Northern NuTS
 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: York, May 11

The next meeting will take place in York on Thursday May 11.

(A later meeting is planned in Sheffield on September 8).

All talks are in the Dusa McDuff Room (G/N/135) also known as the MSc/MMath Study Centre (directly above the coffee room) on the first floor of James College on the University of York Campus West. See below for a description of how to get there.

We plan to go to dinner after the last talk. If you are interested in coming please add your details to this form.


Title and abstracts

Andrew Scoones: On the abc Conjecture in Algebraic Number Fields

While the abc conjecture remains open, much work has been done on weaker versions, and on generalising the conjecture to number fields. Stewart and Yu were able to give an exponential bound for \(\max\{a,b,c\}\) in terms of the radical over the integers, while Györy was able to give an exponential bound for the projective height \(H(a,b,c)\) in terms of the radical for algebraic integers. We generalise Stewart and Yu’s method to give an improvement on Györy’s bound for algebraic integers, before briefly discussing applications to the effective Skolem-Mahler-Lech problem and the XYZ conjecture. We note that independently Györy attained similar results which we will also discuss.

Subhajit Jana: Reciprocity, non-vanishing, and subconvexity of central \(L\)-values

A reciprocity formula usually relates certain moments of two different families of \(L\)-functions that apparently have no connections between them. The first such formula was due to Motohashi who related a fourth moment of Riemann zeta values on the central line with a cubic moment of certain automorphic central \(L\)-values for \(\operatorname{GL}(2)\). In this talk, we describe some instances of reciprocity formulas both in low and high-rank groups and give certain applications to subconvexity and non-vanishing of central \(L\)-values. The talks will be based on a few (ongoing) joint works with Nunes and Blomer-Nelson.

Shreyasi Datta: Friendly measures in a nutty field

I will focus on Diophantine approximation on \(\mathbb{Q}_p\), and more generally on \(\mathbb{Q}_p \times \mathbb{R}\), \(\mathbb{Q}_{p_1} \times \mathbb{Q}_{p_2}\) and so on. This kind of metric number theory has been studied by various mathematicians starting with Mahler, Jarnik and extended heavily by Lutz. After reviewing some recent developments in this area, I want to concentrate on a recent work with Anish Ghosh and Victor Beresnevich. We showed that the pushforward of a \(p\)-adic fractal measure by ‘nice’ functions exhibit ‘nice’ Diophantine properties, settling a conjecture of Kleinbock and Tomanov. In particular, we prove \(p\)-adic analogue of a result by Kleinbock, Lindenstrauss and Weiss on friendly measures. I will talk about how lack of the mean value theorem makes life difficult in the \(p\)-adic fields, and how we can sometimes overcome this problem.

Past Meetings

The last meeting was held in Nottingham on Friday, June 10

The lectures took place in room C05 in the Physics building (22 on the map) at the University of Nottingham.


Title and abstracts

Demi Allen: An inhomogeneous Khintchine-Groshev Theorem without monotonicity

The classical (inhomogeneous) Khintchine-Groshev Theorem tells us that for a monotonic approximating function \(\psi: \mathbb{N} \to [0,\infty)\) the Lebesgue measure of the set of (inhomogeneously) \(\psi\)-well-approximable points in \(\mathbb{R}^{nm}\) is zero or full depending on, respectively, the convergence or divergence of \(\sum_{q=1}^{\infty}{q^{n-1}\psi(q)^m}\). In the homogeneous case, it is now known that the monotonicity condition on \(\psi\) can be removed whenever \(nm>1\), and cannot be removed when \(nm=1\). In this talk I will discuss recent work with Felipe A. Ramírez (Wesleyan, US) in which we show that the inhomogeneous Khintchine-Groshev Theorem is true without the monotonicity assumption on \(\psi\) whenever \(nm>2\). This result brings the inhomogeneous theory almost in line with the completed homogeneous theory. I will survey previous results towards removing monotonicity from the homogeneous and inhomogeneous Khintchine-Groshev Theorem before discussing the main ideas behind the proof our recent result.

