Past Number Theory Seminar

Let $S$ be a set of rational places of odd cardinality containing infinity and a rational prime $p$. We can associate to $S$ a Shimura curve $X$ defined over $\mathbb{Q}$. The Gross--Kohnen--Zagier theorem states that certain generating series of Heegner points of $X$ are modular forms of weight $3/2$ valued in the Jacobian of $X$. We will state this theorem and outline a new approach to proving it using the theory of $p$-adic uniformization and $p$-adic families of modular forms of half-integral weight. This is joint work with Lea Beneish, Henri Darmon, and Lennart Gehrmann.

The 12th of Hilbert's 23 problems posed in 1900 asks for an explicit description of abelian extensions of a given base field. Over the rationals, this is given by the exponential function, and over imaginary quadratic fields, by meromorphic functions on the complex upper half plane.  Darmon and Vonk's theory of rigid meromorphic cocycles, or "RM theory", includes conjectures giving a $p$-adic solution over real quadratic fields. These turn out to be closely linked to purely topological questions about intersections of geodesics in the upper half plane, and to $p$-adic deformations of Hilbert modular forms. I will explain an extension of results of Darmon, Pozzi and Vonk proving some of these conjectures, and some ongoing work concerning analogous results on Shimura curves.

One can get fairly good estimates for primes in short
intervals under the assumption of the Riemann Hypothesis. Weaker
estimates can be shown unconditionally by using a 'zero density
estimate' in place of the Riemann Hypothesis. These zero density
estimates are typically proven by bounding how often a Dirichlet
polynomial can take large values, but have been limited by our
understanding of the number of zeros with real part 3/4. We introduce a
new method to prove large value estimates for Dirichlet polynomials,
which improves on previous estimates near the 3/4 line.

This is joint work (still in progress) with Larry Guth.

We give a purely topological formula for the square class of the central value of the L-function of a symplectic representation on a curve. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds. This is related to the theory of epsilon factors in number theory and Meyer’s signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

The Mixing Conjecture of Michel-Venkatesh has now taken on additional arithmetic significance via Wiles' new approach to modularity. Inspired by this, we present the best currently available method, pioneered by Khayutin's proof for quaternion algebras over the rationals, which we have successfully applied to totally real fields. The talk will overview the method, which brings a suprising combination of ergodic theory, analysis and geometry to bear on this arithmetic problem.

There have been a lot of interests in understanding the behaviour of random multiplicative functions, which are probabilistic models for deterministic arithmetic functions such as the Möbius function and Dirichlet characters. Despite recent advances, the limiting distributions of partial sums of random multiplicative functions remain mysterious even at the conjectural level. In this talk, I shall discuss the so-called $L^2$ regime of twisted sums and provide a precise answer to the distributional problem. This is based on ongoing work with Ofir Gorodetsky.

The talk is based on joint work with Lasse Grimmelt. We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of $\operatorname{SL}_2(\mathbb{R})$ automorphic forms to study Poincaré series over congruence subgroups while keeping track of interactions between multiple orbits. This approach offers increased flexibility over the widely used sums of Kloosterman sums techniques. We give applications to correlations of the divisor function twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical line.

A conjecture of Malle predicts an asymptotic formula for the number of number fields with given Galois group and bounded discriminant. Malle conjectured the shape of the formula but not the leading constant. We present a new conjecture on the leading constant motivated by a version for algebraic stacks of Peyre's constant from Manin's conjecture. This is joint work with Tim Santens.

The compatibility of local and global Langlands correspondences is a central problem in algebraic number theory. A possible approach to resolving it relies on the existence of global Galois representations with prescribed local monodromy.  I will provide a partial solution by relating the question to its topological analogue. Both the topological and arithmetic version can be solved using the same family of projective hypersurfaces, which was first studied by Dwork.

In the 1960's Langlands proposed a generalisation of Class Field Theory. I will review this and describe a new approach using the trace formua as well as some analytic arguments reminiscent of those used in the classical case. In more concrete terms the problem is to prove general modularity theorems, and I will explain the progress I have made on this problem.

Schubert varieties in (twisted) affine Grassmannians and their singularities are of interest to arithmetic geometers because they model the étale local structure of the special fiber of Shimura varieties. In this talk, I will discuss a proof of a conjecture of Haines-Richarz classifying the smooth locus of Schubert varieties, generalizing a classical result of Evens-Mirkovic. The main input is to obtain a lower bound for the tangent space at a point of the Schubert variety which arises from considering certain smooth curves passing through it. In the second part of the talk, I will explain how in many cases, we can prove this bound is actually sharp, and discuss some applications to Shimura varieties. This is based on joint work with Pappas and Kisin-Pappas.

