Skip to main content Skip to navigation

2023-24

A list of members of the group and research intersts is available here

The Number Theory Seminar takes place on Mondays from 3 pm to 4 pm (or on Tuesday if Monday is a bank holiday).

Seminars are held in B3.02 of the Zeeman building (MS.03 if Tuesday) and live-streamed on Teams; sometimes the speaker is online but the talk will still be streamed in B3.02. Sometimes a different room is used, see the entry below.

2023-24 Term 3

Organisers: Simon Myerson, Harry Schmidt and Ju-Feng Wu

22 Apr

Anna Cadoret (Paris)

On the toric locus of motivic local systems

Let $X$ be a smooth, geometrically connected variety over a number field $k$ and let $F_l$ be a motivic $l$-adic local system on $X$ (typically, arising from the $l$-adic cohomology of a smooth proper variety over $X$); such $F$ actually comes in a compatible family of local systems $F_l$, $l$:prime. The toric locus of $F_l$ is the set of points of $X$ where the Zariski-closure of the image of the monodromy is a torus. Various conjectures (of motivic / diophantine nature) predict that the set of closed points of bounded degree in the toric locus is not Zariski dense in $X$. I will explain the following result in favor of this conjecture: under a mild level assumption, there exists a set $S$ of primes of Dirichlet density one such that for all $l$ in $S$ the set of $k$-rational points in the toric locus of $F_l$ is not Zariski-dense. The proof relies on Lawrence-Venkatesh strategy of comparing $v$-adic and complex period maps.

In person

29 Apr

Luca Mastella (Genova)

Vanishing of the $\mathfrak{p}$-part of the Shafarevich-Tate group of a modular form and its consequences for anticyclotomic Iwasawa theory

In this talk I will present some results of my PhD thesis, generalizing to modular forms a theorem on the anticyclotomic Iwasawa theory of elliptic curves of A. Matar and J. Nekovar. In particular we will give the definition of the ($\mathfrak{p}$-part of the) Shafarevich-Tate groups of a modular form $f$ of weight $k > 2$, over an imaginary quadratic field $K$ satisfying the Heegner hypothesis and over its anticyclotomic $\mathbb{Z}_p$-extension $K_\infty$ and we show that if the basic generalized Heegner cycle is non-torsion and not divisible by $p$, then they both vanish.

In person

7 May (Tue)

Rohini Ramadas (Warwick)

Degenerations and irreducibility problems in dynamics

This talk is about an application of combinatorial algebraic geometry to complex/arithmetic dynamics. The $n$-th Gleason polynomial $G_n$ is a polynomial in one variable with $\mathbf{Z}$-coefficients, whose roots correspond to degree-2 self-maps of $\mathbf{C}$ with an $n$-periodic ramification point. $\mathrm{Per}_n$ is an affine algebraic curve, defined over $\mathbf{Q}$, parametrizing degree-2 self-maps of $\mathbb{P}^1$ with an $n$-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is $G_n$ is irreducible over $\mathbf{Q}$? (2) Is $\mathrm{Per}_n$ connected? We show that if $G_n$ is irreducible over $\mathbf{Q}$, then $\mathrm{Per}_n$ is irreducible over $\mathbf{C}$, and is therefore connected. In order to do this, we find a $\mathbf{Q}$-rational smooth point on a projective completion of $\mathrm{Per}_n$ — this $\mathbf{Q}$-rational smooth point represents a special degeneration of degree-2 self-maps.

In person

13 May

Vandita Patel (Manchester)

Values of the Ramanujan $\tau$-function

The infamous Ramanujan $\tau$-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan $\tau$-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

In person

20 May

Dan Loughran (Bath)

The leading constant in Malle's conjecture

A conjecture of Malle predicts an asymptotic formula for the number of number fields with given Galois group and bounded discriminant. Malle conjectured the shape of the formula but not the leading constant. We present a new conjecture on the leading constant motivated by a version for algebraic stacks of Peyre's constant from Manin's conjecture. This is joint work with Tim Santens.

