## London Junior Number Theory Seminar

2023—2024

Organised by Giorgio Navone, Harmeet Singh and Harry Spencer

Held on Tuesdays at 5pm in S-3.20, King's College, London, Strand Campus.

(Note the minus!)

## Our aim

The aim of this seminar is to prepare PhD students in number theory — particularly those just starting out — for attending seminars and talks aimed at a more experienced audience, such as the London Number Theory Seminar. It is also a space where speakers can have the opportunity to share any mathematics that they think will make an impact on the other students. Lastly, it is a social enterprise which brings a community of young number theorists together on Tuesday evenings.

## Links

# Talks

Congruent elliptic curves and twists of modular curves

Let us say that two elliptic curves E, F over Q are congruent modulo a prime if their p-torsion Galois-modules are isomorphic. The Frey–Mazur conjecture states that, when p is large enough with respect to E, any elliptic curve which is congruent to E is isogenous to E. I will explain how to reformulate this question as a problem of finding rational points on a suitable curve. I will then describe in more detail the (somewhat simpler) situation when the Galois image of E is small. Time permitting, I will finally discuss how to apply recent results about the Bloch–Kato conjecture for GL2 × GL2 to this situation.

Newform Eisenstein congruences of local origin

The theory of Eisenstein congruences dates back to Ramanujan’s surprising discovery that the Fourier coefficients of the discriminant function are congruent to the 11th power divisor sum modulo 691. This observation can be explained via the congruence of two modular forms of weight 12 and level 1; the discriminant function and the Eisenstein series, E₁₂. Eisenstein congruences were later used by Ribet in his proof of the converse to Herbrand's theorem.

In this talk, I will first discuss the steps Ribet used in his proof and then compare these steps to my recent work, joint with Dan Fretwell, on congruences between Eisenstein series and newforms of weight k > 2, squarefree level and non-trivial character.

### Riccardo Tosi, 21 May 2024

A geometric approach to irrationality proofs of zeta values

From class number formulae to mixed Tate motives, the values of the Riemann zeta function at positive integers appear in many areas of mathematics, yet their arithmetic properties are still poorly understood. Though conjectured to be transcendental, most of them are not even known to be irrational. In this talk, we will have a look at how irrationality proofs are usually carried out and we will then turn to some recent inputs coming from period integrals over moduli spaces of curves.

### Sven Cats, 14 May 2024

23AndMe

During a course on complex multiplication in Luminy in May 2023, we considered polynomials Υₚ(X,Y), one for each prime number p. The definition being analogous to that of the p-th modular polynomial Φₚ(X,Y), we expected that the coefficients of Υₚ(X,Y) would satisfy a congruence modulo p analogous to Kronecker's congruence. A few of us set out to compute Υₚ(X,Y) for small values of p and noticed not only congruences modulo p, but more surprisingly also modulo 23. This observation led us on an adventure to try to figure out where these congruences came from. In this talk we will summarise our findings and review some class field theory, Galois representations and congruences of modular forms.

Arithmetic Chern Simons Theory and Linking Numbers

Mazur first observed in the '60s a deep analogy between knots in a 3-manifold and primes in a number field. In the '80s Witten showed that knot invariants can be computed using path integrals coming from quantum field theory. More recently, Minhyong Kim and his collaborators combined these ideas to develop the study of arithmetic field theories in order to compute new arithmetic invariants.

In this talk I will introduce an Arithmetic QFT known as Arithmetic Chern Simons Theory, and define a notion of arithmetic linking numbers, analogous to linking numbers of knots.

Exponential sums with random multiplicative coefficients

The study of exponential sums with multiplicative coefficients is classical in analytic number theory. For example, understanding exponential sums with coefficients given by the Liouville function could offer profound insights into the distribution of primes in arithmetic progressions. Unfortunately, our current understanding of these sums is far from what we expect to be the truth. In this talk, we will explore an alternative approach: considering exponential sums with random multiplicative coefficients. We will introduce the relevant theory and discuss recent progress in proving conjecturally sharp lower bounds for the size of a large proportion of these exponential sums.

