Past Number Theory Seminar

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.

I will explain a formalism for anticyclotomic Euler systems for a large class of Galois representations and explain how to prove analogs of Kolyvagins' celebrated "rank one" results. A novelty of this approach lies in the use of primes that split in the CM field. This is joint work with Dimitar Jetchev and Jan Nekovar. I will also describe some higher-dimensional examples of such Euler systems.

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.

For positive integers $k$ and $n$ let $\sigma_k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erd\H{o}s and Kac conjectured that, for every $k$, the number $\alpha_k = \sum_{n\geq 1} \frac{\sigma_k(n)}{n!}$ is irrational. This is known conditionally for all $k$ assuming difficult conjectures like the Hardy-Littlewood prime $k$-tuples conjecture. Before our work it was known unconditionally that $\alpha_k$ is irrational if $k\leq 3$. We discuss some of the ideas in our recent proof that $\alpha_4$ is irrational. The proof involves sieve methods and exponential sum estimates.

In joint work with Jacob Tsimerman we study when the primitive
of a given algebraic function can be constructed using primitives
from some given finite set of algebraic functions, their inverses,
algebraic functions, and composition. When the given finite set is just {1/x}
this is the classical problem of "elementary integrability".
We establish some results, including a decision procedure for this problem.

Representations of finite reductive groups have a rich, well-understood structure, first explored by Deligne--Lusztig. In joint work with Anne-Marie Aubert and Dan Ciubotaru, we show a way to lift some of this structure to representations of p-adic groups. In particular, we consider the relation between Lusztig's nonabelian Fourier transform and a certain involution we define on the level of p-adic groups. This talk will be an introduction to these ideas with a focus on examples.

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves. I will also mention some applications to rational points on certain families of curves.

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

Last summer, there was a lot of activity regarding an old conjecture of van der Waerden, culminating in its solution by Bhargava, and including joint work by Sam Chow and myself on which I want to report in this talk: We showed that the number of irreducible monic integer polynomials of degree n, with coefficients in absolute value bounded by H, which have Galois group different from S_n and A_n, is of order of magnitude O(H^{n-1.017}), providing that n is at least 3 and is different from 7,8,10. Apart from the alternating group and excluding degrees 7,8,10, this establishes the aforementioned conjecture to the effect that irreducible non-S_n polynomials are significantly less frequent than reducible polynomials.

As a model for the primes, in this talk I will address such statistical questions for the sequence of squarefree numbers, i.e., numbers not divisible by the square of any prime, among other related ``sifted'' sequences called B-free numbers. I hope to further motivate and explain our main result that shows, unconditionally, that short interval counts of squarefree numbers do satisfy Gaussian statistics, answering several questions of R.R. Hall.

In his 1976 proof of the converse of Herbrand’s theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree-p extensions of the p-th cyclotomic field when p is an odd prime. After reviewing Ribet’s strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degree-p extensions of Q(N^{1/p}) when N is a prime that is congruent to -1 mod p. This answers a question posed on Frank Calegari’s blog.

Let X be an algebraic variety over an infinite field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. For example, is X(k) empty or not? And if it is not empty, is X(k) dense in X with respect to the Zariski topology?

Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d >= 3, these are the smooth surfaces of degree d in P^d). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general.

I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is finitely generated over Q.

Let X be a smooth projective curve over a finite field equipped with an action of a finite group G. I’ll first briefly introduce the corresponding Artin L-function and a certain equivariant Euler characteristic. The main result will be a precise relation between the epsilon constants appearing in the functional equations of Artin L-functions and that Euler characteristic if the projection X → X/G is at most weakly ramified. This generalises a theorem of Chinburg for the tamely ramified case. I’ll end with some speculations in the arbitrarily wildly ramified case. This is joint work with Helena Fischbacher-Weitz and with Adriano Marmora.

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

In joint work with James Wheeler, we show that if a subset $A$ of $GL_n(\mathbb{F}_q)$ is a $K$-approximate group and the group $G$ it generates is soluble, then there are subgroups $U$ and $S$ of $G$ and a constant $k$ depending only on $n$ such that:

$A$ quickly generates $U$: $U\subseteq A^k$,
$S$ contains a large proportion of $A$: $|A^k\cap S| \gg K^{-k}|A|, and
$S/U$ is nilpotent.

Briefly: approximate soluble linear groups over any finite field are (almost) finite by nilpotent.

The proof uses a sum-product theorem and exponential sum estimates, as well as some representation theory, but the presentation will be mostly self-contained.

We construct a functor from arithmetic schemes (and dominant morphisms) to dynamical systems which allows to recover the Hasse-Weil zeta function of a scheme as a Ruelle type zeta function of the corresponding dynamical system. We state some further properties of this correspondence and explain the relation to the work of Kucharczyk and Scholze who realize the Galois groups of fields containing all roots of unity as (etale) fundamental groups of certain topological spaces. We also explain the main reason why our dynamical systems are not yet the right ones and in what regard they need to be refined.
 

