Northern NuTS
Northern NuTS
Northern NuTS

The Northern Number Theory Seminar is a joint research seminar in number theory. It links the number theory research groups of Durham, Manchester, Nottingham, Sheffield and York.

We expect to meet twice per year in normal times. The seminar is funded by the LMS Grant Scheme 3.

The local organisers are:

## Next meeting

The next online meeting will be on May 26, 2021

### Schedule

The talks will be online using Microsoft Teams. Follow this link : https://tinyurl.com/vr5xvwxu to join the meeting.

#### Title and abstracts

##### Vantida Patel : Shifted powers in Lucas-Lehmer sequences

The explicit determination of perfect powers in (shifted) non-degenerate, integer, binary linear recurrence sequences has only been achieved in a handful of cases. In this talk, we combine bounds for linear forms in logarithms with results from the modularity of elliptic curves defined over totally real fields to explicitly determine all shifted powers by two in the Fibonacci sequence. This is joint work with Mike Bennett (UBC) and Samir Siksek (Warwick).

##### Kirsti Biggs : Ellipsephic efficient congruencing for the Vinogradov system

Ellipsephic sets are subsets of the natural numbers defined by digital restrictions in a given base - such sets have a fractal-like structure which can be seen as a $$p$$-adic analogue of generalised real Cantor sets. The recent work of Maynard on primes with missing digits can be seen as an ellipsephic problem, although in this talk we focus on smaller sets of permitted digits, one motivating example being the set of natural numbers whose digits are squares. I will present an upper bound for the number of ellipsephic solutions to the Vinogradov system of diagonal equations, and highlight the key features of the proof, which uses Wooley's efficient congruencing method.

## Past Meetings

The first online meeting was held on March 24

### Schedule

The talks were over Teams.

• 4:00    Joshua Drewitt (Nottingham)

#### Title and abstracts

##### Yingkun Li: Span of restriction of Hilbert theta series

It is well-known that an elliptic modular form of level 1 and weight divisible by 4 can be written as a linear combination of theta series associated to unimodular lattices. It is natural to ask what happens in the case of Hilbert theta series. In this talk, we will look at a slightly easier question when the Hilbert modular forms are restricted to the diagonal, and give some results concerning their span.

##### Joshua Drewitt: Period functions associated to real-analytic modular forms.

The space of modular iterated integrals of length one sits naturally inside the broader class of real analytic modular forms, recently introduced by F. Brown. In this talk we construct period functions for length one iterated integrals and see how they reflect the period polynomials associated to classical modular forms. To achieve this, we give a brief recap of the theory of period polynomials for modular forms and of the period functions associated to Maass wave forms. This is joint work with Nikolaos Diamantis.

#### Video Lectures in summer 2020

We organised a series of online lectures in collaboration with two other Scheme 3 networks: the ROW seminars and the Arithmetic Statistics Seminars.

The videos are now available on youtube channes of the LMS.

There were three speakers:

• Richard Hatton (Nottingham):
Heegner points and self-points on elliptic curves
• Beth Romano (Oxford):
An introduction to Vinberg theory and related arithmetic invariant theory
• Tiago Fonseca (Oxford):
A crash course on modular forms and cohomology

The third meeting was held in York on Tuesday, October 15, 2019.

### Schedule

All talks were in Topos (024B – next to the coffee room) on the ground floor of James College on the University of York Campus West. See below for a description of how to get there.

#### Title and abstracts

##### Emma Bailey: Moments of moments

I will motivate and present results concerning moments of moments of the Riemann zeta function alongside those of characteristic polynomials of random unitary matrices. The study of moments of moments has been furthered through connections with probability, combinatorics, and representation theory. This is joint work with Jon Keating.

##### Pablo Shmerkin: Multiplying by 2 and by 3: old conjectures and new results

In the 1960s, H. Furstenberg proposed a series of conjectures which, in different ways, aim to capture the heuristic principle that "expansions in different bases have no common structure". I will discuss a sample of these conjectures, some which remain open and some which have been established in recent years. These problems lie at the intersection of Ergodic Theory and Fractal Geometry, but no previous background on either area will be assumed.

##### Patrice Philippon: Around two problems of Hardy

Let $$\alpha>1$$ and $$\lambda$$ be real numbers. About a century ago, Hardy asked: "In what circumstances can it be true that the distance of $$\lambda \alpha^n$$ to the nearest integer tends to $$0$$ as $$n$$ goes to $$\infty$$?". This question is still open. We will present some advances in this topic and around a related problem of Mahler, in the case $$\alpha$$ is algebraic. This is joint work with Purusottam Rath.

The second meeting was held in Nottingham on Wednesday, May 15, 2019.

### Schedule

All talks were in room C04 of the Physics building. See below for a description of how to get there.

