Past Number Theory Seminar

We will describe aspects of the geometry of non-minuscule rigid analytic period domains and their covering spaces, and pose some questions about p-adic period mappings and period images by analogy with the complex analytic theory.

In 1977 Mazur proved that the rational torsion subgroup of the Jacobian of the modular curve $X_0(N)$, $N > 5$ prime, is generated by the linear equivalence class of the difference of the two cusps. More generally, it is conjectured that for a general $N$, the rational torsion subgroup of the Jacobian of $X_0(N)$ is generated by cusps.  In this talk, we'll discuss a generalisation of this to other modular curves, namely certain covers of $X_0(N)$, indexed by subgroups of $(\mathbf{Z}/N\mathbf{Z})^\times$.

It is well-known that one can attach Galois representations to modular forms. In the case of cusp forms, the corresponding l-adic Galois representations are irreducible for every prime l, while in the case of Eisenstein series, the corresponding Galois representations are reducible. The Langlands correspondence is expected to generalise this picture, with cuspidal automorphic representations always giving rise to irreducible Galois representations. In the cuspidal, polarized, regular algebraic setting over a CM field, a construction of Galois representations is known, but their irreducibility is still an open problem in general. I will discuss recent joint work with Zachary Feng establishing new instances of irreducibility, and outline how our methods extend some previous approaches to this problem.

I will briefly introduce the Bloch-Kato conjecture, a very general conjecture relating special values of L-functions to arithmetic, and explain how it generalises many more familiar theorems and conjectures such as the BSD conjecture for elliptic curves. I will then introduce the concept of an "Euler system", which is a powerful tool in proving cases of these conjectures, and survey some recent constructions of Euler systems using the geometry of Shimura varieties.

An old conjecture of Graham asks whether there are infinitely many integers n such that \binom{2n}{n} is coprime to 105. This is equivalent to asking whether there are infinitely many integers which only have the digits 0,1 in base 3, 0,1,2 in base 5, and 0,1,2,3 in base 7. In general, one can ask whether there are infinitely many integers which only have 'small' digits in multiple bases simultaneously. For two bases this was established in 1975 by Erdos, Graham, Ruzsa, and Straus, but the case of three or more bases is much more mysterious. I will discuss recent joint work with Ernie Croot, in which we prove that (assuming the bases are sufficiently large) there are infinitely many integers such that almost all of the digits are small in all bases simultaneously. 

I will review how the equations of general relativity near a spacetime singularity map onto an arithmetic hyperbolic billiard dynamics. The semiclassical quantum states for this dynamics are Maaβ cusp forms on fundamental domains of modular groups. For example, gravity in four spacetime dimensions leads to PSL(2,Z) while five dimensional gravity leads to PSL(2,Z[w]), with Z[w] the Eisenstein integers. The automorphic forms can be expressed, in a dilatation (Mellin transformed) basis as L-functions. The Euler product representation of these L-functions indicates that these quantum states admit a dual interpretation as a "primon gas" partition function. I will describe some physically motivated mathematical questions that arise from these observations.

We'll describe connections between probabilistic models for primes,
the Hardy-Littlewood k-tuples conjectures, the distribution of primes in
very short intervals, the interval sieve, and hypothetical Landau-Siegel
zeros of Dirichlet L-functions.  We will emphasize the role and limitations
of probabilistic ideas.

The cohomology of Shimura varieties plays an important role in Langlands program, serving as a link between automorphic forms and Galois representations. In this talk, we prove a vanishing result for the cohomology of Shimura varieties of abelian type with torsion coefficients, generalizing the previous results of Caraiani-Scholze, Koshikawa, Hamann-Lee, and others. Our proofs utilize the unipotent categorical local Langlands correspondence developed by Zhu and the Igusa stacks constructed by Daniels-van Hoften-Kim-Zhang. This is a joint work with Xinwen Zhu.