Manin's conjecture for P¹ as a compactification of a non-trivial twist of Gₐ

Let X be a Fano variety over a global field F. Choosing an appropriate height function on X, Manin's conjecture gives a prediction for the asymptotic of the number of points of bounded height on a suitable Zariski open dense subset of X. In this talk, I will sketch a proof of the conjecture for X=P¹, viewed as an equivariant compactification of a non-trivial twist of Gₐ, defined over the global function field F=F₂(t). We do so by studying the poles of the height zeta function using harmonic analysis techniques on the group of adelic points of the twist.

A generalisation of Tate's algorithm for hyperelliptic curves

Tate's algorithm tells us that, for an elliptic curve E over a discretely valued field K with residue characteristic >3, the dual graph of the special fibre of the minimal regular model of E over the maximal unramified extension of K can be read off from the valuation of j(E) and Δ(E). This is really important for calculating Tamagawa numbers of elliptic curves, which are involved in the refined Birch and Swinnerton–Dyer conjecture formula. For a hyperelliptic curve C/K, we can ask if we can give a similar algorithm that gives important data related to the curve and its Jacobian from polynomials in the coefficients of a Weierstrass equation for C/K. This talk will be split between being an introduction to cluster pictures of hyperelliptic curves, from which the important data can be gathered, and a presentation of how the cluster picture can be recovered from polynomials in the coefficients of a Weierstrass equation.

### Zerui Tan, 9 April 2024

Computing Frobenius on algebraic de Rham cohomology

The de Rham cohomology groups give an alternative way to compute the cohomology groups on smooth manifolds. In the algebraic setting, we can also define the algebraic de Rham cohomology groups over any field. In particular, we are interested in the reduction algorithms and the Frobenius actions we can compute on the algebraic de Rham cohomology groups of a smooth projective variety over a finite of p-adic field; these problems are of number theoretical interest (e.g. computing invariants and Coleman integrals and more). We intend to give a gentle introduction to algebraic de Rham cohomology groups and how to compute the reduction/Frobenius actions on them (not by hand).

A bridge between genus 2 curves and del Pezzo surfaces of degree 4

A longstanding conjecture of Colliot–Thelene predicts that the Brauer–Manin obstruction explains all violations of the Hasse principle for del Pezzo surfaces. For del Pezzo surfaces of degree 4 (dP4s), conditionally on Schinzel’s hypothesis and finiteness of Sha, this has been proven when the Brauer group is trivial. There is a bridge between dP4s and genus 2 curves that aids computation of the Brauer group, and in some situations, allows us to prove the existence of a rational point. I will explain how we plan on implementing this bridge in Magma, and time permitting, some cute examples of dP4s with nontrivial Brauer group that have a rational point.

Grothendieck group relations for GL2(Fₚ)

In this talk we introduce the Grothendieck group on finite dimensional representations of GL2(Fₚ) over Fₚ. The Grothendieck group relations help us to compute the Jordan–Holder factors of the representations of GL2(Fₚ), which contributes to the weight part of Serre's conjecture. We will also introduce the weight shifting operators that are cohomological analogues of the Hasse invariants and theta operators.

### Harvey Yau, 19 March 2024

Elliptic surfaces and Brauer–Manin obstructions

In this talk I will introduce some of the main theory and ideas about elliptic surfaces, and show how they can be applied, together with some Galois cohomology, to obtain elements of the Brauer group, and examples of Brauer–Manin obstruction on elliptic surfaces.

Representations of GL2(F) and equivariant vector bundles with connection on the Drinfeld upper half-plane

In this talk we introduce the Drinfeld upper half-plane, a non-archimedean analogue of the complex upper half-plane, with its natural action of GL2(F), F a finite extension of Qₚ. We will introduce the notion of an equivariant vector bundle with connection on the Drinfeld upper half-plane, and motivate why these are interesting to study from the point of view of the locally analytic representation theory of GL2(F). Finally, if time permits, we will explain my work in classifying the equivariant vector bundles with connection which come from the Drinfeld tower, and how these are related to the category of smooth representations of D^× where D is the division algebra of centre F and dimension 4. We won't assume a lot of prerequisites, and so this talk should be quite friendly and accessible to everyone.

### Josh Tilley, 5 March 2024

Exploring periods

A period is a complex number arising from the the integral of an algebraic function with algebraic coefficients over a semialgebraic set. Periods generalise algebraic numbers and include many of the fundamental constants in mathematics. They arise as values of the pairing between Betti cohomology and algebraic de Rham cohomology on a variety defined over Q. This talk will be an introduction to periods, basic facts about them, their role in arithmetic algebraic geometry, and the main conjectures in the area. I will discuss their relation to G-functions, and differential equations, and, if there is time, also discuss the connection to motives, and exponential periods.

