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:
 Durham: Thanasis Bouganis
 Manchester: Vandita Patel
 Nottingham: Christian Wuthrich
 Sheffield: Haluk Şengün
 York: Sanju Velani
Next meeting
The next meeting is not yet planned.
Past Meetings
The last meeting was held on September 22nd 2021 in Durham hybrid. The lectures take place in the Scott Logic Lecture Theatre in the Department of Mathematical and Computing Sciences, in the new MCS building. See below for how to get there.
Schedule
 1:00 Larry Rolen (Vanderbilt University, Nashville)
 2:30 Min Lee (Bristol)
 3:30 Dan Clark (Durham)
Title and abstracts
Larry Rolen : Recent problems in partitions and other combinatorial functions
In this talk, I will summarize some recent work with a number of collaborators on conjectured analytic and combinatorial properties of partitions and related functions. In particular, I will look at recent conjectures of Stanton, which ultimately aim to give a deeper understanding into the workings of the rank and crank functions in explaining Ramanujan's congruences, as well as recentprogress in producing such functions explaining congruences for combinatorial functions using GritsenkoSkoruppaZagier's theory of theta blocks (special products built out of Dedekind eta and Jacobi theta functions with ties to Lie theory). I will also discuss how analytic questions about partitions can be used to study Stanton's conjectures, as well as recent conjectures of ChernFuTang and HeimNeuhauser, which are related to the NekrasovOkounkov formula.
Min Lee : Nonvanishing of symmetric cube Lfunctions
The nonvanishing of \(L\)series at the central of the critical strip has long been a subject of great interest. An important example of the significance of nonvanishing is in the case of an \(L\)series corresponding to a modular form of weight 2, where the nonvanishing at the central point has been shown to be equivalent to the finiteness of the group of rational points of the associated elliptic curve. In the case of higher rank \(L\)functions whose Euler product has even degree, such connections between nonvanishing at the central point and the finiteness of certain groups are believed to be true, but the relations remain purely conjectural. The symmetric cube \(L\)series plays a role in one of these conjectures.
Ginzburg, Jiang and Rallis (2001) proved that the nonvanishing at the central point of the critical strip of the symmetric cube \(L\)series of any \(\operatorname{GL}(2)\) automorphic form is equivalent to the nonvanishing of a certain triple product integral. The main purpose of this talk is to use this equivalence to prove that there are infinitely many MaassHecke cuspforms over the imaginary quadratic field of discriminant \(3\) such that the central values of their symmetric cube \(L\)functions do not vanish.
This is a joint work with Jeff Hoffstein and Junehyuk Jung.
Dan Clark : The geometry of moduli spaces of inertially trivial WeilDeligne representations
A recent trend of mathematics is the idea that one can understand better the objects of a particular category, by making sense of these objects as points of a geometric “moduli space” and studying the geometric properties of such a space. I seek to understand the geometry of a certain space, \(S_n\) of WeilDeligne representations, and prove certain smoothness/non smoothness/CohenMacaulay results of this space, and its irreducible components.
The second online meeting was held on May 26, 2021
 2:30 Vandita Patel (Manchester)
 4:00 Kisti Biggs (Chalmers Göteborg)
Title and abstracts
Vantida Patel : Shifted powers in LucasLehmer sequences
The explicit determination of perfect powers in (shifted) nondegenerate, 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 fractallike 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.
The first online meeting was held on March 24
Schedule
The talks were over Teams.
 2:30 Yingku Li (Darmstadt)
 4:00 Joshua Drewitt (Nottingham)
Title and abstracts
Yingkun Li: Span of restriction of Hilbert theta series
It is wellknown 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 realanalytic 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 selfpoints 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.
 1:30 Emma Bailey (Bristol)
 2:45 Pablo Shmerkin (T. Di Tella University and CONICET)
 4:30 Patrice Philippon (Institute de Mathématiques de Jussieu))
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.
 11am NilsPeter Skoruppa (Siegen)
 2pm Rachel Newton (Reading)
 3:30pm Robert Evans (King's College, London)
Title and abstracts
NilsPeter Skoruppa: A short proof of the Macdonald identities
We present a new and short proof of the wellknown Macdonald identities. The proof is rather selfcontained 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 BSDtype formula for elliptic curves twisted by Artin representations. We will see that expected properties of Lfunctions 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 Jfloor of the Hicks building. See below for a description of how to get there.
 11am Farrell Brumley (Paris 13)
 2pm Andreea Mocanu (Nottingham)
 4pm Clark Barwick (Edinburgh)
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.
Durham

The lectures take place in the Scott Logic Lecture Theatre in the MCS building.
It is the lecture room at the left of the main entrance.
The building itself is located at the science campus but now it is at the top of the hill.
The address of the new building is
Durham University Upper Mountjoy Campus, Stockton Rd, Durham DH1 3LE
Directions to get there 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
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.