objects, such as t-motives, iterated extensions, and to Hopf algebras, big Galois representations.
In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has only been proved for degrees d=1, 2, 3. For each d we prove the hyperbolicity of all but (perhaps) finitely many Jensen polynomials. We obtain a general theorem which models such polynomials by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function. This result can be thought of as a proof of GUE for the Riemann zeta function in derivative aspect. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.
Manin’s conjecture predicts the asymptotic behavior of the number of rational points of bounded height on Fano varieties over number fields. We prove this conjecture for a family of nonsplit singular quartic del Pezzo surfaces over arbitrary number fields. For the proof, we parameterize the rational points on such a del Pezzo surface by integral points on a nonuniversal torsor (which is determined explicitly using a Cox ring of a certain type), and we count them using a result of Barroero-Widmer on lattice points in o-minimal structures. This is joint work in progress with Marta Pieropan.
Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.
Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.
We will discuss a basic framework to deal with coherent sheaves on schemes over $\mathbb{Z}$, involving infinite-dimensional results on the geometry of numbers. As an application, we will discuss basic results, old and new, on arithmetic ampleness, such as Serre vanishing, Nakai-Moishezon, and Bertini. This is joint work with Jean-Benoît Bost.
We will briefly revisit Voronoi summation in its classical form and mention some of its many applications in number theory. We will then show how to use the global Whittaker model to create Voronoi type formulae. This new approach allows for a wide range of weights and twists. In the end we give some applications to the subconvexity problem of degree two $L$-functions.
It is an old problem in number theory to count number fields of a fixed degree and having a fixed Galois group for its Galois closure, ordered by their absolute discriminant, say. In this talk, I shall discuss some background of this problem, and then report a recent work with Stanley Xiao. In our paper, we considered quartic $D_4$-fields whose ring of integers has a certain nice algebraic property, and we counted such fields by their conductor.
Exponents of Diophantine approximation are defined to study specific sets of real numbers for which Dirichlet's pigeonhole principle can be improved. Khintchine stated a transference principle between the two exponents in the cases of simultaneous approximation and approximation by linear forms. This shows that exponents of Diophantine approximation are related, and these relations can be studied via so called spectra. In this talk, we provide an optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation for both simultaneous approximation and approximation by linear forms. This is joint work with Nikolay Moshchevitin.
Let $P$ be a random polynomial of degree $d$ such that the leading and constant coefficients are 1 and the rest of the coefficients are independent random variables taking the value 0 or 1 with equal probability. Odlyzko and Poonen conjectured that $P$ is irreducible with probability tending to 1 as $d$ grows. I will talk about an on-going joint work with Emmanuel Breuillard, in which we prove that GRH implies this conjecture. The proof is based on estimates for the mixing time of random walks on $\mathbb{F}_p$, where the steps are given by the maps $x \rightarrow ax$ and $x \rightarrow ax+1$ with equal probability.
Let $f_1,\dots,f_k$ be real polynomials with no constant term and degree at most $d$. We will talk about work in progress showing that there are integers $n$ such that the fractional part of each of the $f_i(n)$ is very small, with the quantitative bound being essentially optimal in the $k$-aspect. This is based on the interplay between Fourier analysis, Diophantine approximation and the geometry of numbers. In particular, the key idea is to find strong additive structure in Fourier coefficients.
There is a conjecture by Colliot-Thelene (about 2005) that under specific hypotheses, a morphism of Q-varieties f : X --> Y has the property that for almost all prime numbers p, the corresponding map X(Q_p) --> Y(Q_p) is surjective. A sharpening of the conjecture was solved by Denef (2016), and later, "if and only if" conditions on f were given by Skorobogatov et al. The plan for the talk is to explain in detail the conjecture and the results mentioned above, and to report on work in progress on a different method to attack the conjecture under quite relaxed hypotheses.
The construction of permutation functions of a finite field is a task of great interest in cryptography and coding theory. In this talk we describe a method which combines Chebotarev density theorem with elementary group theory to produce permutation rational functions over a finite field F_q. Our method is entirely constructive and as a corollary we get the classification of permutation polynomials up to degree 4 over any finite field of odd characteristic.
This is a joint work with Andrea Ferraguti.