One can study periods of algebraic varieties by a process of "fibering out" in which the variety is fibred over a punctured curve $f:X->U$. I will explain this process and how it leads to the classical Picard Fuchs (or Gauss-Manin) differential equations. Periods are computed by integrating solutions of Picard Fuchs over suitable closed paths on $U$. One can also couple (i.e.tensor) the Picard Fuchs connection to given connections on $U$. For example, $t^s$ with $t$ a unit on $U$ and $s$ a parameter is a solution of the connection on $\mathscr{O}_U$ given by $\nabla(1) = sdt/t$. Our "periods" become integrals over suitable closed chains on $U$ of $f(t)t^sdt/t$. Golyshev called the resulting functions of $s$ "motivic Gamma functions". 
Golyshev and Zagier studied certain special Picard Fuchs equations for their proof of the Gamma conjecture in mirror symmetry in the case of Picard rank 1. They write down a generating series, the Apéry series, the knowledge of the first few terms of which implied the gamma conjecture. We show their Apéry series is the Taylor series of a product of the motivic Gamma function times an elementary function of $s$. In particular, the coefficients of the Apéry series are periods up to inverting $2\pi i$. We relate these periods to periods of the limiting mixed Hodge structure at a point of maximal unipotent monodromy. This is joint work with M. Vlasenko. 
 

The famous Birch & Swinnerton-Dyer conjecture predicts that the (algebraic) rank of an elliptic curve is equal to the so-called analytic rank, which is the order of vanishing of the associated L-functions at the central point. In this talk, we shall discuss the analytic rank of automorphic L-functions in an "alternate universe". This is joint work with Kyle Pratt and Alexandru Zaharescu.

Some of the simplest expected cases of Langlands functoriality are the symmetric power liftings Sym^r from automorphic representations of GL(2) to automorphic representations of GL(r+1). I will discuss some joint work with Jack Thorne on the symmetric power lifting for holomorphic modular forms.

I will describe some new-ish results on the average and maximum size of the Riemann zeta function in a "typical" interval of length 1 on the critical line. A (hopefully) interesting feature of the proofs is that they reduce the problem for the zeta function to an analogous problem for a random model, which can then be solved using various probabilistic techniques.

Let E be an elliptic curve over the rationals and p a prime such that E admits a rational p-isogeny satisfying some assumptions. In a joint work with J. Lee and C. Skinner, we prove the anticyclotomic Iwasawa main conjecture for E/K for some suitable quadratic imaginary field K. I will explain our strategy and how this, combined with complex and p-adic Gross-Zagier formulae, allows us to prove that if E has rank one, then the p-part of the Birch and Swinnerton-Dyer formula for E/Q holds true.
 

We will outline the main features of Wooley's efficient congruencing method for the parabola. Then we will go on to prove new bounds for discrete restriction to the curve (x,x^3). The latter is joint work with Trevor Wooley (Purdue).

Stevenhagen conjectured that the density of d such that the negative Pell equation x^2-dy^2=-1 is solvable over the integers is 58.1% (to the nearest tenth of a percent), in the set of positive squarefree integers having no prime factors congruent to 3 modulo 4. In joint work with Peter Koymans, Djordjo Milovic, and Carlo Pagano, we use a recent breakthrough of Smith to prove that the infimum of this density is at least 53.8%, improving previous results of Fouvry and Klüners, by studying the distribution of the 8-rank of narrow class groups of quadratic number fields.

My talk will have two protagonists: (1) Automorphic representations which -- let's be honest -- are very complicated and mysterious, but also (2) Involutions  (=automorphisms of order at most 2) of connected reductive groups -- these are very concrete and can often be represented by diagonal matrices with entries 1,-1 or i, -i. The goal is to explain how difficult questions about (1) can be reduced to relatively easy, concrete questions about (2).
Automorphic representations are representation-theoretic generalizations of modular forms. Like modular forms, automorphic representations are initially defined analytically. But unlike modular forms -- where we have a reinterpretation in terms of algebraic geometry -- for most automorphic representations we currently only have a (real) analytic definition. The Langlands Program predicts that a wide class of automorphic representations admit the same algebraic properties which have been known to hold for modular forms since the 1960's and 70's. In particular, certain complex numbers "Hecke eigenvalues" attached to these automorphic representations are conjectured to be algebraic numbers. This remains open in many cases (especially those cases of interest in number theory and algebraic geometry), in particular for Maass forms -- functions on the upper half-plane which are a non-holomorphic variant of modular forms.
I will explain how elementary structure theory of reductive groups over the complex numbers provides new insight into the above algebraicity conjectures; in particular we deduce that the Hecke eigenvalues are algebraic for an infinite class of examples where this was not previously known. 
After applying a bunch of "big, old theorems" (in particular Langlands' own archimedean correspondence), it all comes down to studying how involutions of a connected, reductive group vary under group homomorphisms. Here I will write down the key examples explicitly using matrices.