Congruences of modular (eigen)forms

Two modular eigenforms are congruent when almost all of their Hecke eigenvalues are congruent modulo some fixed prime p. An interesting phenomenon is that two modular forms of different weights can be congruent to each other: the question of classifying all such weights were congruences occur is referred to as the weight part of Serre's conjecture. I will explain this for modular forms, where the conjecture is solved, and the relation with Galois representations can be made pleasantly explicit. If time allows I'll give a few hints as to why going beyond modular forms is much harder.

The Combinatorial Sieve

Sieve methods are a technique used in analytic number theory to estimate the size of sets, most notably sets of prime numbers. They have been behind some of the biggest breakthroughs surrounding the Twin Prime Conjecture. In this introductory talk, we will explore the axioms of sieve theory and the definitions underlying the combinatorial sieve. In particular we will look at Brun's pure sieve, including some results that can be proven using it.

The Effective Shafarevich Conjecture

Let K be a number field, d a positive integer, and S a finite set of primes of K. One of the crowning achievements of 20th century arithmetic geometry was Faltings's proof that there are only finitely many isomorphism classes of dimension d abelian varieties A/K with good reduction away from S. Whilst several effective algorithms have been developed to explicitly classify elliptic curves with good reduction outside a finite set of primes, effectively solving this problem in higher dimensions remains a challenge. In this talk, I will give a brief survey on some known methods for classifying abelian varieties, and will present some work in progress on classifying isogeny classes of abelian surfaces over Q with good reduction away from 2.

### Alexandros Groutides, 6 February 2024

Integral structures in smooth GL2(Qₚ)-representations

The representation theory of p-adic groups is a topic at crossroads which comes in many different flavours, and in this talk, we will taste some of them. Given an unramified maximal torus H in GL2 over Qₚ, we will report on recent work regarding integral structures in smooth (GL2 × H)(Qₚ)-representations. Inspired by work of Loeffler-Skinner-Zerbes, we will introduce and study certain integral lattices of functions which possess deep integral properties. We will then link them back to the prototypical construction in the style of op. cit and hint towards potential generalisations. If time permits, we will also discuss global applications to automorphic representations attached to modular forms.

Dissecting Jacobians of curves

In this talk, we focus on the isogeny decomposition of Jacobians of curves endowed with an action of a finite group G. In particular, we use basic Galois theory, which I will recall briefly, and representation theory of finite groups. We then use these tools to study the l-adic Tate module as a G-representation. Through examples, we illustrate how the G-module structure of the Tate module reveals ``dissections'' of the Jacobian variety into finer ``motivic'' pieces previously unseen when studying the Jacobian solely up to isogeny. This is a joint project with V. Dokchitser, H. Green and A. Morgan.

(Some) Nonabelian number fields with prescribed norms

Let α be a rational number and let Σ be a family of number fields. For each number field K in Σ, either α is a norm of K, or it is not. We might ask for what proportion of K in Σ that is the case. We will see that this is a natural question to ask, and that it is extremely hard in general. For an abelian group A, the case Σ = {A-extensions} was solved by Frei, Loughran, and Newton. We will discuss new results for the simplest class of nonabelian extensions: so-called "generic" number fields of a given degree.

Étale cohomology for the cohomies

How often have you heard "Ye, it's because this morphism is étale, innit?!" or "in order to carry on we need étale cohomology"? The common reaction is usually discomfort or numbness at best. The aim of this talk is to demystify this topic without going through SGA-kind of details, but focusing on the intuition instead.

Starting with the Weil conjectures as motivation, the main ideas and some properties will be presented and followed, if time permits, by the computation of the cohomology of a curve over an algebraically closed field.

Exploring the Hasse Norm Principle: Unravelling the algebraic mysteries

For a finite extension of number fields K/k, we say that the Hasse Norm Principle holds if an element of k which is a norm everywhere locally is in fact a global norm. The question of which extensions satisfy this property has been of interest ever since Hasse proved his famous norm theorem in 1931. In this poorly-planned talk, I will introduce the Hasse Norm Principle, mention various results on when it holds, and give some indication of how these are proved.

