Number Theory Seminar
The Number Theory Seminar is a 55-minute in-person research seminar, including questions. Talks are suitable for PhD students (including new PhD students) from all fields of number theory.
The seminar is at 15:00 on Mondays (except bank holidays). On seminar days we meet for lunch at 12:30 and coffee at 16:00 in the common room. Seminars are held in B3.02 of the Zeeman building. Sometimes a different room is used, see the entry below.
Colleagues and especially number theory group members are warmly encouraged to suggest speakers by emailing the organisers: Adam Harper , Harry Schmidt , Akshat Mudgal.
We kindly remind members that it is polite to the speaker, to come to talks we might not personally expect to be interested in, and in compensation there will be a good audience for the speakers each of us is interested in!
A list of members of the group and research interests is available.
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16 February 2026 at 15:00 in B302
Speaker: Rodolphe Richard (University of Manchester)
Title: Uniform Kummer theory on hybrid classes of abelian varieties
Abstract: We present uniformity results concerning Galois representations attached to abelian varieties: action on torsion points (Tate module), and Kummer theory (affine Tate module). We obtain uniformity over a "hybrid" equivalence class, which generalises isogeny classes and the set of all abelian varieties with complex multiplication.
This is motivated by applications to arithmetic geometry in mixed shimura varieties (via equidistribution: There are links with Tuesday talk in the ETDS seminar.) -
09 February 2026 at 15:00 in B302
Speaker: Philipp Habegger (University of Basel)
Title: Specializing Linear Recurrence Sequences at Roots of Unity
Abstract: he Skolem-Mahler-Lech Theorem characterizes the vanishing
members of a linear recurrence sequence. Over a number
field, no effective proof of this classical theorem is known. In other
words, given a linear recurrence sequence, we know no algorithm that
is guaranteed to produce precisely the indices where the sequence vanishes.
We consider linear recurrence sequences of rational functions and
study when sequence members vanish at a root of unity. Bilu-Luca and
Ostafe-Shparlinski proved finiteness results for linear recurrence
sequences of order 2. I will report on a new finiteness result
for linear recurrence sequences of order 3. We require lower bounds for
linear forms in
logarithms and the Pila-Zannier strategy that relies on o-minimal geometry.
This is joint work in progress with Alina Ostafe and David Masser. -
02 February 2026 at 15:00 in B3.02
Speaker: Rena Chu (University of Goettingen)
Title: Short character sums evaluated at homogeneous polynomials
Abstract: Let $p$ be a prime. Bounding short Dirichlet character sums is a classical problem in analytic number theory, and the celebrated work of Burgess provides nontrivial bounds for sums as short as $p^{1/4+\varepsilon}$ for all $\varepsilon>0$. In this talk, we will first survey known bounds in the original and generalized settings. Then we discuss the so-called ``Burgess method'' and present new results that rely on bounds on the multiplicative energy of certain sets in products of finite fields.
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26 January 2026 at 15:00 in B3.01
Speaker: Nick Rome (TU Graz)
Title: Counting quadratic points
Abstract: I'll discuss recent joint work with Francesca Balestreieri, Kevin Destagnol, Julian Lyczak and Jennifer Park concerning counting quadratic points of bounded height on varieties. We give a general framework for attacking this kind of problem, including results about summing Euler products over varying quadratic fields. I'll also discuss the connection with Manin's conjecture and discuss a new family of varieties for which we are able to prove the conjecture.
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19 January 2026 at 15:00 in B3.01
Speaker: Steve Lester (King's College London)
Title: The hyperbolic lattice point problem
Abstract: In this talk I will discuss the hyperbolic circle problem for SL₂(ℤ). Given two points z, w that lie in the hyperbolic upper half‑plane, the problem is to determine the number of SL₂(ℤ) translates of w that lie in the hyperbolic disk centred at z with radius arcosh(R/2) for large R. Selberg proved that the error term in this problem is O(R^{2/3}). I will describe some recent work in which we improve the error term to o(R^{2/3}) as R tends to infinity, under the condition that z, w are CM-points of different, square-free discriminants. This is joint work with Dimitrios Chatzakos, Giacomo Cherubini, and Morten Risager.
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12 January 2026 at 15:00 in B3.02
Speaker: Sean Prendiville (University of Lancaster)
Title: Regularity as an alternative to transference
Abstract: One way to approach certain number-theoretic problems involving sparse sets, such as the primes or squares, is to "model" these sets with dense sets of integers, then import results from the dense regime. This "transference principle" has been well-used in the last two decades, for instance in showing that relativley dense sets of primes contain three-term arithmetic progressions. We discuss the related method of "arithmetic regularity". We aim to make the case that regularity and transference are two sides of the same coin, but that regularity is more robust, and has the potential to yield results which are not amenable to transference.