In person

28 May (Tue)

Group meeting, no seminar  

3 Jun

Andrew Graham (Bonn)

In person

10 Jun

Cameron Wilson (Glasgow)

Diagonal quadric surfaces with a rational point

Following work of Serre on the solubility of conics, a current problem of interest in the study of Diophantine equations is to count the number of varieties in families which have a rational point. In this talk I will give an overview of recent works in this area, before focussing on the particular example of diagonal quadric surfaces parameterised by {Y: wx=yz}. This family was first studied by Browning, Lyczak, and Sarapin, who showed that it exhibits an uncommonly large number of soluble members and attributed this phenomenon to the existence of thin sets on Y. They predicted that the “typical” behaviour should hold outside of these thin set, in the style of modern formulations of the Batyrev--Manin conjectures. In recent work I have shown that unusual behaviour occurs even with the removal of these thin sets by providing an asymptotic formula for the corresponding counting problem. Finally, I will outline the character sum methods used to prove this result and introduce an adaptation of the large sieve for quadratic characters.

In person

17 Jun

Jie Lin (Essen)

Teams

24 Jun

Kevin Kwan (UCL)

In person

2023-24 Term 2

Organisers: Simon Myerson, Harry Schmidt and Ju-Feng Wu

8 Jan

Martí Roset Julià (McGill)

The Gross—Kohnen—Zagier theorem via $p$-adic uniformization

Let $S$ be a set of rational places of odd cardinality containing infinity and a rational prime $p$. We can associate to $S$ a Shimura curve $X$ defined over $\mathbb{Q}$. The Gross—Kohnen—Zagier theorem states that certain generating series of Heegner points of $X$ are modular forms of weight $3/2$ valued in the Jacobian of $X$. We will explain this theorem and outline a new approach to prove it using the theory of $p$-adic uniformization and $p$-adic families of modular forms of half-integral weight. This is joint work in progress with Lea Beneish, Henri Darmon, and Lennart Gehrmann.

In person

15 Jan

Julian Lawrence Demeio (Bath)

The Grunwald Problem for solvable groups

Let $K$ be a number field. The Grunwald problem for a finite group (scheme) G/K asks what is the closure of the image of $H^1(K,G) \to \prod_{v \in M_K} H^1(K_v,G)$. For a general $G$, there is a Brauer—Manin obstruction to the problem, and this is conjectured to be the only one. In 2017, Harpaz and Wittenberg introduced a technique that managed to give a positive answer (BMO is the only one) for supersolvable groups. I will present a new fibration theorem over quasi-trivial tori that, combined with the approach of Harpaz and Wittenberg, gives a positive answer for all solvable groups. This is work in progress. Partial results were also obtained independently by Harpaz and Wittenberg

In person

22 Jan

Martin Orr (Manchester)

Endomorphisms of abelian varieties in families

Elliptic curves have played a central role in the development of algebraic number theory and there is an elegant theory of the endomorphisms of elliptic curves. Generalising to the higher-dimensional analogues of elliptic curves, called abelian varieties, more complex phenomena occur. When we consider abelian varieties varying in families, there are often only finitely many members of the family whose endomorphism ring is larger than the endomorphism ring of a generic member. The Zilber-Pink conjecture, generalising the André-Oort conjecture, predicts precisely when this finiteness occurs. In this talk, I will discuss some of the progress which has been made on the Zilber-Pink conjecture, including results of Daw and myself about families with multiplicative degeneration.

In person

B3.01

23 Jan

Alexander Molyakov (ENS Paris)

The Hasse principle for intersections of two quadrics

One of the first non-trivial examples of geometrically rational varieties is given by geometrically integral non-conical intersections of two quadrics in the projective space $\mathbb{P}^n\;(n\geqslant 4)$. In 1987 Colliot-Thélène, Sansuc and Swinnerton-Dyer proved the smooth Hasse principle for such a variety $X\subset \mathbb{P}^n$ over a number field when $n\geqslant 8$, they also conjectured that the smooth Hasse principle holds starting with the dimension $n=6$. Thirty years later, Heath-Brown established the Hasse principle for smooth intersections of two quadrics in $\mathbb{P}^7$. In the talk we will discuss the recent progress on this problem for singular intersections in $\mathbb{P}^7$. (Based on arXiv:2305.00313)

