Past Number Theory Seminar

Abstract. I will talk about projects in which we combine heuristics with computational data to develop a theory in problems where it was previously hard to be confident of the guesses that there are in the literature.

1/ "Speculations about the number of primes in fast growing sequences". Starting from studying the distribution of primes in sequences like $2^n-3$, Jon Grantham and I have been developing a heuristic to guess at the frequency of prime values in arbitrary linear recurrence sequences in the integers, backed by calculations.

If there is enough time I will then talk about:

2/ "The spectrum of the $k$th roots of unity for $k>2$, and beyond".  There are many questions in analytic number theory which revolve around the "spectrum", the possible mean values of multiplicative functions supported on the $k$th roots of unity. Twenty years ago Soundararajan and I determined the spectrum when $k=2$, and gave some weak partial results for $k>2$, the various complex spectra.  Kevin Church and I have been tweaking MATLAB's package on differential delay equations to help us to develop a heuristic theory of these spectra for $k>2$, allowing us to (reasonably?) guess at the answers to some of the central questions.

I will explain how to construct an Euler system for imaginary quadratic fields using Eisenstein series and their cohomology classes. This illustrates a template for a construction that should yield many new Euler systems.

The elliptic Gamma function — a generalization of the q-Gamma function, which is itself the q-analog of the ordinary Gamma function — is a meromorphic special function in several variables that mathematical physicists have shown to satisfy modular functional equations under SL(3,Z). In this talk I will present evidence (numerical and theoretical) that this function often takes algebraic values that satisfy explicit reciprocity laws and that are related to derivatives of Hecke L-functions at s=0. Thus this function conjecturally allows to extend the theory of complex multiplication to complex cubic fields as envisioned by Hilbert's 12^th problem. This is joint work with Nicolas Bergeron and Pierre Charollois.

I will explain a new construction of an Euler system for the symmetric square of an eigenform and its connection with L-values. The construction makes use of some simple Eisenstein cohomology classes for Sp(4) or, equivalently, SO(3,2). This is an example of a larger class of similarly constructed Euler systems.  This is a report on joint work with Marco Sangiovanni Vincentelli.

The main object of study of the talk is the balanced triple product p-adic L-function; this is a p-adic L-function associated with a triple of families of (quaternionic) modular forms. The first instances of these functions appear in the works of Darmon-Lauder-Rotger, Hsieh, and Greenberg-Seveso. They have proved to be effective tools in studying cases of the p-adic equivariant Birch & Swinnerton-Dyer conjecture. With this aim in mind, we discuss the construction of a new p-adic L-function, extending Hsieh's construction, and allowing classical weight one modular forms in the chosen families. Such improvement does not come for free, as it coincides with the increased dimension of certain Hecke-eigenspaces of quaternionic modular forms with non-Eichler level structure; we discuss how to deal with the problems arising in this more general setting. One of the key ingredients of the construction is a p-adic extension of the Jacquet-Langlands correspondence addressing these more general quaternionic modular forms. This is joint work in progress with Aleksander Horawa.

Many problems in number theory are equivalent to determining all of the rational points on some curve or family of curves. In general, finding all the rational points on any given curve is a challenging (even unsolved!) problem. 

The focus of this talk is rational points on so-called Erdős-Selfridge curves. A deep conjecture of Sander, still unproven in many cases, predicts all of the rational points on these curves. 

I will describe work-in-progress proving new cases of Sander's conjecture, and sketch some ideas in the proof. The core of the proof is a `mass increment argument,' which is loosely inspired by various increment arguments in additive combinatorics. The main ingredients are a mixture of combinatorial ideas and quantitative estimates in Diophantine geometry.

The closest thing we have to a general method for finding primes in sets is to use sieve methods to turn the problem into some other (hopefully easier) arithmetic questions about the set.

Unfortunately this process is still poorly understood - we don’t know ‘how much’ arithmetic information is sufficient to guarantee the existence of primes, and how much is not sufficient. Often arguments are rather ad-hoc.

I’ll talk about work-in-progress with Kevin Ford which shows that many of our common techniques are not optimal and can be refined, and in many cases these new refinements are provably optimal.

