Past Junior Number Theory Seminar

In 1973, Serre defined $p$-adic modular forms as limits of modular forms, and constructed the Leopoldt-Kubota $L$-function as the constant term of a limit of Eisenstein series. This was extended by Deligne-Ribet to totally real number fields, and Lauder and Vonk have developed an algorithm for interpolating $p$-adic $L$-functions of such fields using Serre's idea. We explain what an $L$-function is and why you should care, and then move on to giving an overview of the algorithm, extensions, and applications.

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 $h$.

The action of the automorphisms of a formal group on its deformation space is crucial to understanding periodic families in the homotopy groups of spheres and the unsolved Hecke orbit conjecture for unitary Shimura varieties. We can explicitly pin down this squirming action by geometrically modelling it as coming from an action on a moduli space, which we construct using inverse Galois theory and some representation theory (a Hurwitz space). I will show you pretty pictures.

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum 1/(n\log n)$ for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-\lambda}$ over a primitive set is maximized by the primes if and only if $\lambda$ is at least the critical exponent $\tau_1\approx1.14$.
A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. In joint work with C. Pomerance and T.H. Chan, we study the critical exponent $\tau_k$ for which the primes are maximal among $k$-primitive sets. In particular we prove that $\tau_2<0.8$, which directly implies the Erdős conjecture for 2-primitive sets.

Multiple zeta values, originally considered by Euler, generalise the Riemann zeta function to multiple variables. While values of the Riemann zeta function at odd positive integers are conjectured to be algebraically independent, multiple zeta values satisfy many algebraic and linear relations, even forming a Q-algebra. While families of well understood relations are known, such as the associator relations and double shuffle relations, they only conjecturally span all algebraic relations. As multiple zeta values arise as the periods of mixed Tate motives, we obtain further algebraic structures, which have been exploited to provide spanning sets by Brown. In this talk we will aim to define a new set of relations, known to be complete in low block degree.

To achieve this, we will first review the necessary algebraic set up, focusing particularly on the motivic Lie algebra associated to the thrice punctured projective line. We then introduce a new filtration on the algebra of (motivic) multiple zeta values, called the block filtration, based on the work of Charlton. By considering the associated graded algebra, we quickly obtain a new family of graded motivic relations, which can be shown to span all algebraic relations in low block degree. We will also touch on some conjectural ungraded `lifts' of these relations, and if we have time, compare to similar approaches using the depth filtration.

Over the past few decades, there has been a lot of interest in partial sums of Dirichlet characters. Montgomery and Vaughan showed that these character sums remain a constant size on average and, as a result, a lot of work has been done on the distribution of the maximum. In this talk, we will investigate the distribution of these character sums themselves, with the main goal being to describe the limiting distribution as the prime modulus approaches infinity. This is motivated by Kowalski and Sawin’s work on Kloosterman paths.
 

A semiprime is a natural number which can be written as the product of two primes. Using elementary methods, we'll explore an asymptotic expansion for the counting function of semiprimes $\pi_2(x)$, which generalises previous findings of Landau, Delange and Tenenbaum.  We'll also obtain an efficient way of computing the constants involved. In the end, we'll look towards possible generalisations for products of $k$ primes.

For every positive integer N and every α ∈ [0,1), let B(N, α) denote the probabilistic model in which a random set A of (1,...,N) is constructed by choosing independently every element of (1,...,N) with probability α. We prove that, as N → +∞, for every A in B(N, α) we have |AA| ~ |A|^2/2 with probability 1-o(1), if and only if (log(α^2(log N)^{log 4-1}))(√loglog N) → ∞. This improves on a theorem of Cilleruelo, Ramana and Ramar\'e, who proved the above asymptotic between |AA| and |A|^2/2 when α =o(1/√log N), and supplies a complete characterization of maximal product sets of random sets.

I will briefly introduce the Chabauty-Kim argument for effective finiteness results on "topologically rich enough" curves. I will then introduce the Fontaine-Mazur conjecture and show how it provides an effective proof of Faltings' Theorem.

In the case of non-CM elliptic curves minus a point, following work of Federico Amadio Guidi, I'll show how the relevant input for effective finiteness is provided by the vanishing of adjoint Selmer groups proven by Newton and Thorne.