For $\ell$ an odd prime number and $d$ a squarefree integer, a notable problem in arithmetic statistics is to give pointwise bounds for the size of the $\ell$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. This is in general a difficult problem, and unconditional pointwise bounds are only available for $\ell = 3$ due to work of Pierce, Helfgott—Venkatesh and Ellenberg—Venkatesh. The current record due to Ellenberg—Venkatesh is $h_3(d) \ll_\epsilon d^{1/3 + \epsilon}$. We will discuss how to improve this to $h_3(d) \ll d^{0.32}$. This is joint work with Peter Koymans.

Let $p$ be an odd prime. Let $K/\mathbf{Q}_p$ be a finite unramified extension. Let $\rho: G_K \to \mathrm{GL}_2(\overline{\mathbf{F}}_p)$ be a continuous representation. We prove that $\rho$ has a crystalline lift of small irregular weight if and only if it has multiple crystalline lifts of certain specified regular weights. The inspiration for this result comes from recent work of Diamond and Sasaki on geometric Serre weight conjectures. We also discuss applications to partial weight one modularity.

The BSD conjecture predicts that a rational elliptic curve $E$ has infinitely many points if and only if its $L$-function vanishes at $s=1$.

There are $p$-adic versions of similar phenomena. If $E$ is $p$-ordinary, there is, for example, a $p$-adic analytic analogue $L_p(E,s)$ of the $L$-function, and if $E$ has good reduction, then it has infinitely many rational points iff $L_p(E,1) = 0$. However if $E$ has split multiplicative reduction at $p$ - that is, if $E/\mathbf{Q}_p$ admits a Tate uniformisation $\mathbf{C}_p^{\times}/q^{\mathbf{Z}}$ - then $L_p(E,1) = 0$ for trivial reasons, regardless of $L(E,1)$; it has an 'exceptional zero'. Mazur--Tate--Teitelbaum's exceptional zero conjecture, proved by Greenberg--Stevens in '93, states that in this case the first derivative $L_p'(E,1)$ is much more interesting: it satisfies $L_p'(E,1) = \mathrm{log}(q)/\mathrm{ord}(q) \times L(E,1)/(\mathrm{period})$. In particular, it should vanish iff $L(E,1) = 0$ iff $E(\mathbf{Q})$ is infinite; and even better, it has a beautiful and surprising connection to the Tate period $q$, via the 'L-invariant' $\mathrm{log}(q)/\mathrm{ord}(q)$.

In this talk I will discuss exceptional zero phenomena and L-invariants, and a generalisation of the exceptional zero conjecture to automorphic representations of GL(3). This is joint work in progress with Daniel Barrera and Andrew Graham.

In the 1980s, Mazur and Tate proposed refinements of the Birch–Swinnerton-Dyer conjecture that also capture congruences between twists of Hasse–Weil L-series by Dirichlet characters. In this talk, I will report on new results towards these refined conjectures, obtained in joint work with Matthew Honnor. I will also outline how the results fit into a more general approach to refined conjectures on special values of L-series via an enhanced theory of Euler systems. This final part will touch upon joint work with David Burns.

If $E/\mathbb{Q}$ is an elliptic curve, and $F/\mathbb{Q}$ is a finite Galois extension, then $E(F)$ is not merely a finitely generated abelian group, but also a Galois module. If we fix a finite group $G$, and let $F$ vary over all $G$-extensions, then what can we say about the statistical behaviour of $E(F)$ as a $\mathbb{Z}[G]$-module? In this talk I will report on joint work with Adam Morgan, in which we investigate the simplest non-trivial special case of this very general question. Our work has surprising connections to questions about frequency of failure of the Hasse principle for genus 1 hyperelliptic curves, and to work of Heath-Brown on 2-Selmer group distributions in quadratic twist families.

We discuss ongoing work with Joseph Leung in which we obtain estimates for sums of Fourier coefficients of GL(2) and certain GL(3) automorphic forms along the values of irreducible binary cubics.

