# Junior Number Theory Seminar

Hello,

Welcome to Warwick's Junior Number Theory Seminar, where graduate students in the area of Number Theory give accessible talks about topics they are interested in, and share the outcomes of their research with their peers.

This term, all talks will be held in **B3.01 at 11 am on Monday **(except when stated otherwise).

Do you want to give a talk in this seminar? Let us know :)

This seminar is organised by Alvaro Gonzalez HernandezLink opens in a new window, James RawsonLink opens in a new window and Katerina SanticolaLink opens in a new window. If you have any question, do not hesitate to get in contact with us (if possible, when you send us an email please add all of us as recipients)!

## Term 1

## Date |
## Speaker |
## Affiliation |
## Title |

Week 1 - 2nd October |
No Talk | ||

Week 2 - 9th October |
Abdul Alfaraj | University of Bath |
On the finiteness of perfect powers in elliptic divisibility sequences |

Week 3 - 16th October |
Isabel (Izzy) Rendell | King's College London |
Rational points on modular curves |

Week 4 - 23rd October |
Arshay Sheth | University of Warwick |
The Hilbert-Polya dream: finding determinant expressions of zeta functions |

Week 5 - 30th October |
Maryam Nowroozi | University of Warwick | Perfect Powers in Elliptic Divisibility Sequences |

Week 6 - 6th November |
Alexandros Konstantinou | University College London | Unveiling the power of isogenies: From Galois theory to the Birch and Swinnerton-Dyer conjecture |

Week 7 - 13th November |
Amelia Livingston | University College London | The Langlands correspondence for algebraic tori |

Week 8 - 20th November |
Robin Ammon | University of Glasgow | Cohen--Lenstra Heuristics for Ray Class Groups |

Week 9 - 27th November |
Benjamin Bedert | University of Oxford | On Unique Sums in Abelian Groups [Rescheduled] |

Week 10 - 4th December |
Harvey Yau | University of Cambridge | An introduction to the Brauer-Manin obstruction |

## Abstracts

**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.