Term 3

Abstracts (Term 3)

Week 1 - 22nd April

Lilybelle Cowland Kellock (University College London) - [Rescheduled] 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 $\geq 5$, the dual graph of the special fibre of the minimal regular model of $E$ over $K^{\text{unr}}$ can be read off from the valuation of $j(E)$ and $\Delta_E$. 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.

Week 2 - 29th April

Benjamin Bedert (University of Oxford) - On Unique Sums in Abelian Groups

In this talk, we will study the problem in additive combinatorics of determining for a finite Abelian group $G$ the size of its smallest subset $A\subset G$ that has no unique sum, meaning that for every two $a_1,a_2\in A$ we can write $a_1+a_2=a’_1+a’_2$ for different $a’_1,a’_2\in A$. We begin by using classical rectification methods to obtain the previous best lower bounds of the form $|A|\gg \log p(G)$, which stood for 50 years. Our main aim is to outline the proof of a recent improvement and discuss some of its key notions such as additive span, additive dimension and the density increment method. This talk is based on Bedert, B. On Unique Sums in Abelian Groups. Combinatorica (2023).

Week 3 - 6th May

Beatriz Barbero Lucas (University College Dublin) - Obtaining new quantum codes from Generalized Monomial-Cartesian Codes

Week 4 - 13th May

Bijay Raj Bhatta (University of Manchester) - Height bounds on Lattices with skew-Hermitian forms over Type IV algebras

In this talk, we will discuss about the certain cases of Zilber-Pink conjecture on unlikely intersections in moduli space $\mathcal{A}_g$ of principally polarised abelian varieties of dimension g. In particular, we will talk about the case (mainly proving the effective height bound) when the associated endomorphism algebra is of Type IV (Albert types). This extends the work of Daw and Orr who proved Type I and II cases assuming Galois bounds.

Week 5 - 20th May

Zachary Feng (University of Oxford) - Eigenvarieties and $p$-adic propagation of automorphy

Functoriality is a key feature in Langlands’ conjectured relationship between automorphic representations and Galois representations; it predicts that certain Galois representations are automorphic, i.e. should come from automorphic representations. We discuss the idea of $p$-adic propagation of automorphy, which seeks to establish the automorphy of everything in a “neighbourhood” given the automorphy of something in that neighbourhood. The “neighbourhoods” that we consider will be the irreducible components of a $p$-adic analytic space called the eigenvariety, which parameterizes $p$-adic automorphic representations. This technique was introduced by Newton and Thorne in their proof of symmetric power functoriality, and can be adapted to investigate similar problems.

Week 6 - Tuedsay 28th May

Harmeet Singh (University College London) - The Hasse norm principle

For an extension of number fields K/k, we say that the Hasse norm principle holds if an element of k that is a norm everywhere locally is also a norm globally. The study of when this holds has been of interest ever since Hasse proved his famous norm theorem in 1931. In this down-to-earth talk, I'll begin by introducing the Hasse norm principle, discuss some of the instances when it holds, and provide an overview of the main ideas that go into proving these results.

Week 7 - 3rd June

Mike Daas (Leiden University) - A $p$-adic analogue of a formula by Gross and Zagier

In their 1984 paper “On singular moduli”, Gross and Zagier proved an explicit factorisation formula for the norm of the difference between two CM-values of the classical $j$-function. In 2022, it was conjectured by Giampietro and Darmon that the CM-values of certain $p$-adic theta-functions on Shimura curves should obey similar factorisation patterns. In this talk, we explore the classical result about the $j$-function, discuss its proofs and outline how the study of infinitesimal deformations of $p$-adic Hilbert Eisenstein series was used to settle the conjectures about the theta-function. This $p$-adic analytic approach bears resemblance to some of the newly developed methods in modern RM-theory.

Week 8 - 10th June

Giorgio Navone (Kings College London) - Transcendental Brauer groups of certain K3 surfaces

The talk is a gentle introduction to the Brauer-Manin obstruction for algebraic K3 surfaces. In particular we will recall the most important definitions, like Brauer groups, the Brauer-Manin set, and Skorobogatov's conjecture. Then we will present ongoing work computing the transcendental Brauer group of a family of K3 surfaces, constructed from a planar cubic curve in a similar fashion to Kummer surfaces. After discussing the similarities and differences of the two problems, we will conclude by proposing an independent open question whose solution would be extremely relevant to this topic.

Week 9 - 17th June

Alexandros Groutides (University of Warwick) - Integral structures in spherical GL(2)-representations distinguished by an unramified maximal torus

Loeffler-Skinner-Zerbes (2020) and Loeffler (2021) introduce a ''local essence'' to norm relations. This involves zeta integrals and certain ''integral'' elements which inherit this sacred name out of global considerations. As such, a local integral analogue would be desirable. Due to the nature of these objects, it is natural to hope that a deeper underlying integral theory should exist at a representation-theoretic level; at least for nice enough pairs of groups (H,G). After a gentle introduction to local representation theory, we will briefly formulate and address the ''Heegner case'' of (Res_{E/Q}(G_m), GL(2)) with E/Q quadratic. To finish off, we will discuss how this relates to the integral behaviour of automorphic representations attached to modular forms.