We show the primes have level of distribution 66/107 using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level 3/5 of Maynard. We also extend this level to 5/8, assuming Selberg's eigenvalue conjecture. As a result, we obtain new upper bounds for twin primes and for Goldbach representations of even numbers $a$. For the Goldbach problem, this is the first use of a level of distribution beyond the 'square-root barrier', and leads to the greatest improvement on the problem since Bombieri--Davenport from 1966.

Kaufman constructed a family of Fourier-decaying measures on the set of badly approximable numbers. Pollington and Velani used these to show that Littlewood’s conjecture holds for a full-dimensional set of pairs of badly approximable numbers. We construct analogous measures that have implications for inhomogeneous diophantine approximation. In joint work with Agamemnon Zafeiropoulos and Evgeniy Zorin, our idea is to shift the continued fraction and Ostrowski expansions simultaneously.

A recent conjecture proposed by Harris and Venkatesh relates the action of derived Hecke operators on the space of weight one modular forms to certain Stark units. In this talk, I will explain how this can be rephrased as a conjecture about "tame" analogues of triple product periods for a triple of mod p eigenforms of weights (2,1,1). I will then present an elliptic counterpart to this conjecture relating a tame triple product period to a regulator for global points of elliptic curves in rank 2. This conjecture can be proved in some special cases for CM weight 1 forms, with techniques resonating with the so-called Jochnowitz congruences. This is joint work in preparation with Henri Darmon. 

In this talk, we present an algorithm to compute p-adic heights on hyperelliptic curves with good reduction. Our algorithm improves a previous algorithm of Balakrishnan and Besser by being considerably simpler and faster and allowing even degree models. We discuss two applications of our work: to apply the quadratic Chabauty method for rational and integral points on hyperelliptic curves and to test the p-adic Birch and Swinnerton-Dyer conjecture in examples numerically. This is joint work with Steffen Müller.

I will report on current work in progress on the construction of anticyclotomic p-adic L-functions for Rankin--Selberg products. I will explain how by p-adically interpolating the branching law for the spherical pair (U(n)xU(n+1), U(n)) we can construct a p-adic L-function attached to cohomological automorphic representations of U(n) x U(n+1), including anticyclotomic variation. Due to the recent proof of the unitary Gan--Gross--Prasad conjecture, this p-adic L-function interpolates the square root of the central L-value. Time allowing, I will explain how we can extend this result to the Coleman family of an automorphic representation.

This talk will cover the greatest hits of pointwise ergodic theory, beginning with Birkhoff's theorem, then Bourgain's work, and finishing with more modern directions.

Is there a partition of the natural numbers into finitely many pieces, none of which contains a Pythagorean triple (i.e. a solution to the equation x²+y²=z²)? This is one of the simplest questions in arithmetic Ramsey theory which is still open. I will present a recent partial result, showing that in any finite partition of the natural numbers there are two numbers x,y in the same cell of the partition, such that x²+y²=z² for some integer z which may be in a different cell. 

The proof consists, after some initial maneuvers inspired by ergodic theory, in controlling the behavior of completely multiplicative functions along certain quadratic polynomials. Considering separately aperiodic and "pretentious" functions, the last major ingredient is a concentration estimate for functions in the latter class when evaluated along sums of two squares.

The talk is based on joint work with Frantzikinakis and Klurman.

It was asked by E. Szemerédi if, for a finite set $A\subset \mathbf{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In this paper we show that this maximum is at least of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\varepsilon}$ for some $\varepsilon>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$. Our proof consists of three parts: combinatorial, analytical and number theoretical.

(Joint work with Kartik Prasanna)

Siegel modular forms are higher-dimensional analogues of modular forms. While each rational elliptic curve corresponds to a single holomorphic modular form, each abelian surface is expected to correspond to a pair of Siegel modular forms: a holomorphic and a generic one. We propose a conjecture that explains the appearance of these two forms (in the cohomology of vector bundles on Siegel modular threefolds) in terms of certain higher algebraic cycles on the self-product of the abelian surface. We then prove three results:
(1) The conjecture is implied by Beilinson's conjecture on special values of L-functions. Amongst others, this uses a recent analytic result of Radzwill-Yang about non-vanishing of twists of L-functions for GL(4).
(2) The conjecture holds for abelian surfaces associated with elliptic curves over real quadratic fields.
(3) The conjecture implies a conjecture of Prasanna-Venkatesh for abelian surfaces associated with elliptic curves over imaginary quadratic fields.

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.