In recent joint work with J. Merikoski, we developed a new way to employ $\mathrm{SL}_2(\mathbb{R})$  spectral methods to number-theoretical counting problems, entirely avoiding Kloosterman sums and the Kuznetsov formula. The main result is an asymptotic formula for an automorphic kernel, with error terms controlled by two new kernels. This framework proves particularly effective when averaging over the level and leads to improvements in equidistribution results involving quadratic polynomials. In particular, we show that the largest prime divisor of $n^2 + h$ is infinitely often larger than $n^{1.312}$, recovering earlier results that had relied on the Selberg eigenvalue conjecture. Furthermore, we obtain, for the first time in this setting, strong uniformity in the parameter $h$.
 

The study of periods of automorphic forms is a key theme in the Langlands program and has become an important tool to tackle various problems in Number Theory and Arithmetic Geometry.  For instance, Waldspurger formula and its generalisations have created a fertile ground for numerous arithmetic applications. In recent years, the conjectures of Sakellaridis and Venkatesh (and then Ben-Zvi, Sakellaridis, and Venkatesh) in the context of spherical varieties has led to a deeper understanding of automorphic periods and their relation to special values of $L$-functions. In this talk, I present work in progress aimed at looking at certain non-spherical cases. Precisely, I will describe a new integral representation of the degree 12 "exterior square x standard" $L$-function on generic cusp forms on $\mathrm{GU}(2,2) \times \mathrm{GL}(2)$ (or $\mathrm{GL}(4) \times \mathrm{GL}(2)$) and how it can be used to relate the non-vanishing of its central value to a certain cohomological period.  If time permits, I will describe how the same strategy applies to the case of $\mathrm{GSp}(6) \times \mathrm{GL}(2)$. This is joint work with Armando Gutierrez Terradillos.

We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics.  This is a joint work with Shreyasi Datta.

We resolve Manin's conjecture for all Châtelet surfaces over Q
(surfaces given by equations of the form x^2 + ay^2 = f(z)) -- in other
words, we establish asymptotics for the number of rational points of
increasing height. The key analytic ingredient is estimating sums of
Fourier coefficients of modular forms along polynomial values.

A way to study rational points on a variety is by looking at their image in the p-adic points. Some natural questions that arise are the following: is there any obstruction to weak approximation on the variety? Which primes might be involved in it? I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation, with particular emphasis on the case of K3 surfaces.

There is an idea, going back to work of Krasner, that p-adic fields tend to function fields as absolute ramification tends to infinity. We will present a new way of rigorizing this idea, as well as give applications to the local Langlands correspondence of Fargues–Scholze.

I will explain the content of Geometric Langlands (which is a theorem over the ground fields of characteristic 0 but still a conjecture in positive characteristic) and show how it implies a description of the space of automorphic functions in terms of Galois data. The talk will mostly follow a joint paper with Arinkin, Kazhdan, Raskin, Rozenblyum and Varshavsky from 2022.

We show how to explicitly compute equations for everywhere locally soluble 3-coverings of Jacobians of genus 2 curves with a rational Weierstrass point, using the notion of visibility introduced by Cremona and Mazur.  These 3-coverings are abelian surface torsors, embedded in the projective space $\mathbb{P}^8$ as degree 18 surfaces. They have points over every $p$-adic completion of $\mathbb{Q}$, but no rational points, and so are counterexamples to the Hasse principle and represent non-trivial elements of the Tate-Shafarevich group.  Joint work in progress with Tom Fisher.

I will describe the contents of a joint project with Pablo Destic and Nuno Hultberg. In the paper we confirm a conjecture of Roberto Gualdi regarding a formula for the average height of the intersection of twisted (by roots of unity) hyperplanes in a toric variety. I will introduce the 'GVF analytification' of a variety, which is defined similarly as the Berkovich analytification, but with norms replaced by heights. Moreover, I will discuss some motivations coming from (continuous) model theory and Arakelov geometry.

