###### About the London Junior Number Theory seminar

This weekly seminar from and for PhD students aims to bridge the gap between the adult seminar in London (here) and run-of-the-mill number theory. Talks usually give an introduction to some research-related topic but are colloquium-style and accessible for a general number-theory-inclined audience.

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If you would like to suggest a talk (that you could give or want someone else to give), then do get in touch!

Spring 2021

The talks will be held over Zoom this term. You can register here. Following the talk, we will be having a virtual Number Theory social for anyone interested.

Schedule

- 12/01 Ashvni Narayanan (LSGNT – Imperial) : Galois Cohomology of Elliptic Curves

*Abstract: I will give a brief overview of the book Galois Cohomology of Elliptic Curves, by Prof J Coates and Prof R Sujatha. After a few basic results of Galois cohomology, we shall see an application of Iwasawa theory to a Z_p-extension obtained by attaching the p-power torsion points of an elliptic curve E over a number field F. Finally, if time permits, we will take a look at some specific examples, that is, Iwasawa theory for curves of conductors 11 and 294.*

- 19/01 Ashwin Iyengar (LSGNT – King’s) : TBA

*Abstract: TBA *

- 26/01: Jordan Docking (LSGNT – UCL) : Extracting rank information: elliptic curves and beyond

*Abstract: The rational points of an abelian variety A form a finitely-generated group of the form A( Q) = Δ x Z^{r}, where Δ is the finite torsion part, and r the rank. This rank is notoriously difficult to pin down in any generality. Even in the classical case of elliptic curves, one relies on the conjectural finiteness of the Tate–Shafarevich group for an algorithm which is guaranteed to terminate. In higher-dimensions, one faces the further challenge that such algorithms are largely undeveloped. We review the current approaches, theorems, and conjectures that facilitate some extraction of rank information. In particular, we look at how isogenies play an intimate role in the parity of the rank, and some general setups (not just for elliptic curves) to exploit this to get some cold, hard numbers out the other side.*

- 2/02: TBA

*Abstract: TBA *

- 9/02: Jared Duker Lichtman : TBA

*Abstract: TBA *

- 16/02: TBA

*Abstract: TBA *

- 23/02: TBA

*Abstract: TBA *

Autumn 2020

The talks will be held over Zoom this term. You can register here. Following the talk, we will be having a virtual Number Theory social for anyone interested.

Schedule

- 06/10 Lambert A’Campo (LSGNT – King’s) : Modular Forms and Class Groups

*Abstract: In 1976 K. Ribet discovered a new connection between modular forms and the p-part of the class group of the p-th cyclotomic field. This lead to an improved form of Kummer’s criterion for irregular primes and eventually was used by Mazur-Wiles in 1984 to prove the Iwasawa Main Conjecture. The protagonists of Ribet’s 1976 article are modular forms, Galois representations, Bernoulli numbers and a large part of modern number theory revolves around them. The aim of the talk is to briefly introduce them and explain Ribet’s proof (using a few strong results as black boxes). Hopefully it will leave you wanting more and motivate a few of the subjects in the following year.*

- 13/10 Abigail Burton (LSGNT – Imperial) : An Introduction to Hida Theory

*Abstract: We are interested in the question of whether, given a modular form, we can assign to it a family of modular forms which varies p-adic analytically over weights. In the 1980’s, Haruzo Hida gave a partial answer to this question though his theory of ordinary parts. His work has since grown and found numerous applications, notably in Wiles’ proof of the Iwasawa main conjecture for totally real fields in 1990. We will give a short introduction to his work, starting by defining the space of \Lambda-adic forms and giving an example of p-adic interpolation in the case of Eisenstein series, then defining the ordinary projector, and ending by stating some of the main results in Hida Theory. It is intended to be very accessible with no prerequisites.*- 20/10 Louis Jaburi (Imperial): Reductive groups and their geometry

*Abstract: In this talk I want to present the well-behaved properties of reductive groups and notions that come along with them, such as maximal Tori, Borel subgroups and Parabolic subgroups. At first the definition of a reductive group over some field might seem not that intuitive and rather group theoretic. But a deeper study of this field has shown that reductive groups have a good representation theory, can be classified fairly well via root data, and also yield geometric structures via their corresponding flag varieties. I will try to touch on all of these subjects via examples such as Tori, GL _{n} and Sp_{4}. *

27/10 Art Waeterschoot (KU Leuven): Introduction to Semistable Reduction

*Abstract: We will explain the link between the Deligne-Mumford semistable reduction theorem for curves and Grothendieck’s semistable reduction theorem for abelian varieties. These results are basically equivalent and there are many known approaches. First we will sketch a explicit proof in characteristic zero of the Deligne-Mumford theorem and outline some key ideas of the Artin-Winters proof for the general case. Time permitting we will discuss applications to Néron-Ogg-Shafarevich-like inertial criteria. The talk will be an introduction to the subject and no prior knowledge other than basic algebraic geometry will be needed.*

03/11 Dan Clark (Durham) : Complex multiplication, and it’s connection to the Hilbert class field of imaginary quadratic number fields.

