This is the webpage for the London JNT seminar of 2024/2025, organised by Calle Sönne, Edwina Aylward, and Albert Lopez Bruch. Here you can find previous and upcoming talks. Last year's website can be viewed here.
This seminar provides a space for London-based PhD students in number theory to learn about the research of their peers, as well as having the chance to present their own work. The aim is to keep the talks at a understandable level, and to prepare students for the London number theory seminar. In addition, it is a chance for us to gather together, socialize, and enjoy the snacks!
When: Tuesdays at 5:00pm (see talks for exact dates).
Where: Room S-3.20, Strand Building, Kings College London.
Point-counting on varieties and random multiplicative functions.
24th June 2025, Thiago Solovera e Nery (Duisburg-Essen)
Abstract: Local models are schemes which model the singularities of integral Shimura varieties. In this talk, we explore the geometry and combinatorics involved in this theory, and explain the connection between the special fibre of this scheme with the wonderful compactification of the associated reductive group. When this group is unramified, this is a result of X. He. We then focus on the difficulties arising for ramified groups and what we can say about what happens in this case.
Geometrization conjecture of Fargues-Scholze and remarks on an extended version.
17th June 2025, Drimik Roy (IMJ-PRG)
Abstract: I'll explain some of the main results of the work of Fargues-Scholze, the history of the subject and some remarks on an extended geometrization program, which aims to geometrize T. Kaletha's work on rigid inner forms and isocrystals.
Point-counting on varieties and random multiplicative functions.
10th June 2025, Besfort Shala (University of Bristol)
Abstract: I will start with a survey on sums of random multiplicative functions, focusing on distributional questions and almost sure upper bounds and Ω-results. In this context, I will describe previous work with Jake Chinis on a central limit theorem for correlations of Rademacher multiplicative functions, as well as ongoing work on establishing almost sure sharp bounds for them. All of these probabilistic results arise from point-counting on varieties and their quadratic twists, and I will primarily focus on this connection.
Subconvex bounds for SO(4) x SO(3).
3rd June 2025, Blanca Gil Rosell (Aarhus University)
Abstract: In this talk, I will briefly introduce automorphic forms, L-functions, and the subconvexity problem. Then I will talk about my current project (still on progress), which concerns establishing a subconvex bound for L-functions attached to orthogonal groups SO(4) x SO(3). This work is based on the paper "Subconvex bounds for U(n+1) x U(n) in horizontal aspects”, by Yueke Hu and Paul Nelson (2023+), where they establish a subconvex bound valid in certain horizontal aspects for L-functions attached to automorphic representations of unitary groups U(n+1) x U(n). The ultimate goal is to find an analogue for orthogonal groups SO(n+1) x SO(n), but for now we will focus on the case n=3.
Abelian number fields with restricted ramification and rational points on stacks.
27th May 2025, Julie Tavernier (University of Bath)
Abstract: A conjecture by Malle gives a prediction for the number of number fields of bounded discriminant. In this talk I will give an asymptotic formula for the number of abelian number fields of bounded height whose ramification has been restricted and provide an explicit formula for the leading constant in terms of sums of Euler products. I will then describe how counting these number fields can be viewed as a problem of counting rational points on the stack BG (I'm not assuming any knowledge of stacks!) and how the existence of such number fields is controlled by a Brauer-Manin obstruction. Finally, I will briefly discuss how this stacky viewpoint can be used to prove a result in the vein of the Malle-Bhargava heuristics via the equidistribution of rational points on BG.
Constructing K3 surfaces as quotients of abelian surfaces.
20th May 2025, Alvaro Gonzalez Hernandez (University of Warwick)
Abstract: You have probably sat through many talks about elliptic curves, but have you ever wondered about what happens when you go beyond dimension one and study abelian surfaces? In this talk, I will give a crash course on how to work with these surfaces, drawn from what I have learned over the past three years.
My interest in abelian surfaces is motivated by my current work constructing examples of generalised Kummer surfaces, which are K3 surfaces that are quotients of abelian surfaces by finite automorphisms. I will explain why these surfaces are of interest to number theorists, and how they give rise to some interesting phenomena that only occur in positive characteristic.
Perfect powers in elliptic divisibility sequences.
13th May 2025, Maryam Nowroozi (University of Warwick)
Abstract: Let E be an elliptic curve over the rationals given by an integral Weierstrass model and let P be a rational point of infinite order. The multiple nP has the form (A_n / B_n^2, C_n / B_n^3) where A_n, B_n, C_n are integers with A_nC_n and B_n coprime and B_n positive. The sequence (B_n) is called the elliptic divisibility sequence generated by P . In this talk we answer the question posed in 2007 by Everest, Reynolds and Stevens: does the sequence (B_n) contain only finitely many perfect powers?
Rational points on K3 surfaces of degree 2.
6th May 2025, Júlia Martinez Marin (University of Bristol)
Abstract: K3 surfaces can be considered a 2-dimensional analogue of elliptic curves. However, their rational points are not yet well understood. In this talk, we focus on K3 surfaces of degree 2, arising as double covers of P^2 branched along a smooth sextic curve, and we use elliptic curves to produce infinitely many rational points on them.
Unramified geometric class field theory.
8th April 2025, Ken Lee (University of Oxford)
Abstract: Roughly speaking, class field theory for a number field K describes the abelianization of its absolute Galois group in terms of the idele class group of K. Geometric class field theory is what we get when K is instead the function field of a smooth projective geometrically connected curve X over a finite field. In this talk, I give a precise statement of geometric class field theory in the unramified case and describe how one can prove it by showing the Picard stack of X is the “free dualizable commutative group stack on X”. A key part is to show that the usual “divisor class group exact sequence“ can be done in families to give the adelic uniformization of the Picard stack by the moduli space of Cartier divisors on X.
The Hasse norm principle: degrees of failure.
1st April 2025, Harmeet Singh (KCL)
Abstract: I’m going to explain why I am basically a finite group theorist.
Gauss Manin Connection and p-adic Differential Equations in Number Theory.
25th March 2025, Zerui Tan (KCL)
Abstract: In this talk, we’ll unravel the origins of the Gauss Manin connection and explore how it helps us tackle families of integrals : a quest to uncover p-adic differential equations linked to families of Coleman integrals. Some dark technical lemmas will also be needed to obtain local bounds and beat the number of zeros of these analytic functions, which gives us arithmetic treasures. You will see a lot of numbers at the end of the talk, definitely number theory!
Moments of automorphic Rankin-Selberg L-functions.
18th March 2025, Jakub Dobrowolski (QMUL)
Abstract: Subconvexity problems have interested people for around 100 years, with Hardy and Littlewood's first positive result for a zeta function in 1923. Since then, the problem has been generalised to automorphic L-functions and has become a key area of analytic number theory. In this talk, we will present an approach to proving subconvexity using a moment estimate for a GL(2)xGL(2) Rankin-Selberg L-function.
Modular curves and isolated points.
11th March 2025, Kenji Terao (University of Warwick)
Abstract: Modular curves are objects of central importance in arithmetic geometry, parametrizing elliptic curves with particular Galois representations. They form a key part of the proof of results such as Fermat's Last Theorem and Mazur's torsion theorem. On the other hand, isolated points are "exceptional" low-degree points on curves, which lie outside the infinite families of low-degree points effected by the geometry of the curve. In this talk, I will aim to give a gentle introduction to these two concepts, the former via a stroll through the world of moduli spaces, the latter via an examination of Faltings's proof of the Mordell conjecture. In particular, little to no prior knowledge will be assumed. Time permitting, I will conclude with some recent advances on the intersection of these two notions.
Integral Hecke operators and applications on the modular curve X_0(p).
4th March 2025, Yicheng Yang (KCL)
Abstract: In general, Hecke operators on coherent cohomologies of Shimura varieties can be constructed via cohomological correspondence. We will motivate by the case of integral models of modular curves and show some integral analogue of classical results as applications.
Explicit descent for Jacobians of hyperelliptic curves.
25th February 2025, Lee Berry (KCL)
Abstract: The Mordell–Weil rank of the Jacobian of a curve is a fundamental invariant in arithmetic geometry, and its effective computation is invaluable in methods to determine rational points on higher-genus curves. In this talk, I will explain how explicit 2-descent can be used to determine effective upper bounds for the Mordell-Weil rank of Jacobians of hyperelliptic curves, and outline recent progress on explicit second descent to further refine these bounds.
Period-index problem with an arithmetic geometry twist.
18th February 2025, Giorgio Navone (KCL)
Abstract: This talk will be an introduction to the period-index problem for curves, focusing in particular on the case of the Weil-Châtelet group of an elliptic curve. We will then see an interesting application to quadratic points on del Pezzo surfaces of degree 4 over p-adic fields.
Quadratic Chabauty for Modular Curves.
11th February 2025, Isabel Rendell (KCL)
Abstract: Quadratic Chabauty is a specific case of the Chabauty-Kim method, which has been used to determine the rational points on certain modular curves in recent years, the first such curve being the ‘cursed curve’ due to Balakrishnan-Dogra-Müller-Tuitman-Vonk in 2017. Chabauty-Kim is a non-abelian generalisation of the Chabauty-Coleman method, and so the first part of the talk will introduce these topics. The second part will concern Quadratic Chabauty itself and its relation to p-adic heights. The current method requires an explicit plane model of the curve, which prevents generalisation of results. Therefore, we would like to replace this with something more ‘modular’: a natural candidate is q-expansions of modular forms. I will end the talk by discussing recent work towards ‘model-free’ methods due to Chen-Kedlaya-Lau.
Tensor product functoriality via p-adic propagation.
04th February 2025, Zachary Feng (University of Oxford)
Abstract: Tensor product functoriality predicts that the tensor product of two automorphic Galois representations should be automorphic. We will motivate this statement, and describe forthcoming work establishing this conjecture in some cases.
The unramified Fontaine-Mazur and Boston conjectures.
28th January 2025, Simon Gabriel Alonso (LSGNT)
Abstract: In 1995 Fontaine and Mazur published a series of conjectures about the "geometricity" of Galois representations. In this talk I will present one of those conjectures about unramified representations, as well as Boston's strenghtening of it.
In 2011 Allen and Calegari proved that the two conjectures are equivalent in the residually irreducible case. I will present this proof and quickly comment on how to prove a similar result in the residually reducible case.
Low-lying zeros of families of L-functions.
21st January 2025, Catinca Mujdei (UCL)
Abstract: The Katz-Sarnak philosophy aims to describe the distribution of the zeros of a family F of L-functions near the central point s=1/2, when the L-functions are ordered by their analytic conductor. It is conjectured that these distributions are governed by a symmetry group G(F) related to random matrix theory. I will discuss some results that provide encouraging evidence towards these conjectures for certain families.
O-minimality and Diophantine applications.
14th January 2025, Haoran Liang (KCL)
Abstract: Formulated around 1980’s by van der Dries, the theory of o-minimality emerged as a field at the border of logic (model theory), geometry and topology. Over a decade later, Pila and Zannier realized that some of the ideas of o-minimality had powerful arithmetic applications. Nowadays, o-minimality has surprisingly wide reach towards number theory and arithmetic geometry, e.g. via the Pila—Wilkie theorem, which is used in proofs of Lang’s conjecture, the Manin—Mumford conjecture, and the Andre—Oort conjecture. In this talk, we will embark on a brisk tour of this flourishing area and explore some of its Diophantine applications alluded to above. No prerequisites on model theory will be assumed.
The notes for this talk can be found here.
Towards non-abelian Selmer group Chabauty.
10th December 2024, Corijn Rudrum (KCL)
Abstract: A result of Chabauty from 1941 forms the basis for several explicit methods for computing the set of rational points on certain curves. After a quick introduction to more classical Chabauty methods, I will give a brief overview of Stoll's Selmer group Chabauty method from 2017, whose main advantage compared to the others is that it does not require computation of explicit generators for the Mordell-Weil group of the Jacobian. I will also discuss work in progress on a "non-abelian" version of this method inspired by Kim's generalisation of the Chabauty-Coleman method.
Lower bounds for the size of discrete sums of linear transformations.
3rd December 2024, Albert Lopez Bruch (KCL)
Abstract: Estimating the size of discrete sumsets is a central problem in additive combinatorics that has encouraged extensive research over the last few decades. A notable instance of this problem involves studying the sizes of sums of dilates of finite sets of integers. In 2008, Bukh established a sharp lower bound and proposed a higher-dimensional generalization for discrete sums of linear transformations, inspired by the classical Brunn-Minkowski inequality. In this talk, I will motivate this conjecture and outline the ideas behind recent progress. The talk has no prerequisites, so it should be accessible to everyone!
Reduction mod p and Models of Curves.
26th November 2024, Jakab Schrettner (UCL)
Abstract: Many number theoretic questions boil down to the study of certain varieties defined over number fields or local fields. One might try to study these objects by studying their reductions, defined over simpler (finite) fields. In this talk I will explain some aspects of this theory, applied to elliptic curves and Weierstrass models, to general projective curves and their minimal regular models, and (if time permits) to abelian varieties and Néron models.
A Variant of the Eichler-Shimura Isomorphism for Modular Forms with Prescribed Inertial Type.
19th November 2024, Nada Baessa (KCL)
Abstract: Modular symbols allow us to compute vector spaces of modular forms. A key step in this process is the Eichler-Shimura isomorphism between modular forms and a certain cohomology group. However we are often interested in modular forms with certain local information given by inertial type. For these modular forms, current methods become inefficient as the level increases. In this talk we will present a variant of the Eichler-Shimura isomorphism for modular forms with a prescribed inertial type.
Zeta functions of Simple Algebras.
12th November 2024, Alberto Acosta Reche (UCL)
Abstract: In his 1950 PhD thesis, John Tate used adelic methods to give an elegant proof of the analytic continuation and functional equation of L-functions attached to Hecke characters. In 1972, Roger Godement and Hervé Jacquet generalized his arguments to prove analogous properties for L-functions attached to automorphic representations of the group of invertible elements of any simple algebra over a global field (e.g. GL(n) over a number field). In this talk we will explain the main result of this theory and illustrate it with some examples.
Towards a fundamental domain for Bianchi groups.
29th October 2024, Sara Varljen (KCL)
Abstract: The modular group PSL2(Z) is a very familiar object when dealing with classical modular forms. If, instead of considering Z, we take the ring of integers O_k of an imaginary quadratic field k, we obtain Bianchi groups. They act on the hyperbolic 3-space, similarly to how PSL2(Z) acts on the hyperbolic plane, and are key to defining Bianchi modular forms.
In this talk I will give some geometric background on Bianchi groups and see how the fundamental domain for the action can be defined from a computational point of view, in analogy to the classical case. The talk should be very accessible to anyone who has ever come across the classical tessellation of the hyperbolic plane.
Associating Galois representations to mod p Hilbert modular forms.
22nd October 2024, Calle Sonne (Imperial)
Abstract: From the point of view of Serre's conjecture, it is interesting to be able to construct mod p Galois representations associated to mod p Hilbert modular forms. This could also be seen as an instance of the "mod p Langlands philosophy" (whatever that would mean). For most of this talk, I will be talking about modular curves, modular forms and mod p modular forms (and how to associate Galois representation to them). Towards the end of this talk, I will explain how this theory generalizes to the much more complicated setting of Hilbert modular varieties and Hilbert modular forms.
Quaternionic modular forms mod p.
15th October 2024, Yiannis Fam (KCL)
Abstract: In a 1987 letter, Serre relates the actions of Hecke operators on modular forms mod p and functions on a certain quaternion algebra depending on p. We explain a version of this result over other quaternion algebras, working in the context of modular forms over Shimura curves.
A survey of the inverse Galois problem.
8th October 2024, Edwina Aylward (UCL)
Abstract: The inverse Galois problem for a finite group G and field K asks whether there is an extension L of K with Galois group G. This is expected to be true for all groups when K is the rational numbers, but remains one of the biggest open problems in number theory today. In this talk, I will give an overview of this problem, considering its history as well as different ways of approaching it. The talk will be extremely light on prerequisites and should be accessible to anyone who knows what a Galois group is!
Any of the organizers can be contacted via email address firstname.lastname.23@ucl.ac.uk. To receive updates on the talks via email, join the mailing list: https://www.mailinglists.ucl.ac.uk/mailman/listinfo/juniornumbertheory.