This talk will be the first in a spin-off series on the Lawrence-Venkatesh approach to showing that every hyperbolic curve$/K$ has finitely many $K$-points. In this talk, we will give the overall outline of the approach and prove several of  the preliminary results, such as Faltings' finiteness theorem for semisimple Galois representations.

In 1966 Chen Jingrun showed that every large even integer can be written as the sum of two primes or the sum of a prime and a semiprime. To date, this weakened version of Goldbach's conjecture is one of the most remarkable results of sieve theory. I will talk about the big ideas which paved the way to this proof and the ingenious trick which led to Chen's success. No prior knowledge of sieve theory required – all necessary techniques will be introduced in the talk.

We will discuss the problem of deciding (algorithmically) whether a variety over a local field K has a K-rational point, surveying some known results. I will then allow K to be an infinite extension (of some arithmetic interest) of a local field and present some recent work.
 

Let C be an elliptic or hyperelliptic curve over a p-adic field K. Then C is equipped with a Galois representation, given by the action of the absolute Galois group of K on the Tate module of C. The behaviour of this representation depends on the reduction type of C. We will focus on the case of C having bad reduction, and acquiring potentially good reduction over a wildly ramified extension of K. We will show that, if C is an elliptic curve, the Galois representation can be completely determined in this case, thus allowing one to fully classify Galois representations attached to elliptic curves. Furthermore, the same can be done for a special family of hyperelliptic curves, obtaining a result which is surprisingly similar to that for the corresponding elliptic curves case.
 

We will survey an analogy between random integers and random permutations, which goes back to works of Erdős and Kac and of Billingsley.
This analogy inspired results and proofs about permutations, originating in the setting of integers, and vice versa.
Extensions of this analogy will be described, involving the generalized Ewens measure on permutations, based on joint work with D. Elboim.
If time permits, an analogous analogy, this time between random polynomials over a finite field and random permutations, will be discussed and formalized, with some applications.
 

Chowla's conjecture from the 1960s is the assertion that the Möbius function does not correlate with its own shifts. I'll discuss some recent works where with collaborators we have made progress on this conjecture.

In this talk, I will talk about two seemingly disjoint topics - Vinogradov’s mean value theorem, a classically important topic of study in additive number theory concerning solutions to a specific system of diophantine equations, and Incidence geometry, a collection of combinatorial results which focus on estimating the number of incidences between an arbitrary set of points and curves. I will give a brief overview of these two topics along with some basic proofs and applications, and then point out how these subjects connect together.

Local-to-global principles are a key tool in arithmetic geometry. Through a theorem of Siegel on representations of totally positive numbers as sums of squares in number fields we give a concrete introduction to the Hasse principle, and briefly talk about other local-to-global principles. No prerequisites from algebraic number theory are assumed, although some familiarity is helpful for context.

I will discuss some of Mazur's work about congruences between Eisenstein series and cusp forms, and then end with an application to class groups of fields $\mathbb{Q}(N^{1/p})$, where $N$ and $p$ are primes.  I will only assume some algebraic number theory.  In particular, nothing about modular forms will be assumed.
 

Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions. In this talk, I will present work (joint with Rachel Newton) developing explicit methods to study this principle for non-Galois extensions. As a key application, I will describe how these methods can be used to characterize the HNP for extensions whose normal closure has Galois group A_n or S_n. I will additionally discuss some recent generalizations of these methods to study the Hasse principle and weak approximation for multinorm equations as well as consequences in the statistics of these local-global principles.

Sieve methods are analytic tools that we can use to tackle problems in additive number theory. This talk will serve as a gentle introduction to the area. At the end we will discuss recent progress on a variation on the prime $k$-tuples conjecture which involves sums of two squares. No knowledge of sieves is required!

We discuss the problem in representation theory of decomposing restricted representations. We start classically with the symmetric groups via Young diagrams and Young tableaux, and then move into the world of Lie groups. These problems have connections with both physics and number theory, and if there is time I will discuss the Gan-Gross-Prasad conjectures which predict results on restrictions for algebraic groups over both local and global fields. The pre-requisites will build throughout the talk, but it should be accessible to anyone with some knowedge of both finite groups and Lie groups.