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08 December 2025 at 15:00 in B3.02
Speaker: Lasse Grimmelt (University of Cambridge)
Title: Sums, Sieves, and Power Saving
Abstract: A modern perspective on studying primes in additive contexts is the following: Replace the difficult prime indicator by a simpler model, while not changing the count you are interested about. Clearly the model must have some similarity with the primes, as for example the sum of three primes is generically odd, the same must hold for the model. This swap inevitably introduces an error term and one exciting area is to push this error term into the so-called power saving range. This is remarkable, given we do not even have such a saving for the prime counting function itself.
In the first part, I will explain from a modern point of view how Montgomery and Vaughan proved a power-saving exceptional-set result for the binary Goldbach problem, and introduce the model Green used for his power-saving version of Sárközy in shifted primes. Afterwards, I’ll describe joint work with J. Teräväinen where we use numbers free of small prime factors as a model to study sums of almost twin-primes. -
01 December 2025 at 15:00 in B302
Speaker: Alex Bartel (Glasgow)
Title: Isospectral manifolds via orders in quaternion algebras
Abstract: I will report on joint work with Aurel Page on a number/representation theoretic approach to the question "Can you hear the shape of a drum". We use quaternion algebras over number fields to construct pairs of manifolds that "sound the same", but differ from each other in subtle ways. I will not assume that you already care about this question.
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24 November 2025 at 15:00 in B3.02
Speaker: Adam Morgan (University of Cambridge)
Title: TBA
Abstract: TBA
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17 November 2025 at 15:00 in B3.02
Speaker: David Hokken (Universiteit Utrecht)
Title: TBA
Abstract: TBA
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10 November 2025 at 15:00 in B3.02
Speaker: Cathy Swaenepoel (Paris Cite)
Title: Prime numbers with an almost prime reverse
Abstract: Let b ≥ 2 be an integer. For any integer n ≥ 0, we call `reverse' of n in base b the integer obtained by reversing the digits of n. The existence of infinitely many prime numbers whose reverse is also prime is an open problem. In this talk, we will present a joint work with Cécile Dartyge and Joël Rivat, in which we show that there are infinitely many primes with an almost prime reverse. More precisely, we show that there exist an explicit integer \Omega_b > 0 and c_b > 0 such that, for at least c_b b^ℓ / ℓ^2 primes p ∈ [b^{ℓ-1},b^ℓ[, the reverse of p has at most \Omega_b prime factors. Our proof is based on sieve methods and on obtaining a result in the spirit of the Bombieri-Vinogradov theorem concerning the distribution in arithmetic progressions of the reverse of prime numbers.
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03 November 2025 at 15:00 in B302
Speaker: Ross Paterson (University of Bristol)
Title: Quadratic Twists as Random Variables
Abstract: For each square-free integer D, and each elliptic curve E, the 2-Selmer groups of E and its quadratic twist E_D naturally live in the same space. We are motivated to study their independence as E varies. We shall present a heuristic in this direction, and some results in support of it.
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27 October 2025 at 15:00 in B3.02
Speaker: Hung Bui (University of Manchester)
Title: Weighted central limit theorem for central values of L-functions.
Abstract: A classical result of Selberg says that \log|\zeta(1/2 + it)| has a Gaussian limit distribution. We expect the same thing holds for \log|L(1/2, \chi)| for \chi being over the primitive Dirichlet characters modulo q, as q tends to infinity. Proving such a result remains completely out of reach, as it would imply 100% of these central L-values are non-zero, which is a well-known open conjecture. In this talk, I will describe how one can establish a weighted central limit theorem for the central values of Dirichlet L-functions. Under the Generalized Riemann Hypothesis, one can also obtain a weighted central limit theorem for the joint distribution of the central L-values corresponding to twists of two distinct primitive Hecke eigenforms. This is joint work with Natalie Evans, Stephen Lester and Kyle Pratt.
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20 October 2025 at 15:00 in B3.02
Speaker: Holly Krieger (University of Cambridge)
Title: Uniformity in arithmetic dynamics
Abstract: The periodic points of a discrete algebraic dynamical system control its local and global dynamical behaviour. When we impose an arithmetic structure on such a system, we do not generally expect periodic points to be rational. The central open conjecture in arithmetic dynamics asks whether this arithmetic structure imposes uniform constraints on the possible periods of points for families of algebraic dynamical systems. In this talk, we will discuss this conjecture, how it generalizes the torsion conjecture—in particular, the celebrated theorems of Mazur and Merel on rational torsion of elliptic curves—and survey some recent progress on and strategies for attacking this problem.
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13 October 2025 at 15:00 in B3.02
Speaker: Thomas Bloom (University of Manchester)
Title: TBA
Abstract: TBA
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06 October 2025 at 15:00 in B3.02
Speaker: Chris Hughes (University of York)
Title: Discrete moments of the Riemann zeta function
Abstract: I will discuss some new results on moments of zeta'(rho), the derivative of the Riemann zeta function evaluated at the zeta zeros. Despite being a complex function evaluated at complex points, it turns out to be real and positive on average. We will discuss this from both theoretical and heuristic viewpoints.
Click on a title to view the abstract!