In person

B3.02

29 Jan

Veronika Ertl-Bleimhofer (IMPAN)

Conjectures on $L$-functions for varieties over function fields and their relations

(Joint work with T. Keller (Groningen) and Y. Qin (Regensburg))
We consider versions for smooth varieties $X$ over finitely generated fields $K$ in positive characteristic $p$ of several conjectures that can be traced back to Tate, and study their interdependence. In particular, let $A/K$ be an abelian variety. Assuming resolutions of singularities in positive characteristic, I will explain how to relate the BSD-rank conjecture for $A$ to the finiteness of the $p$-primary part of the Tate-Shafarevich group of $A$ using rigid cohomology. If time permits, I will discuss what is needed for a generalisation.

In person

B3.01

5 Feb

Jonathan Bober (Bristol)

Murmurations

I will survey the phenomenon first discovered by He, Lee, Oliver, and Pozdnyakov (arXiv:2204.10140) in the context of elliptic curves, described as "murmurations". We now see that this type of phenomenon seems to be present in many families of L-functions; there are low-order biases in the coefficients of L-functions which fluctuate regularly as a function of the index of the coefficient divided by the conductor of the L-function. In a few instances these phenomena can be proved.

In person

B3.02

12 Feb

Luis Santiago Palacios (Bordeaux)

Geometry of the Bianchi eigenvariety at non-cuspidal points

An important tool to study automorphic representations in the framework of the Langlands program is to produce $p$-adic variation. Such variation is captured geometrically in the study of certain "moduli spaces" of $p$-adic automorphic forms, called eigenvarieties.In this talk, we first introduce Bianchi modular forms, that is, automorphic forms for $\mathrm{GL}_2$ over an imaginary quadratic field, and then discuss its contribution to the cohomology of the Bianchi threefold. After that, we present the Bianchi eigenvariety and state our result about its geometry at a special non-cuspidal point. This is a joint work in progress with Daniel Barrera (Universidad de Santiago de Chile).

In person

B3.02

19 Feb

Damaris Schindler (Göttingen)

Density of rational points near manifolds

Given a bounded submanifold $M$ in $R^n,$ how many rational points with common bounded denominator are there in a small thickening of $M$? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? I will discuss some recent work in this direction and arithmetic applications such as Serre's dimension growth conjecture as well as applications in Diophantine approximation. For this I'll focus on joint work with Shuntaro Yamagishi, as well as joint work with Rajula Srivastava and Niclas Technau.

In person

B3.02

26 Feb

Chris Lazda (Exeter)

Comparisons between overconvergent isocrystals and arithmetic D-modules

According to a philosophy of Grothendieck, every good cohomology theory should have a six functor formalism. Arithmetic D-modules were introduced by Berthelot to provide the theory of rigid cohomology with exactly such a formalism. However, it is not clear that cohomology groups computed via the theory of arithmetic D-modules coincide with the analogous rigid cohomology groups. In this talk I will describe an 'overconvergent Riemann-Hilbert correspondence' that can be used to settle this question.

In person

B3.01

4 Mar

Lars Kuehne (Dublin)

Diophantine Equations, Unlikely Intersections, and Uniformity

Diophantine geometry is a modern-day incarnation of mathematicians' perennial interest in solving algebraic equations in integers. Its fundamental idea is to study the geometric objects defined by algebraic equations in order to understand their integral solutions ("geometry determines arithmetic"). One of its major achievements, Faltings' theorem, states that a large class of algebraic equations has only finitely many primitive integral solutions, namely those associated with smooth, proper curves of genus > 1. In the last few years, work by Dimitrov, Gao, Habegger, and myself has led to rather "uniform" bounds on the number of these solutions. Even more, our techniques yield purely geometric statements (uniform Manin-Mumford conjecture in characteristic 0). I will conclude with an optimistic outlook towards more general unlikely intersection problems in the framework of the conjectures of Zilber and Pink.