The geodesic cycles (resp. Eisenstein classes) for SL(2,Z) are special classes in the homology (resp. cohomology) of modular curve (for SL(2,Z)) defined by the closed geodesics (resp. Eisenstein series). It is known that the pairing between these geodesic cycles and Eisenstein classes gives the special values of partial zeta functions of real quadratic fields, and this has many applications. In this talk, I would like to report on some recent observations on the size of the homology subgroup generated by geodesic cycles and their applications. This is a joint work with Ryotaro Sakamoto.

The study of polynomials whose coefficients lie in a given set $S$ (the most notable examples being $S=\{0,1\}$ or $\{-1,1\}$) has a long history leading to many interesting results and open problems. We begin with a brief general overview of this topic and then focus on the following old problem of Littlewood. Let $A$ be a set of positive integers, let $f_A(x)=\sum_{n\in A}\cos(nx)$ and define $Z(f_A)$ to be the number of zeros of $f_A$ in $[0,2\pi]$. The problem is to estimate the quantity $Z(N)$ which is defined to be the minimum of $Z(f_A)$ over all sets $A$ of size $N$. We discuss recent progress showing that $Z(N)\geqslant (\log \log N)^{1-o(1)}$ which provides an exponential improvement over the previous lower bound. 

A closely related question due to Borwein, Erd\'elyi and Littmann asks about the minimum number of zeros of a cosine polynomial with $\pm 1$-coefficients. Until recently it was unknown whether this even tends to infinity with the degree $N$. We also discuss work confirming this conjecture.

A number of results on classical problems in analytic number theory rely on bounds for multilinear forms of Kloosterman sums, which in turn use deep inputs from the spectral theory of automorphic forms. We’ll discuss our recent work available at arxiv.org/abs/2404.04239, which uses this interplay between counting problems, exponential sums, and automorphic forms to improve results on the greatest prime factor of $n^2+1$, and on the exponents of distribution of primes and smooth numbers in arithmetic progressions.
The key ingredient in this work are certain “large sieve inequalities” for exceptional Maass forms, which improve classical results of Deshouillers-Iwaniec in special settings. These act as on-average substitutes for Selberg’s eigenvalue conjecture, narrowing (and sometimes completely closing) the gap between previous conditional and unconditional results.

I will report on work with Andrew Graham in which we prove new results towards the Bloch--Kato conjecture for automorphic forms on $\mathrm{GSp}_4 \times \mathrm{GL}_2$.

Given an elliptic curve over the rationals, a natural problem is to find an explicit point of infinite order over a given number field when there is expected to be one. Geometric constructions are known in only two different settings. That of Heegner points, developed since the 1950s, which yields points over abelian extensions of imaginary quadratic fields. And that of Stark-Heegner points, from the late 1990s: here the points constructed are conjectured to be defined over abelian extensions of real quadratic fields. I will describe a new analytic formula which encompasses both of these, and conjecturally yields points in many other settings. This is joint work with Henri Darmon and Victor Rotger.

Let $K$ be an unramified extension of $\mathbb{Q}_p$ for a prime $p > 3$. The reduced part of the Emerton-Gee stack for $\mathrm{GL}_{2}$ can be viewed as parameterizing two-dimensional mod $p$ Galois representations of the absolute Galois group of $K$. In this talk, we will consider the extremely non-generic irreducible components of this reduced part and see precisely which ones are smooth or normal, and which have Gorenstein normalizations. We will see that the normalizations of the irreducible components admit smooth-local covers by resolution-rational schemes. We will also determine the singular loci on the components, and use these results to update expectations about the conjectural categorical $p$-adic Langlands correspondence. This is based on recent joint work with Ben Savoie.

If $n$ is congruent to 0 or 4 modulo 6, there are infinitely many primes of the form $x^2 + ny^2$ with both $x$ and $y$ prime. (Joint work with Mehtaab Sawhney, Columbia)

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.