After reviewing old work with Teitelbaum, in which we constructed the character variety X of the additive group o_L in a finite extension L/Q_p and established the Fourier isomorphism for the distribution algebra of o_L, I will briefly report on more recent work with Berger and Xie, in which we establish the theory of (\varphi_L,\Gamma_L)-modules over X and relate it to Galois representations. Then I will discuss an ongoing project with Venjakob. Our goal is to use this theory over X for Iwasawa theory.
I will give a gentle introduction to joint work in progress with George Boxer, Frank Calegari, and Vincent Pilloni, in which we prove that all abelian surfaces over totally real fields are potentially modular. We also prove that infinitely many abelian surfaces over Q are modular.
I will talk about some recent work with Chantal David and Matilde Lalin about the mean value of L-functions associated to cubic characters over F_q[t] when q=1 (mod 3). I will explain how to obtain an asymptotic formula with a (maybe a little surprising) main term, which relies on using results from the theory of metaplectic Eisenstein series about cancellation in averages of cubic Gauss sums over functions fields.
Serre's uniformity question concerns the possible ways the Galois group of Q can act on the p-torsion of an elliptic curve over Q. In this talk I will survey what is known about this question, and describe two recent results related to the Chabauty-Kim method. The first, which is joint work with Jennifer Balakrishnan, Steffen Muller, Jan Tuitman and Jan Vonk, completes the classification of elliptic curves over Q with split Cartan level structure. The second, which is work in progress with Samuel Le Fourn, Samir Siksek and Jan Vonk, concerns the applicability of the Chabauty-Kim method in determining the elliptic curves with non-split Cartan level structure.
Let L be a lattice in R^n and let Z in R^(m+n) a parameterized family of subsets Z_T of R^n. Starting from an old result of Davenport and using O-minimal structures, together with Martin Widmer, we proved for fairly general families Z an estimate for the number of points of L in Z_T, which is essentially best possible.
After introducing the problem and stating the result, we will present applications to counting algebraic integers of bounded height and to Manin’s Conjecture.
Mazur observed that in many cases where an elliptic curve E has a non-trivial element C in its Tate-Shafarevich group, one can find another elliptic curve E' such that ExE' admits an isogeny that kills C. For elements of order 2 and 3 one can prove that such an E' always exists. However, for order 4 this leads to a question about rational points on certain K3-surfaces. We show how to explicitly construct these surfaces and give some results on their rational points.
This is joint work with Tom Fisher.
The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.
In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, and also posed some questions: how many cusp forms of a given level are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Preston Wake, we give an answer to these questions in terms of cup products (and Massey products) in Galois cohomology. Time permitting, we may be able to indicate some partial generalisations of Mazur's results to square-free level.
We count monic quartic polynomials with prescribed Galois group, by box height. Among other things, we obtain the order of magnitude for quartics, and show that non-quartics are dominated by reducibles. Tools include the geometry of numbers, diophantine approximation, the invariant theory of binary forms, and the determinant method. Joint with Rainer Dietmann.
l-adic cohomology was built to provide an etale cohomology with coefficients in a field of characteristic 0. This, via the Grothendieck trace formula, gives a cohomological interpretation of L-functions - a fundamental tool in Deligne's theory of weights developed in Weil II. Instead of l-adic coefficients one can consider coefficients in ultra products of finite fields. I will state the fundamental theorem of Weil II for curves in this setting and explain briefly what are the difficulties to overcome to adjust Deligne's proof. I will then discuss how this ultra product variant of Weil II allows to extend to arbitrary coefficients previous results of Gabber and Hui, Tamagawa and myself for constant $\mathbb{Z}_\ell$-coefficients. For instance, it implies that, in an $E$-rational compatible system of smooth $\overline{\mathbb{Q}}_\ell$-sheaves all what is true for $\overline{\mathbb{Q}}_\ell$-coefficients (semi simplicity, irreducibility, invariant dimensions etc) is true for $\overline{\mathbb{F}}_\ell$-coefficients provided $\ell$ is large enough or that the $\overline{\mathbb{Z}}_\ell$-models are unique with torsion-free cohomology provided $\ell$ is large enough.
A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Efthymios Sofos.
This talk will report on the definition of some motivic cohomology classes and the proof that they satisfy the norm relations expected of Euler systems, emphasizing a connection with the local Gan-Gross-Prasad conjecture.