Week 10 - 24th June

Eda Kirimli (University of Bristol) - Isogeny graphs of abelian surfaces

Isogeny-based cryptography is an active research area in post-quantum cryptography, and its security depends on variants of isogeny problems, namely the problem of finding an explicit isogeny between two abelian varieties. Although much of the research focused on isogenies of elliptic curves so far, it is interesting to understand isogeny graphs in dimension 2 in full generality, and the recent attacks using isogenies in higher dimensions switched the attention to dimension two. Unlike known methods, we use very geometrical and arithmetical objects, called refined Humbert invariants. We will propose possible promising applications of abelian surfaces with these invariants for isogeny-based cryptography.

Term 2

Abstracts (Term 2)

Week 1 - 8th January

Kenji Terao (University of Warwick) - Isolated points on modular curves

As is well known, Faltings's theorem settles the question of determining when a curve, defined over a number field, has infinitely many rational points. However, Faltings's work can also be used to understand when such a curve has infinitely many higher degree points, a study which gives rise to the notion of isolated points. In this talk, we will study some techniques for finding isolated points on curves, and see how they can be applied to the more structured world of modular curves.

Week 2 - 15th January

Sven Cats - Higher descent on elliptic curves

Let $E$ be an elliptic curve over a number field $K$ and $n \geq 2$ an integer. We recall that elements of the $n$-Selmer group of $E/K$ can be written as certain $n$-coverings of $E/K$. Writing the elements in this way is called conducting an explicit $n$-descent. One of the applications of explicit $n$-descent is in finding generators of large height for $E(K)$ and from this point of view one would like to be able to take $n$ as large as possible. Already for $n \geq 5$ it is computationally challenging to apply the known algorithms for explicit $n$-descent. We discuss two ways around this: Improving a $p$-isogeny descent to a $p$-descent and combining $n$- and $(n+1)$-descents to $n(n+1)$-descent.

Week 3 - 22nd January

Cedric Pilatte - Graph eigenvalues and the logarithmic Chowla conjecture in degree 2

The Liouville function $\lambda(n)$ is defined to be +1 if $n$ is a product of an even number of primes, and -1 otherwise. The statistical behaviour of $\lambda$ is intimately connected to the distribution of prime numbers. In many aspects, the Liouville function is expected to behave like a random sequence of +1 and -1. For example, the two-point Chowla conjecture predicts that the average of $\lambda(n)\lambda(n+1)$ over $n < x$ tends to zero as $x$ goes to infinity. In this talk, I will discuss quantitative bounds for a logarithmic version of this problem.

Week 4 - 29th October

Sebastien Monnet - 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.

Week 5 - 5th February

Yan Yau Cheng - Arithmetic Chern Simons Theory

Mazur first observed in the 60s a deep analogy between the embedding of a knot in a 3-manifold and primes in a number field. Witten showed that knot invariants can be obtained by computations from quantum field theory. Using ideas from this analogy, Minhyong Kim and his collaborators developed the study of arithmetic field theories. This talk will be an introduction to Arithmetic Field Theories, in particular focusing on Arithmetic Chern-Simons Theory.

Week 6 - 12th February

Jackie Vorhos - On the average least negative Hecke eigenvalue

In this talk we discuss the first sign change of Fourier coefficients of newforms, or equivalently Hecke eigenvalues. We will see this to be an analogue of the least quadratic non-residue problem, of which the average was investigated by Erdős in 1961. In fact, we will see that the average least negative prime Hecke eigenvalue holds the same (finite) value as the average least quadratic non-residue, under GRH. This is mainly due to the fact that Hecke eigenvalues at primes are equidistributed with respect to the Sato-Tate measure, a consequence of the Sato-Tate conjecture that was proven in 2011. We further explore the so-called vertical Sato-Tate conjecture to show the average least Hecke eigenvalue has a finite value unconditionally.

Week 7 - 19th February

Khalid Younis - The distribution of smooth numbers

A number is said to be y-smooth if all of its prime factors are at most y. In much the same way as one studies primes, one can ask how many smooth numbers there are less than a large quantity x, whether they are spread evenly among arithmetic progressions, or how they are distributed in short intervals. In this talk, we will address some of these questions, with a focus on recent work on short intervals. In doing so, we will explore the connection with zeros of the Riemann zeta function.

Week 8 - 26th February

Seth Hardy - 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 would 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.

Term 1

Abstracts (Term 1)

Week 2 - 9th October

Abdul AlfarajLink opens in a new window (University of Bath) - On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences

Elliptic divisibility sequences are sequences generated by the denominators of the y-coordinates of multiples of some fixed rational point on an elliptic curve defined over the rational numbers. We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves belonging to a certain family. We achieve this by using the modularity of elliptic curves over real quadratic number fields. This work was part of my MASt project at Warwick, which was supervised by Samir Siksek.

