Past Junior Number Theory Seminar

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).

The large sieve is a powerful analytic tool in number theory, with many beautiful and diverse applications. In its most general form it resembles an approximate Bessel's inequality, and this clear modern theory rests on the combined effort of countless mathematicians in the mid-twentieth century -- Linnik, Roth, Selberg, Montgomery, Vaughan, and Bombieri, to name a few. However, it is hardly obvious to the beginner why this rather abstract inequality should be called 'large', or 'sieve'. In this introductory talk, aimed particularly at new graduate students, we discuss the rudimentary theory of the large sieve, some particular applications to sieving problems, and (at least one) proof. 

I will describe a sketch of the proof of Grothendieck conjecture on fundamental groups.
 

Given an integer $d$ such that $2 \leq d \leq 50$, we want to
answer the question: When is the sum of
$d$ consecutive cubes a perfect power? In other words, we want to find all
integer solutions to the equation
$(x+1)^3 + (x+2)^3 + \cdots + (x+d)^3 = y^p$. In this talk, we present some
of the techniques used to tackle such diophantine problems.

I will explain how to find an explicit embedding of the Kummer variety of a higher genus curve into projective space and discuss applications of such an embedding to the study of rational points on Jacobians of curves, as well as the original curves.

Using the "trichotomy principle" by Boris Zilber I will give model theoretic proofs of appropriate versions of Torelli theorem and Borel-Tits theorem. The first one has interesting applications to anabelian geometry, I won't assume any prior knowledge in model theory.

Consider the following question. Given a $k$-colouring of the positive integers, must there exist a solution to $x+y=z^2$ with $x,y,z$ all the same colour (and not all equal to 2)? Using $10$ colours a counterexample can be given to show that the answer is "no". If one instead asks the same question over $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$, the answer turns out to be "yes", provided $p$ is large enough in terms of the number of colours used.  I will talk about how to prove this using techniques developed by Ben Green and Tom Sanders. The main ingredients are a regularity lemma, a counting lemma and a Ramsey lemma.

Last year, G. Kings and D. Rossler related the degree zero part of the polylogarithm
on abelian schemes pol^0 with another object previously defined by V. Maillot and D.
Rossler. More precisely, they proved that the canonical class of currents constructed
by Maillot and Rossler provides us with the realization of pol^0 in analytic Deligne
cohomology.
I will show that, adding some properness conditions, it is possible to give a
refinement of Kings and Rossler’s result involving Deligne-Beilinson cohomology
instead of analytic Deligne cohomology.

Many theorems and conjectures in prime number theory are equivalent to finding solutions to certain linear equations in primes -- witness Goldbach's conjecture, the twin prime conjecture, Vinogradov's theorem, finding k-term arithmetic progressions, etcetera. Classically these problems were attacked using Fourier analysis -- the 'circle' method -- which yielded some success, provided that the number of variables was sufficiently large. More recently, a long research programme of Ben Green and Terence Tao introduced two deep and wide-ranging techniques -- so-called 'higher order Fourier analysis' and the 'transference principle' -- which reduces the number of required variables dramatically. In particular, these methods give an asymptotic formula for the number of k-term arithmetic progressions of primes up to X. In this talk we will give a brief survey of these techniques, and describe new work of the speaker, partially ongoing, which applies the Green-Tao machinery to count prime solutions to certain linear inequalities in primes -- a 'higher order Davenport-Heilbronn method'. 

A question by Poonen asks whether iterating the étale-Brauer set can give a finer obstruction set. We tackle the algebraic version of Poonen's question and give, in many cases, a negative answer.

Much of the arithmetic behaviour of an elliptic curve can be understood by examining its mod p reduction at some prime p. In this talk, we will aim to explain some of the ways we can define the mod p reduction, and the classifications of which reduction types occur.