In person

B3.02

11 Mar

Sara Checcoli (Institut Fourier in Grenoble)

A little bit of little points: around property (N)

The height of an algebraic number is a positive real-valued function measuring its "arithmetic complexity". While numbers of height 0 are completely classified by a theorem of Kronecker, many questions remain open regarding numbers of small but nonzero height. One of the questions I will address is: given a set of algebraic numbers, does it contain only a finite number of elements of bounded height? When, for every possible bound, the answer is 'yes,' we say that the set has the Northcott property (N). This property was introduced in 2001 by Bombieri and Zannier and has since been studied by various authors. A well-known result by Northcott implies that every number field has property (N), but determining its validity for infinite extensions of the rationals is generally a challenging task. In this talk, I will provide an overview of what is known about this problem and present new results obtained in joint projects with Arno Fehm and, more recently, with Gabriel A. Dill.

In person

B3.01

2023-24 Term 1

Organisers: Simon Myerson, Ju-Feng Wu and Han Yu

2 Oct

Riccardo Maffucci (Coventry)

Limiting Theorems for Arithmetic Eigenfunctions

This is joint work with A. Rivera. Several recent papers study the ensemble of Laplace Toral eigenfunctions, and their randomisation 'arithmetic waves,' introduced in 2007 by Oravecz-Rudnick-Wigman, and by now a 'classical' setting. These waves are related to the arithmetic of writing a number as the sum of $d$ squares, where $d$ is the dimension. One question is the nodal volume of the wave, in the high-energy limit.

Rivera and I considered a wider class of 'arithmetic fields,' eigenfunctions of a certain operator. Instead of a sum of squares, we work with (homogeneous) forms in a certain degree and dimension. If the dimension is much larger than the degree, we have a precise asymptotic for the variance of nodal volume, in the high-energy limit. We also prove the limiting distribution to be Gaussian. To solve the problem, we found the precise order of magnitude for the 'correlations,' i.e., tuples of lattice points summing up to $0$.

In person

9 Oct

Postponed Maria-Romina Ivan (Cambridge)

Monochromatic Sums and Products over the Rationals

Hindman’s Theorem states that whenever the natural numbers are finitely coloured there exists an infinite sequence all of whose finite sums are the same colour. By considering just powers of 2, this immediately implies the corresponding result for products: whenever the naturals are finitely coloured there exists a sequence all of whose products are the same colour.

But what happens if we ask for both the sums and the products to all have the same colour? It turns out that this is not true: it has been known since the 1970s that there is a finite colouring of the naturals for which no infinite sequence has the set of all of its sums and products monochromatic.

In this talk, we will investigate what happens to this question if we move from the naturals to a larger space such as the dyadic rationals, the rationals, or even the reals.

Joint work with Neil Hindman and Imre Leader.

In person

16 Oct

Xenia Dimitrakopoulou (Warwick)

Anticyclotomic $p$-adic $L$-functions for families of $U_n \times U_{n+1}$

I will report on current work in progress on the construction of anticyclotomic $p$-adic $L$-functions for Rankin--Selberg products. I will explain how by $p$-adically interpolating the branching law for the spherical pair $\left(U_n, U_n \times U_{n+1}\right),$ we can construct a $p$-adic $L$-function attached to cohomological automorphic representations of $U_n \times U_{n+1}$, including anticyclotomic variation. Due to the recent proof of the unitary Gan--Gross--Prasad conjecture, this $p$-adic $L$-function interpolates the square root of the central $L$-value. Time allowing, I will explain how we can extend this result to the Coleman family of an automorphic representation.

In person

23 Oct

David Lilienfeldt (Leiden)

The Gross-Zagier formula for generalized Heegner cycles

In the 1980s, Gross and Zagier famously proved a formula equating on the one hand the central value of the first derivative of the Rankin-Selberg convolution $L$-function of a weight $2$ eigenform with the theta series of a class group character of an imaginary quadratic field, and on the other hand the height of a Heegner point on the corresponding modular curve. This equality was a key ingredient in the proof of the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rationals in analytic rank $0$ and $1$. Two important generalizations present themselves: to allow eigenforms of higher weight, and to further allow Hecke characters of infinite order. The former generalization is due to Shou-Wu Zhang. The latter one is the subject of this talk and requires the calculation of the Beilinson-Bloch heights of generalized Heegner cycles. This is joint work with Ari Shnidman.