In this talk I will give an introduction to random multiplicative functions, and cover the recent developments in this area. I will also explain how RMF's are connected to some of the important open problems in Analytic Number Theory.

In 1983, Faltings proved Mordell's conjecture on the finiteness of $K$-points on curves of genus >1 defined over a number field $K$ by proving the finiteness of isomorphism classes of isogenous abelian varieties over $K$. The "first" major step from Mordell's conjecture to what Faltings did came 15 years earlier when Parshin showed that a certain conjecture of Shafarevich would imply Mordell's conjecture. In this talk, I'll focus on motivating and sketching Parshin's argument in an accessible manner and provide some heuristics on how to get from Faltings' finiteness statement to the Shafarevich conjecture.

A remarkable theorem due to Khovanskii asserts that for any finite subset $A$ of an abelian group, the cardinality of the $h$-fold sumset $hA$ grows like a polynomial for all sufficiently large $h$. However, neither the polynomial nor what sufficiently large means are understood in general. We obtain an effective version of Khovanskii's theorem for any $A \subset \mathbb{Z}$ whose convex hull is a simplex; previously such results were only available for $d = 1$. Our approach also gives information about the structure of $hA$, answering a recent question posed by Granville and Shakan. The work is joint with Leo Goldmakher at Williams College.

The Dirichlet class number formula gives an expression for the residue at s=1 of the Dedekind zeta function of a number field K in terms of certain quantities associated to K. Among those is the regulator of K, a certain determinant involving logarithms of units in K. In the 1980s, Don Zagier gave a conjectural expression for the values at integers s $\geq$ 2 in terms of "higher regulators", with polylogarithms in place of logarithms. The goal of this talk is to give an algebraic-geometric interpretation of these polylogarithms. Time permitting, we will also discuss a similar picture for Hasse--Weil L-functions of elliptic curves.
 

The Riemann hypothesis (RH) is one of the great open problems in mathematics. It arose from the study of prime numbers in an analytic context, and—as often occurs in mathematics—developed analogies in an algebraic setting, leading to the influential Weil conjectures. RH for curves over finite fields was proven in the 1940’s by Weil using algebraic-geometric methods. In this talk, we discuss an alternate proof of this result by Stepanov (and Bombieri), using only elementary properties of polynomials. Over the decades, the proof has been whittled down to a 5 page gem! Time permitting, we also indicate connections to exponential sums and the original RH.
 

We will discuss some problems around independence of l in compatible systems of Galois representations, mostly focusing on the independence of l of algebraic monodromy groups. We will explain how these problems fit into the context of the Langlands program, and present results both in characteristic zero and in positive characteristic settings.

Understanding the size of the rational points on a curve of higher genus is one of the major open problems in the theory of Diophantine equations. In this talk I will discuss the related problem of understanding how close together rational points can get. I will also discuss the relation to the subject of (generalised) Wieferich primes.

The goal of this talk is to introduce a way to use the philosophy of Random Matrix Theory to understand, pose, and maybe even solve problems about p-adic matrices.

A long time ago, Grothendieck made some conjectures. This has resulted in some things.

Gauss noted quadratic reciprocity to be among his favourite results, and any undergrad will quickly pick up on just how strange it is despite a plethora of elementary proofs. By 1930, E. Artin had finalized Artin reciprocity which wondrously subsumed all previous generalizations, but was still confined to abelian contexts. An amicable non-abelian reciprocity remains a driving force in number-theoretic research.

In this talk, I'll recount Artin reciprocity and show it implies quadratic and cubic reciprocity. I'll then talk about some candidate non-abelian reciprocities, and in particular, which morals of Artin reciprocity they preserve.

Given a number field K, it is natural to ask whether it has a finite extension with ideal class number one. This question can be translated into a fundamental question in class field theory, namely the class field tower problem. In this talk, we are going to discuss this problem as well as its solution due to Golod and Shafarevich using methods of group cohomology.
 

L-functions have become a vital part of modern number theory over the past century, allowing comparisons between arithmetic objects with seemingly very different properties. In the first part of this talk, I will give an overview of where they arise, their properties, and the mathematics that has developed in order to understand them. In the second part, I will give a sketch of the beautiful result of Herbrand-Ribet concerning the arithmetic interpretations of certain special values of the Riemann zeta function, the prototypical example of an L-function.