Topics to be covered include the classical reduction types (good/multiplicative/additive), the Kodaira-Neron reduction types that refine them, and the Raynaud parametrisation of a semistable abelian variety. Time permitting, we may also discuss joint work with Vladimir Dokchitser classifying the semistable reduction types of 2-dimensional abelian varieties.

I will discuss the many appearances of the class of IP sets in classical theorems of combinatorial number theory and ergodic theory. Our point of departure will be the celebrated theorem of Hindman on partition regularity of IP sets, which is crucial for the introduction of IP-limits. We then discuss how existence of certain IP-limits translates into recurrence statements, which in turn give rise to results in number theory via the Furstenberg correspondence principle. Throughout the talk, the methods of ergodic theory will play an important role - however, no prior familiarity with them is required.

Starting from Hilbert's 10th problem, I will explain how to characterise the set of integers by non-solubility of a set of polynomial equations and discuss related challenges. The methods needed are almost entirely elementary; ingredients from algebraic number theory will be explained as we go along. No knowledge of first-order logic is necessary.

The study of rational points on K3 surfaces has recently seen a lot of activity. We discuss how to compute the Picard rank of a K3 surface over a number field, and the implications for the Brauer-Manin obstruction.

I will introduce classical results on finiteness theorem with a way of connecting them to idea of covering spaces. I will talk about the proof of FLT under this connection.

In this talk I will define algebraic automorphic forms, first defined by Gross, which are objects that are conjectured to have Galois representations attached to them. I will explain how this fits into the general picture of the Langlands program and, giving some examples, briefly describe one method of proving certain cases of the conjecture. 

I will introduce the general idea of p-adic Hodge theory from the view point of a beginner. Also, I will give a sketch of the proof of the crystalline comparison theorem in the case of good reduction using 'almost mathematics'.

Many hard problems in Diophantine geometry have analogues over function fields which are less hard. I will give some examples.

We will give a sketch overview of Scholze's theory of perfectoid spaces and the tilting equivalence, starting from Huber's geometric approach to valuation theory. Applications to weight-monodromy and p-adic Hodge theory we will only hint at, preferring instead to focus on examples which illustrate the philosophy of tilting equivalence.
 

After some generalities on étale fundamental groups and anabelian geometry, I will explore some of the current results on the section conjecture, including those of Koenigsmann and Pop on the birational section conjecture, and a recent unpublished result of Mohamed Saidi which reduces the section conjecture for finitely generated fields over the rationals to the case of number fields.

In this talk I will describe two instances of Langlands functoriality concerning the group $\mathrm{Sp}_{2n}$. I will then very briefly explain how this enables one to attach Galois representations to automorphic representations of (inner forms of) $\mathrm{Sp}_{2n}$. 

In 1937 Vinogradov showed that every sufficiently large odd number is the sum of three primes, using bounds on the sums of additive characters taken over the primes. He was improving, rather dramatically, on an earlier result of Schnirelmann, which showed that every sufficiently large integer is the sum of at most 37 000 primes. We discuss a natural analogue of this question in the multiplicative group (Z/pZ)* and find that, although the current unconditional character sum technology is too weak to use Vinogradov's approach, an idea from Schnirelmann's work still proves fruitful. We will use a result of Selberg-Delange, an application of a small sieve, and a few easy ideas from additive combinatorics. 

In this talk I will discuss the notion of o-minimality, which can be approached from either a model-theoretic standpoint, or an algebraic one.  I will exhibit some o-minimal structures, focussing on those most relevant to number theorists, and attempt to explain how o-minimality can be used to attain an assortment of results.

In this talk I will describe Arthur's classification of automorphic representations of symplectic and orthogonal groups using automorphic representations of $\mathrm{GL}_N$.

We will discuss Raynaud's classical theory on Néron models of Jacobians of curves, and mention some tropical aspects of the theory that help us understand modular curves from a modern non-Archimedean viewpoint. There will be an annoyingly large number of examples illustrating the key principles throughout.