In person

30 Oct

Rachel Newton (King's College London)

Distribution of genus numbers of abelian number fields

Let $K$ be a number field and let $L/K$ be an abelian extension. The genus field of $L/K$ is the largest extension of $L$ which is unramified at all places of $L$ and abelian as an extension of $K$. The genus group is its Galois group over $L$, which is a quotient of the class group of $L$, and the genus number is the size of the genus group. We study the quantitative behaviour of genus numbers as one varies over abelian extensions $L/K$ with fixed Galois group. We give an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time. This is joint work with Christopher Frei and Daniel Loughran.

In person

6 Nov

Victor Souza (Cambridge)

The number of monochromatic solutions to multiplicative equation
Abstract: Given an r-colouring of the integer interval [2, N], what is the minimum number of monochromatic solutions of the equation xy = z?
We answer this question, obtained a sharp answer for 2, 3 or 4 colours.
We also explore related questions and conjectures.
 
Joint work with L. Aragão, J. Chapman and M. Ortega.
In person

13 Nov

Manuel Hauke (York)

On the connections between metric Diophantine approximation and Birkhoff sums for irrational rotations

In the theory of metric Diophantine approximation, the existence of unusually large partial quotients destroys the possibility of many classical limit laws (such as Strong law of large numbers, Central Limit Theorem, ...). I will describe how this phenomenon can be applied to obtain precise bounds for the typical and worst-case oscillation of Birkhoff sums $ \sum_{n=1}^N f(n\alpha), N \to \infty$ for almost every $\alpha$ where $f$ is a $1$-periodic function with discontinuities or singularities.

Further, I will present a Khintchine-type behaviour of $ \sum_{n=1}^N f(n\alpha)$ and the (non-)existence of limit laws that again resembles the behaviour known from metric Diophantine approximation.

This is joint work with L. Fruehwirth.

In person

20 Nov

Alisa Sedunova (LIMS)

The multiplication table constant and sums of two squares

Let $r_1(n)$ be the number of integers up to $x$ that can be written as the square of an integer plus the square of a prime. We will show that its mean is asymptotic to $(\pi/2) x \log x$ minus a secondary term of size $x/(\log x)^{1+d+o(1)}$, where d is the multiplication table constant. Detailed heuristics suggest very precise asymptotic for the secondary term as well. In particular, our proofs imply that the main contribution to the mean value of $r_1(n)$ comes from integers with “unusual” number of prime factors, i.e, those with $\omega(n) \sim 2 \log \log x$ (for which $r_1(n) \sim (\log x)^{\log 4-1}$), where $\omega(n)$ is the number of district prime factors of $n$. This is a joint work with Andrew Granville and Cihan Sabuncu.

In person

27 Nov

Maria-Romina Ivan (Cambridge)

Monochromatic Sums and Products over the Rationals

Hindman’s Theorem states that whenever the natural numbers are finitely coloured there exists an infinite sequence all of whose finite sums are the same colour. By considering just powers of 2, this immediately implies the corresponding result for products: whenever the naturals are finitely coloured there exists a sequence all of whose products are the same colour.

But what happens if we ask for both the sums and the products to all have the same colour? It turns out that this is not true: it has been known since the 1970s that there is a finite colouring of the naturals for which no infinite sequence has the set of all of its sums and products monochromatic.

In this talk, we will investigate what happens to this question if we move from the naturals to a larger space such as the dyadic rationals, the rationals, or even the reals.

Joint work with Neil Hindman and Imre Leader.

In person

4 Dec

Thomas Karam (Oxford)

Title: Distinguishing subsets of the cube with mod-$p$ linear forms.

Abstract: Let $A_1$,..,$A_s$ be a sequence of dense subsets of the Boolean cube $\{0,1\}^n$ and let $p$ be a prime. We show that if $s$ is superpolynomial in $n$ then we can find distinct $i$,$j$ such that the two distributions of every mod-$p$ linear form on $A_i$ and $A_j$ are almost positively correlated. We also prove that if $s$ is merely sufficiently large independently of $n$ then we may require the two distributions to have overlap bounded below by a positive quantity depending on $p$ only.

In person

Postponed

Alexander Molyakov (ENS Paris)

Title TBC

In person