Hoheisel used zero-density results to prove that for all x large enough there is a prime number in the interval $[x−x^{\theta}, x]$ with $θ < 1$. The connection between zero-density estimates and primes in short intervals was explicitly described in the work of Ingham in 1937. The approach of Ingham combined with the zero-density estimates of Huxley (1972) provides us with the distribution of primes in $[x−x^{\theta}, x]$ with $\theta > 7/12$. Further improvement upon the value of \theta was achieved by combining sieves with the weighted zero-density estimates in the works of Iwaniec and Jutila, Heath-Brown and Iwaniec, and Baker and Harman. The last work provides the best result achieved using zero-density estimates. We will discuss the main ideas of the paper by Baker and Harman and simplify some parts of it to show a more explicit connection between zero-density results and the sieved sums, which are used in the paper. This connection will provide a better understanding on which parts should be optimised for further improvements and on what the limits of the methods are. This project is still in progress.

I will talk about some current work with Jesse Jaasaari and Steve Lester where we investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture in the weight aspect for Siegel cusp forms of degree 2 and full level. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for Saito–Kurokawa lifts as the weight tends to infinity. As an application, we prove the equidistribution of zero divisors.

Given a finite set A of integer lattice points in d dimensions, let NA denote the N-fold iterated sumset (i.e. the set comprising all sums of N elements from A). In 1992 Khovanskii observed that there is a fixed polynomial P(N), depending on A, such that the size of the sumset NA equals P(N) exactly (once N is sufficiently large, that is). In addition to this 'size stability', there is a related 'structural stability' property for the sumset NA, which Granville and Shakan recently showed also holds for sufficiently large N. But what does 'sufficiently large' mean in practice? In this talk I will discuss some perspectives on these questions, and explain joint work with Granville and Shakan which proves the first explicit bounds for all sets A. I will also discuss current work with Granville, which gives a tight bound 'up to logarithmic factors' for one of these properties. 

Assuming the Riemann Hypothesis, Soundararajan established almost sharp upper bounds for all positive moments of the Riemann zeta-function. This result was later improved by Harper, who proved upper bounds of the right order of magnitude. I will describe some of the ideas in their proofs, and then discuss recent joint work with Alexandra Florea, where we consider negative moments of the Riemann zeta-function. For example, we can obtain asymptotic formulas for negative moments when the shift in the zeta function is large enough, confirming a conjecture of Gonek.  We also obtain an upper bound for the average of the generalised Mobius function.

Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov established the Hasse principle for Kummer varieties associated to a 2-covering of a principally polarised abelian variety A, under certain large image assumptions on the Galois action on A[2]. However, their method stops short of treating the case where the image is the full symplectic group, due to the possible failure of the Shafarevich--Tate group to have square order in this setting. I will explain work in progress which overcomes this obstruction by combining second descent ideas of Harpaz with new results on the 2-parity conjecture. 

I will present results on short sums of arithmetic functions (in particular the von Mangoldt and divisor functions) twisted by polynomial exponential phases or more general nilsequence phases. These results imply the Gowers uniformity of suitably normalised versions of these functions in intervals of length X^c around X for suitable values of c (depending on the function and on whether one considers all or almost all short sums). I will also discuss an application to an averaged form of the Hardy-Littlewood conjecture. This is based on joint works with Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao and Terence Tao.

Wiles' modularity lifting theorem was the central argument in his proof of modularity of (semistable) elliptic curves over Q, and hence of Fermat's Last Theorem. His proof relied on two key components: his "patching" argument (developed in collaboration with Taylor) and his numerical isomorphism criterion.

In the time since Wiles' proof, the patching argument has been generalized extensively to prove a wide variety of modularity lifting results. In particular Calegari and Geraghty have found a way to generalize it to prove potential modularity of elliptic curves over imaginary quadratic fields (contingent on some standard conjectures). The numerical criterion on the other hand has proved far more difficult to generalize, although in situations where it can be used it can prove stronger results than what can be proven purely via patching.