Week 3 - 16th October

Isabel (Izzy) RendellLink opens in a new window (King's College London) - 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 and assume as few prerequisites as possible and demonstrate methods by examples.

Week 4 - 23th October

Arshay ShethLink opens in a new window (University of Warwick) - The Hilbert-Polya dream: finding determinant expressions of zeta functions

The Hilbert-Polya dream, which seeks to express the Riemann zeta function as a characteristic polynomial of an operator on a Hilbert space, is one possible approach to prove the Riemann Hypothesis. While this approach has never been successfully carried out, its core principle- finding determinant expression of zeta functions- has manifested itself in several different areas of number theory in the last century. In this talk, we will attempt to give a panoramic survey of the Hilbert-Polya dream.

Week 5 - 30th October

Maryam NowrooziLink opens in a new window (University of Warwick) - Perfect Powers in Elliptic Divisibility Sequences

The problem of determining all perfect powers in a sequence has always been interesting to mathematicians. The problem we are interested in is to prove that there are finitely many perfect powers in elliptic divisibility sequences. Abdulmuhsin Alfaraj proved that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the from $y^2=x(x^2+b)$, where $b$ is any positive integer. The main goal of our project is to generalize this result for elliptic divisibility sequences generated by any non-integral point on all elliptic curves$y^2=x^3+ax^2+bx+c$. This is a joint work with Samir Siksek.

Week 6 - 6th November

Alexandros KonstantinouLink opens in a new window (University College London) - Unveiling the power of isogenies: From Galois theory to the Birch and Swinnerton-Dyer conjecture

In this talk, we have a two-fold aim. Firstly, we illustrate a method for constructing isogenies using basic Galois theory and representation theory of finite groups. By exploiting the isogenies thus constructed, we shift our focus to the second aspect of our talk: the investigation of ranks of Jacobians with emphasis on predictions made by the Birch and Swinnerton-Dyer conjecture. Finally, we showcase the utility of our approach for studying ranks through various applications. These include a unified framework for studying classical isogenies and ranks, as well as a new proof for the parity conjecture for elliptic curves defined over number fields. This is joint work with V. Dokchitser, H. Green and A. Morgan.

Week 7 - 13th November

Amelia Livingston (University College London) - The Langlands correspondence for algebraic tori

This talk is an introduction to the easiest case of the Langlands correspondence. The correspondence "for $\mathrm{GL}_1$" reduces to class field theory, and using elementary techniques from group cohomology, Langlands extended this from $\mathrm{GL}_1$ to any algebraic torus. This setting involves no analysis, and provides a friendly first look at a couple of the objects involved in more general cases of the Langlands program.

Week 8 - 20th November

Robin AmmonLink opens in a new window (University of Glasgow) - Cohen--Lenstra Heuristics for Ray Class Groups

Even though it is an important object in number theory, for a long time the structure of the ideal class group of a number field seemed very mysterious. To get a better understanding of it, H. Cohen and H. Lenstra started to study the statistical behaviour of class groups, taking a new perspective in number theory. They realised that the behaviour appears to be governed by a fundamental principle about the distribution of random mathematical objects, and made influential conjectures about the distribution of ideal class groups known as the Cohen--Lenstra heuristics.

In my talk, I will give an introduction to the Cohen--Lenstra heuristics and discuss the ideas of arithmetic statistics and the fundamental principle that underlie them. I will then talk about work in progress that aims to generalise Cohen and Lenstra's conjectures to ray class groups.

Week 9 - 27th November

Benjamin BedertLink opens in a new window (University of Oxford) - On Unique Sums in Abelian Groups

In this talk, we will study the old problem in additive combinatorics of determining for a finite Abelian group $G$ the size of its smallest subset $A\subset G$ that has no unique sum, meaning that for every two $a_1,a_2\in A$ we can write $a_1+a_2=a'_1+a'_2$ for different $a'_1,a'_2\in A$. We begin by using classical rectification methods to obtain the previous best lower bounds of the form $|A|\gg \log p(G)$. Our main aim is to outline the proof of a recent improvement and discuss some of its key notions such as additive dimension and the density increment method. This talk is based on Bedert, B. On Unique Sums in Abelian Groups. Combinatorica (2023).

Week 10 - 4th December

Tsz Kiu Harvey YauLink opens in a new window (University of Cambridge) - An introduction to the Brauer-Manin obstruction

To study the rational points on a variety, one useful tool is to study it over a completion of the rationals, and in many cases this suffices to prove there are no rational points. However, sometimes this method is insufficient to prove the nonexistence of rational points, and many such examples have been found over the years. The Brauer-Manin obstruction provides a general explanation for these examples, and was first described by Y. Manin. This talk will give an introduction to the topic and construct some explicit examples of the obstruction on curves and surfaces.