Let $[N] = \{1,..., N\}$ and let $A$ be a subset of $[N]$. A result of Sárközy in 1978 showed that if the difference set $A-A = \{ a - a’: a, a’ \in A\}$ does not contain any number which is one less than a prime, then $A = o(N)$. The quantitative upper bound on $A$ obtained from Sárközy’s proof has be improved subsequently by Lucier, and by Ruzsa and Sanders. In this talk, I will discuss my work on this problem. I will give a brief introduction of the iteration scheme and the Hardy-Littlewood method used in the known proofs, and our major arc estimate which leads to an improved bound.

A subset of the integers larger than 1 is called $\textit{primitive}$ if no member divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ over $n$ in a primitive set $A$ is universally bounded for any choice of $A$. In 1988, he famously asked if this universal bound is attained by the set of prime numbers. In this talk we shall discuss some recent progress towards this conjecture and related results, drawing on ideas from analysis, probability, & combinatorics.

Given a prime p and an elliptic curve E (say over Q), one can associate a "mod p Galois representation" of the absolute Galois group of Q by considering the natural action on p-torsion points of E.

In 1972, Serre showed that if the endomorphism ring of E is "minimal", then there exists a prime P(E) such that for all p>P(E), the mod p Galois representation is surjective. This raised an immediate question (now known as Serre's uniformity conjecture) on whether P(E) can be bounded as E ranges over elliptic curves over Q with minimal endomorphism rings.

I'll sketch a proof of this result, the current status of the conjecture, and (time permitting) some extensions of this result (e.g. to abelian varieties with appropriately analogous endomorphism rings).

Arakelov geometry studies schemes X over ℤ, together with the Hermitian complex geometry of X(ℂ).
Most notably, it has been used to give a proof of Mordell's conjecture (Faltings's Theorem) by Paul Vojta; curves of genus greater than 1 have at most finitely many rational points.
In this talk, we'll introduce some of the ideas behind Arakelov theory, and show how many results in Arakelov theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.

Given a morphism of schemes of characteristic p, we can construct a morphism from the exterior algebra of Kahler differentials to the cohomology of De Rham complex, which is an isomorphism when the original morphism is smooth.

The goal of the talk is to present a proof of the following statement:
Let (K,v) be an algebraic extension of (Q_p,v_p) whose completion is perfectoid. We show that K is relatively decidable to its tilt K^♭, i.e. if K^♭ is decidable in the language of valued fields, then so is K. 
In the first part [of the talk], I will try to cover the necessary background needed from model theory and the theory of perfectoid fields.

In 1941, Chabauty gave a way to compute the set of rational points on specific curves. In 2004, Minhyong Kim showed how to extend Chabauty's method to a bigger class of curves using anabelian methods. In the talk, I will explain Chabauty's method and give an outline of how Kim extended those methods.

The principal ideal theorem (1930) guaranteed that any number field K would embed into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will finish the proof of their cohomological result and thus fully justify how it settled the class field tower problem.

The principal ideal theorem (1930) ascertained that any number field K embeds into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will sketch the proof of their cohomological result and explain how it settled the class field tower problem.

A particularly active area of research in modern algebraic number theory is the study of a class of numbers, called periods. In their simplest form, periods are integrals of rational functions over domains defined by rational in equations. They form a ring, which encompasses all algebraic numbers, logarithms thereof and \pi. They arise in the study of modular forms, cohomology and quantum field theory, and conjecturally have a sort of Galois theory.

We will take a whirlwind tour of these numbers, before discussing non-periods. In particular, we will sketch the construction of an explicit non-period, often forgotten about.

We outline what we expect from a good cohomology theory and introduce some of the most common cohomology theories. We go on to discuss what properties each should encode and detail attempts to fit them into a common framework. We build evidence for this viewpoint through several worked number theoretic examples and explain how many of the key conjectures in number theory fit into this theory of motives.

Let S be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by S. In this talk we establish a necessary and sufficient condition for C to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set S. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree 2 polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions (this is a joint work with A.Ferraguti and R.Schnyder). Time permitting, we will also describe how to use character sum techniques to bound the size of the graph determined by the generating set (this is a joint work with D.R. Heath-Brown).