The Brauer–Manin obstruction

A Diophantine equation with a rational solution always has a real solution and a p-adic solution for each prime number p, but the converse is not always true — why?

### Yicheng Yang, 28 November 2023

Local and global Langlands correspondence for GLn

Roughly speaking, the Langlands reciprocity conjectures predict a correspondence between automorphic representations and Galois representations. In this talk I will introduce some basic concepts and statement for local and global Langlands for GLn and give some rough ideas for arguments in GL2.

### James Kiln, 21 November 2023

An introduction to eigencurves

Since Coleman and Mazur’s first construction of an eigencurve in 1998, these rigid analytic spaces have become an invaluable tool in the study of the relationship between modular forms and Galois representations. In this talk, I will introduce the notion of a rigid analytic space, construct Coleman and Mazur’s eigencurve, and mention how understanding the geometry of these spaces can help give insight into the Langlands program.

Roth's Theorem and the graph regularity method

A landmark result in additive combinatorics is Roth's Theorem on arithmetic progressions. It states that any sufficiently dense subset of the natural numbers contains a 3-term arithmetic progression. In this talk we will look at a graph theoretic proof of Roth's Theorem. We will discuss Szemerédi's graph regularity lemma, which roughly speaking finds a structural decomposition of an arbitrary graph into parts that behave in a "random-like" way. It then allows us to model an arbitrary graph by a random graph. Time permitting we will discuss how this method might be generalised to tackle Szemerédi's Theorem on k-term arithmetic progressions. This talk is entirely self-contained and has no prerequisites.

Étale fundamental groups

In this introductory talk, we explore the concept of étale fundamental groups, a powerful tool that extends the traditional fundamental group to the realm of schemes. We introduce an alternate definition of the topological fundamental group, which provides a framework for its generalisation to schemes and reveals a striking correspondence between Galois theory and the theory of covering spaces. Maybe rather surprisingly, we will try to avoid material related to schemes and will mostly focus on the motivation of the definition of the étale fundamental group and therefore very few prerequisites — outside undergraduate topology and Galois theory — are required.

Rational points on modular curves

The problem of finding rational points on modular curves is of great interest in number theory and arithmetic geometry, with many different methods in use in the subject. This will be an introductory talk where will see some key related theorems due to Faltings, Coleman and Mazur. I will discuss some methods for finding rational points, and how they can relate to other areas such as points on elliptic curves and the congruent number problem. Throughout the talk I will try to assume as few prerequisites as possible (you don’t need to know what a modular curve is!) and demonstrate methods by examples.

Bounding class numbers using elliptic curves

Paraphrased into modern language, a conjecture made by Gauss states that the number of imaginary quadratic fields with a given class number is finite. Several proofs of this fact were given in the previous century, but most of these results are ineffective. In this talk we look at effective lower bounds for such class numbers obtained by counting rational points on elliptic curves. The key ingredient here is a homomorphism from the rational points on an associated elliptic curve to the class group, discovered by Buell and rediscovered by Soleng. Any previous encounter with elliptic curves should be more than enough to be able to follow this talk.

The Petersson Trace Formula

In this introductory talk, we introduce modular forms and discuss the Hilbert space of cusp forms through an example — the Poincaré series. The interplay between the Poincaré series and the (Petersson) inner product leads to our main result. This is the Petersson trace formula, which gives a spectral expansion of the Fourier coefficients of cusp forms. The trace formula has several interesting consequences, and here we will briefly discuss an application to the distribution of critical values of L-functions attached to modular forms.

From class groups to elliptic curves or:

How I learned to stop worrying and love heuristics

We give some philosophical discussion on how frequently different structures ought to appear `in nature', and use this to motivate the Cohen--Lenstra heuristics for class groups. We then discuss analogous heuristics for ranks of elliptic curves over the rationals, justifying the well-known minimalists' conjecture and considering whether ranks ought to be bounded. As this is the first talk of the year, we require (almost) no prerequisites and will be exceptionally light on proof.

## Would you like to attend and/or give a talk?

If you'd just like to attend, then you are welcome to just turn up! If you don't have access to KCL, then make a note of one of our emails (as below), in case you have trouble entering the university.

firstname [dot] lastname [dot] 22 [at] ucl [dot] ac [dot] uk

If you'd like to give a talk, then please get in touch with any of the organisers.