#### Title and abstracts

##### Nils-Peter Skoruppa: A short proof of the Macdonald identities

We present a new and short proof of the well-known Macdonald identities. The proof is rather self-contained since it uses merely some easy combinatorial arguments, a basic (though subtle) geometrical fact about root systems and a bit of 'modular forms magic' (presented in the language of Jacobi forms). The proof suggests the possibility that there might be more, not yet dicovered Macdonald type identities and a method to systematically search for such. We shall discuss the algorithmic problems and computational complexity connected to such a search. Finally we mention some applications of the Macdonald identities to elliptic curves over the rationals.

##### Rachel Newton: Number fields with prescribed norms

Let $$G$$ be a finite abelian group, let $$k$$ be a number field, and let $$x$$ be an element of $$k$$. We count Galois extensions $$K/k$$ with Galois group $$G$$ such that $$x$$ is a norm from $$K/k$$. In particular, we show that such extensions always exist. This is joint work with Christopher Frei and Daniel Loughran.

##### Robert Evans: Factorising BSD: Artin twists of elliptic curves

I shall discuss the possible existence of a BSD-type formula for elliptic curves twisted by Artin representations. We will see that expected properties of L-functions give rise to some surprising arithmetic predictions for elliptic curves. This is joint work with Vladimir Dokchitser and Hanneke Wiersema.

The first meeting was held in Sheffield on Wednesday, February 20, 2019.

### Schedule

All talks were in room J11 on the J-floor of the Hicks building. See below for a description of how to get there.

#### Title and abstracts

##### Farrell Brumley: Concentration properties of theta lifts

I will present some results on the concentration properties of automorphic forms obtained through the theta correspondence. Among other things, the method relies on a distinction principle for these lifts, which detect their functorial origin via the non vanishing of orthogonal periods. The examples we treat are in higher rank, and shed light on a purity conjecture of Sarnak. This is joint work with Simon Marshall.

##### Andreea Mocanu: Newform theory for Jacobi forms of lattice index

I will give a brief introduction to Jacobi forms, including some examples and their relation to other types of modular forms. After that, I will discuss some of the ingredients that go into developing a theory of newforms for Jacobi forms of lattice index, namely Hecke operators, level raising operators and orthogonal groups of discriminant modules.

##### Clark Barwick: Primes, knots, and exodromy

Half a century ago, Barry Mazur and David Mumford suggested a remarkable dictionary between prime numbers and knots. I will explain how the story of exodromy permits one to make this dictionary precise, and I will describe some applications.

## Travel

Here a list of how to get to the meetings.

#### Nottingham

There are trams leaving from Nottingham train station (go upstairs) every 7 minutes. You want a tram in the direction of Toton Lane; get off at the University of Nottingham stop. You need to buy a ticket from the machine before you get on the tram. The trip takes about 15 minutes.
The Maths building is number 20 on the campus map. Talks are usually held in the Physics building, opposite the Maths building, number 22 on the map. The tram stop is the green circle on the map near the South Entrance. A nice place for lunch is the Lakeside Arts cafe (number 49 on the map, near the lake and tram stop) or the staff club (number 8).

#### Sheffield

The School of Mathematics and Statistics is in the Hicks Building which is building 121 on this map You can also look at this map from the train station, but be aware that the walking route to the Hicks building is not obvious.
If you are arriving by train, the easiest way to get to the Hicks Building is by taking the tram. There is a tram stop right behind the train station. You need to take the Blue Line Tram in the direction of "Malin Bridge". A ticket costs £1.70 - you pay the conductor on the tram. It takes about 10 minutes to get to the "University of Sheffield" tram stop where you should get off. Cross over the road at the traffic lights at the front of the tram (not to be confused with crossing over the tram lines) and go UP the Leavygreave Rd. The entrance to the Hicks Building (a very tall building) will be in front of you.

#### York

The Department of Mathematics is in James College on the University of York Campus West (the original campus). It is C3 building 21 on this map. Click here to plan your journey. There are buses from the city centre and railway station approximately every 5 minutes during the day, and it takes around 20 minutes to reach the campus. If you are taking a taxi, ask to be dropped at the Roger Kirk Centre.

## Expenses

If you have a UK bank account you can get reimbursements for travel by using this form. Otherwise ask me.

Then send the form and receipts (try to avoid having the machine swallow your train tickets when you exit the station, please) to

Christian Wuthrich
School of Mathematical Sciences
University of Nottingham
Nottingham NG7 2RD

The seminar is committed to making sure all parents working in mathematics should be able to attend our meetings without being hindered by childcare costs. The LMS has a Childcare Supplementary Grant Scheme that can be cover extra costs caused by attending the meeting. Further information about this scheme can be found on this LMS website. Some of the host Universities, like Nottingham University, also have their own scheme to help. Please ask any of the organisers.