# Junior Number Theory 2024-2025

Hello,

Welcome to Warwick's Junior Number Theory Seminar!

The seminar takes place during term time on **Mondays **from **11am-12pm** in **B3.01 **(terms 1 and 2) or **B3.02** (term 3) in the Zeeman Building.

The seminar is organised by Alexandros Groutides, Seth Hardy and Kenji Terao.

If you want to give a talk in this seminar, contact one of the organisers!

## Term 1

## Date |
## Speaker |
## Title |
---|---|---|

30th September |
Martí Oller Riera (Cambridge) | Arithmetic Statistics and Dynkin Diagrams |

7th October |
Ned Carmichael (KCL) | Sums of Arithmetical Functions |

14th October |
Julie Tavernier (Bath) | Counting number fields whose conductor is the sum of two squares |

21st October |
Yicheng Yang (LSGNT) | Higher Hida theory on the modular curve |

28th October |
Besfort Shala (Bristol) | Chowla's Conjecture for Random Multiplicative Functions |

4th November |
Edwina Aylward (LSGNT) | Rank Predictions in Elliptic Curves: Parity vs. L-Value Approaches |

11th November |
Nicola Ottolini (Rome Tor Vergata) | Unlikely (and singular) intersections in Diophantine geometry |

18th November |
Arshay Sheth (Warwick) | - |

25th November |
Constantinos Papachristoforou (Sheffield) | - |

2nd December |
James Rawson (Warwick) | - |

## Abstracts

**Week 1 - 30th September **

**Martí Oller Riera (University of Cambridge) - Arithmetic Statistics and Dynkin Diagrams**

** **

Arithmetic Statistics is an area of Number Theory that has massively grown in the past decades. One of the most impressive recent results in the area is the work of Bhargava and Shankar, which proved that the average rank of elliptic curves is bounded, a result that was later generalised to Jacobians of hyperelliptic curves. We will begin the talk by giving an overview of the proof of these results, and later in the talk we will explain how such results can be interpreted and generalised using Dynkin diagrams.

**Week 2 - 7th October**

**Ned Carmichael (Kings College London) - Sums of Arithmetical Functions**

** **

In this talk, we introduce the Dirichlet divisor problem, overviewing some classical questions and results. A well-known problem in analytic number theory, this asks for the best possible asymptotic approximations to sums of the divisor function. Finally, we consider an analogous problem where the divisor function is replaced by coefficients of cusp forms, and explain some results in this new setting (with comparison to the classical case).

**Week 3 - 14th October**

**Julie Tavernier (University of Bath) - Counting number fields whose conductor is the sum of two squares**

** **

A conjecture by Malle proposes an asymptotic formula for the number of number fields of bounded discriminant and given Galois group. In this talk we will consider abelian extensions of fixed Galois group G whose conductor is bounded and the sum of two squares, and show how one can employ techniques from harmonic analysis and class field theory to count such extensions. We then explain how this can be extended to a more general class of field extensions whose conductor satisfies some frobenian condition and conclude by mentioning how some of these results could be interpreted in terms of geometric objects.

**Week 4 - 21st October**

**Yicheng Yang (LSGNT) - Higher Hida theory on the modular curve**

In the 80's, Hida introduced an ordinary projector on modular forms and constructed p-adic families of ordinary modular forms. In the talk we will give a construction of integral Hecke operators and explain the idea of higher Hida theory which works for all cohomological degrees and can be generalised to higher dimensional.

**Week 5 - 28th October**

**Besfort Shala**** (Bristol) - ****Chowla's Conjecture for Random Multiplicative Functions**

Let f be a Steinhaus or Rademacher random multiplicative function. Even though a lot is known about partial sums of f over positive integers due to work of Harper, the exact distribution is still elusive. However, partial sums of f over some restricted sets of positive integers are better understood and are expected/known to mimic a sum of independent random variables, in the sense that they satisfy a central limit theorem and/or law of iterated logarithm. There are several results in this direction such as summing over: (i) integers with a restricted number of prime factors (Hough, Harper), (ii) integers in a short interval (Chatterjee-Sound, Sound-Xu), (iii) values of integer polynomials in the Steinhaus case (Klurman-Shkredov-Xu). The latter may be viewed as a random analogue for Chowla's conjecture or the related Elliott's conjecture. In this talk, I will give a survey of the above results and discuss joint work with Jake Chinis, where we address the random analogue of Chowla's conjecture for Rademacher multiplicative functions. If time permits, I will also discuss some open questions in this area, some of which we address in ongoing joint work with Christopher Atherfold.

**Week 6 - 4th November**

**Edwina Aylward (LSGNT) - Rank Predictions in Elliptic Curves: Parity vs. L-Value Approaches**

Finding points of infinite order on elliptic curves is tough work. The parity conjecture has long been used to predict the existence of such points without the need to compute them directly. This has led to the discovery of many intriguing phenomena that we cannot explain by other means. After discussing some examples of these occurrences, I will turn to an alternative approach for predicting positive rank in families of elliptic curves. This method relies on Deligne's conjecture concerning the Galois theoretic properties of L-values, as well as the Birch—Swinnerton-Dyer conjecture. I will then compare the two approaches, and end with discussing whether this new method can predict positive rank in cases where the parity conjecture falls short.

**Week 7 - 11th November**

**Nicola Ottolini (Roma Tor Vergata) - Unlikely (and singular) intersections in Diophantine geometry**

Starting from Mordell, many conjectures have been put forward (and proved) about how geometry influences the behaviour of diophantine problems. It turns out that many of them can be put in a common framework about varieties that for dimensional reasons we do not expect to intersect. Whenever they do we say that this intersection is "unlikely". After a brief introduction on the geometric objects involved and some examples of problem of this type, we will turn to a slight variation of this problem, where we look at varieties who we expect to intersect transversally, and consider when they do so in a singular way, i.e. they are actually tangent to each other. If time permits, I will also sketch a common proof strategy due to Pila and Zannier.