Number Theory Seminar
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 livestreamed 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.
202324 Term 2
Organisers: Simon Myerson, Harry Schmidt and JuFeng 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 halfintegral 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 quasitrivial 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 higherdimensional 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 ZilberPink conjecture, generalising the AndréOort conjecture, predicts precisely when this finiteness occurs. In this talk, I will discuss some of the progress which has been made on the ZilberPink 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 nontrivial examples of geometrically rational varieties is given by geometrically integral nonconical intersections of two quadrics in the projective space $\mathbb{P}^n\;(n\geqslant 4)$. In 1987 ColliotThélène, Sansuc and SwinnertonDyer 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, HeathBrown 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 ErtlBleimhofer (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 BSDrank conjecture for $A$ to the finiteness of the $p$primary part of the TateShafarevich 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 Lfunctions; there are loworder biases in the coefficients of Lfunctions which fluctuate regularly as a function of the index of the coefficient divided by the conductor of the Lfunction. 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 noncuspidal 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 noncuspidal 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 Dmodules According to a philosophy of Grothendieck, every good cohomology theory should have a six functor formalism. Arithmetic Dmodules 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 Dmodules coincide with the analogous rigid cohomology groups. In this talk I will describe an 'overconvergent RiemannHilbert correspondence' that can be used to settle this question. 
In person B3.01 
4 Mar 
Lars Kuehne (Dublin) 
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 realvalued 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 wellknown 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 
202324 Term 1
Organisers: Simon Myerson, JuFeng 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 OraveczRudnickWigman, 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 highenergy 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 highenergy 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 

Postponed MariaRomina Ivan (Cambridge)


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 RankinSelberg 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 GanGrossPrasad 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 GrossZagier 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 RankinSelberg 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 SwinnertonDyer 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 ShouWu Zhang. The latter one is the subject of this talk and requires the calculation of the BeilinsonBloch 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 rcolouring 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 worstcase 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 Khintchinetype 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 $r_1(n) \sim (\log x)^{\log 41}$), 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 
MariaRomina 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 