In this talk I will present joint work with Srikanth Iyengar and Chandrashekhar Khare which proves a generalization of the numerical criterion to the context considered by Calegari and Geraghty (and contingent on the same conjectures). This allows us to prove integral "R=T" theorems at non-minimal levels over imaginary quadratic fields, which are inaccessible by Calegari and Geraghty's method. The results provide new evidence in favor of a torsion analog of the classical Langlands correspondence.

Let K be a number field and let L/K be an abelian extension. The genus field of L/K is the largest extension of L which is unramified at all places of L and abelian as an extension of K. The genus group is its Galois group over L, which is a quotient of the class group of L, and the genus number is the size of the genus group. We study the quantitative behaviour of genus numbers as one varies over abelian extensions L/K with fixed Galois group. We give an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time. This is joint work with Christopher Frei and Daniel Loughran.

In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this talk, we'll discuss arithmetic problems over F_q[T] and their connections to matrix integrals, focusing on variations on the divisor function problem with symplectic and orthogonal distributions. Joint work with Matilde Lalín.

Néron models are mathematical objects which play a very important role in contemporary arithmetic geometry. However, they usually behave badly, particularly in respect of exact sequences and base change, which makes most problems regarding their behaviour very delicate. Chai introduced the base change conductor, a rational number associated with a semiabelian variety $G$ which measures the failure of the Néron model of $G$ to commute with (ramified) base change. Moreover, Chai conjectured that this invariant is additive in certain exact sequences. We shall introduce a new method to study the Néron models of Jacobians of proper (possibly singular) curves, and sketch a proof of Chai's conjecture for semiabelian varieties which are also Jacobians. 

I will talk about work in progress with Ana Caraiani aimed at proving Ihara’s lemma for quaternionic Shimura varieties, generalising the strategy of Manning-Shotton for Shimura curves. As an arithmetic motivation, in the first part of the talk I will recall an approach to the Birch and Swinnerton-Dyer conjecture based on congruences between modular forms, relying crucially on Ihara’s lemma.

Abstract: We give a generalized Kuznetsov formula that allows one to impose additional conditions at finitely many primes.  The formula arises from the relative trace formula. I will discuss applications to spectral large sieve inequalities and subconvexity. This is work in progress with M.P. Young and Y. Hu.

The Zilber-Pink conjecture predicts that there should be only finitely
many algebraic numbers t such that the three elliptic curves with
j-invariants t, -t, 2t are all isogenous to each other.  Using previous
work of Habegger and Pila, it suffices to prove a height bound for such
t.  I will outline the proof of this height bound by viewing periods of
the elliptic curves as values of G-functions.  An innovation in this
work is that both complex and p-adic periods are required.  This is
joint work with Christopher Daw.

For a family of finite sets whose cardinalities are naturally called class numbers, the Gauss problem asks to determine the subfamily in which every member has class number one. We study the Siegel moduli space of abelian varieties in characteristic $p$ and solve the Gauss problem for the family of central leaves, which are the loci consisting of points whose associated $p$-divisible groups are isomorphic. Our solution involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $4$. This geometric Gauss problem is closely related to an arithmetic Gauss problem for genera of positive-definite quaternion Hermitian lattices, which we also solve.

It is known that the Brauer group of a smooth, projective surface
defined by an equality of two homogeneous polynomials in characteristic 0, is
finite up to constants. I will report on new methods to determine these Brauer
groups, or at least their algebraic parts, as long as the coefficients are in a
certain sense generic. This generalises previous results obtained over the
years by Colliot-Thélène--Kanevsky--Sansuc, Bright, Uematsu and Santens.
(Joint work with A. N. Skorobogatov.)

If $Q(x_0,x_1,x_2)$ is a quadratic form, how many solutions, of size at most $B$, does $Q=0$ have? How does this depend on $Q$? We apply the answers to the surface $y_0 Q_0 +y_1 Q_1 = 0$ in $P^1 x P^2$. (Joint work with Dan Loughran.)
 

Classical ramification theory deals with complete discrete valuation fields k((X)) with perfect residue fields k. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow non-discrete valuations and imperfect residue fields k. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.

Degree p extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups.