London Junior Number Theory Seminar 2024/2025

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.

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.

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.

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?

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.

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!

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.

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.

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.

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.

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.

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!

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.

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.

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.