*Abstract: Elliptic curves are a central object of study in number theory. It should be no surprise that there are many discoveries to be made when one looks at the ring of endomorphisms of an elliptic curve. For most curves, this is simply Z. But for some special curves, it is strictly larger. These are the curves with ‘complex multiplication’. In this talk, we will study these curves a little bit, with an eye to a beautiful result about in the class field theory of imaginary quadratic fields.*

10/11 – Dominik Bullach (LSGNT – King’s) – L-functions and Galois module theory

*Abstract: The deep connection between special values of L-functions and arithmetic is one of the most fascinating branches of modern number theory. For most people the way in is via the analytic class number formula or the conjecture of Birch and Swinnerton-Dyer. In this talk, I will instead use classical Galois module theory as a motivation. This will lead to (a special case of) the so-called equivariant Tamagawa Number Conjecture (eTNC). At the very end I will mention joint work with Martin Hofer that proves new cases of this conjecture.*

- 17/11 – Gabriel Micolet (LSGNT – King’s) – A solid introduction to condensed mathematics

*Abstract: Introducing topology into algebra has been an essential aspect of mathematics in the last century. This however has, among others, the drawback that categories of topological algebraic structures are not abelian (eg. the lack of cokernels). A remedy to this has been recently introduced by Clausen and Scholze, in the form of condensed mathematics which enlarges the notion of topological spaces and yields categories of algebraic objects with excellent categorical properties. I intend to introduce the basics of the theory, notably the notion of solid abelian groups and the existence of completed tensor products. If time permits, I will discuss how this theory can lead to a new (categorical) framework for functional analysis. This will be an overview and I will avoid focusing on the most technical details!*

24/11 – Miriam Norris (LSGNT – King’s) – A very brief introduction to modular representation theory

*Abstract: For K a number field it is natural to look at the representations of the Galois group of a finite extension not only over K but also over its ring of integers. Taking reduction modulo a prime ideal of these integral representations we get examples of representations over fields of finite characteristic naturally occurring in number theory. For G a finite group and K a field of characteristic 0, Mashke’s theorem says that every finite dimensional KG-module is semisimple. This is true also if K is a field of characteristic p, where p is a prime not dividing the order of G. However if K is a field of prime characteristic p that divides the order of G then Mashke’s theorem no longer holds and the building blocks of KG-modules can no longer be assumed to be simple. The study of such KG- modules is called modular representation theory and in this talk we will give a whistle stop tour of some of its key introductory results.*

01/12 – Mohamed Tawfik (Reading) – The Arithmetic of Elliptic Curves

*Abstract: In 1922, Mordell proved that the group of rational points E(Q) of an elliptic curve E/Q is a finitely generated Abelian group, written in the form E(Q) \cong E(Q)_{tors} x Z^r. While the finite part is well understood, the infinite part is much more mysterious. In this talk, we discuss ranks of quadratic twists of pairs of elliptic curves and show how to obtain quadratic twists of pairs of elliptic curves with high ranks.*

08/12 – Andy Graham (LSGNT – UCL) – The Bloch-Kato Conjecture

*Abstract: Let V be a p-adic Galois representation which arises from geometry (i.e. as a subquotient of the etale cohomology of some variety). Then attached to V is an arithmetic object, the Bloch–Kato Selmer group, and an analytic object, the complex valued L-function. The conjecture of Bloch–Kato predicts a relation between these two objects and can be viewed as a generalisation of the Birch–Swinnerton-Dyer conjecture. In this talk I will describe this conjecture and survey some known cases.*

15/12 – Matthew Honnor (King’s) – The Gross Regulator Matrix

*Abstract: Abstract: Let F be a totally real field and \chi a totally odd character on the absolute Galois group of F. The Gross-Stark conjecture predicts the value of the leading term of the p-adic L-function attached to the character \chi. A key part of this value is the determinant of the Gross-Regulator matrix. In this talk I will introduce this conjecture and its relation to the Gross-Stark units. Formulas have been conjectured by Dasgupta and Spiess for the values of this determinant and the Gross-Stark units. At the end I will mention forthcoming work showing that these two formulas are compatible when F is of degree 3.*