Sadiah Zahoor: Congruences between Modular forms of integer and half-integer weight

The theory of half-integral weight modular forms may be adopted to prove 'congruences' between Selmer type groups. William McGraw and Ken Ono used this approach by passing from 'congruences' between modular forms of integer weight to congruences between modular forms of half-integer weight. For example, recall the famous 'congruence modulo \(11\)' between the normalised Discriminant function \(\Delta\) of weight \(12\) and the newform \(f\) of weight \(2\) attached to Elliptic curve of conductor \(11\). Using Shimura's correspondence and Kohnen’s isomorphism, which connects modular forms of weight \(2k\) for a positive integer \(k\) with half-integer modular forms of weight \(k+\tfrac{1}{2}\), our congruence descends to a congruence modulo \(11\) between half integer modular forms of weight \(\tfrac{3}{2}\) and \(\tfrac{13}{2}\).

The talk shall begin with a brief introduction to modular forms of integer and half-integer weight leading to statement and overview of the main result I have been working on. I will also give an overview of current progress and generalisation of Theorem of McGraw and Ono to Hilbert modular forms of integer and half-integer weight.

Charles F. Doran: \(K_2\) and Quantum Curves

A 2015 conjecture of Codesido-Grassi-Mariño in topological string theory relates the enumerative invariants of toric Calabi-Yau 3-folds to the spectra of operators attached to their mirror curves. In a recent paper with Matt Kerr and Soumya Sinha Babu, we deduce and prove consequences of this conjecture for the integral regulators of \(K_2\) classes on these curves. (While the conjecture and the deduction process both entail forms of local mirror symmetry, the consequences/theorems do not: they only involve the curves themselves.) In particular, in the seminar talk I will discuss our result that relates zeroes of the higher normal function to the spectra of the operators for curves of genus one, suggesting a new link between analysis and arithmetic geometry.

The previous meeting was held in Manchester on March 30 2022

The lectures took place in the Alan Turing Building Room G.207 (Ground floor) at the University of Manchester. See below for how to get there.


Title and abstracts

Vaidehee Thatte : Understanding the Defect via Ramification Theory

Classical ramification theory deals with complete discrete valuation fields \(k((X))\) with perfect residue fields \(k\). Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow non-discrete valuations and imperfect residue fields \(k\). Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.

Degree \(p\) extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups.

Alice Pozzi : Rigid meromorphic cocycles and \(p\)-adic variations of modular forms

A rigid meromorphic cocycle is a class in the first cohomology of the group \(\operatorname{SL}_2\bigl(\mathbb{Z}[\tfrac{1}{p}]\bigr)\) acting on the non-zero rigid meromorphic functions on the Drinfeld \(p\)-adic upper half plane by Möbius transformation. Rigid meromorphic cocycles can be evaluated at points of “real multiplication”, and their values conjecturally lie in composita of abelian extensions of real quadratic fields, suggesting striking analogies with the classical theory of complex multiplication.

In this talk, we discuss the proof of this conjecture for a special class of rigid meromorphic cocycles. Our proof connects the values of rigid meromorphic cocycles to the study of certain \(p\)-adic variations of Hilbert modular forms. This is joint work with Henri Darmon and Jan Vonk.

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.


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 recent progress in producing such functions explaining congruences for combinatorial functions using Gritsenko-Skoruppa-Zagier'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 Chern-Fu-Tang and Heim-Neuhauser, which are related to the Nekrasov-Okounkov formula.

Min Lee : Non-vanishing of symmetric cube L-functions

The non-vanishing 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 non-vanishing is in the case of an \(L\)-series corresponding to a modular form of weight 2, where the non-vanishing 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 non-vanishing 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 non-vanishing 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 non-vanishing of a certain triple product integral. The main purpose of this talk is to use this equivalence to prove that there are infinitely many Maass-Hecke 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 Weil-Deligne 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 Weil-Deligne representations, and prove certain smoothness/non smoothness/Cohen-Macaulay results of this space, and its irreducible components.

The second online meeting was held on May 26, 2021

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.

The first online meeting was held on March 24


The talks were over Teams.

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:

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


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.


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.


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.


Here a list of how to get to the meetings.


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


Talks will be held in the Alan Turing Building Room G.207 (Ground floor). The Alan Turing Building (ATB) is where the Department of Mathematics is based. It is building 46 on this map .
Access by train: From Manchester Piccadilly, ATB is a short 20 minute walk, or a short 15 minute bus ride.
Access by Car: There are many places to park nearby, however, these can get very busy. Charles Street Car Park is a public car park which usually has space, see the map . ATB is then a short 13 minute walk away, or 9 minute bus journey.


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).


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.


The Department of Mathematics is in James College on the University of York Campus West (the original campus). It is C3 building 23 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.


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.