Junior Number Theory Seminar 20222023
Hi all!
Welcome to the Warwick Junior Number Theory Seminar, where graduate students share the outcomes of their research to their peers.
This term, all talks will be held in B3.02 at 11 am on Monday (except when stated otherwise).
Do you want to give a talk in this seminar? Feel free to contact one of the organisers!
This seminar was organised by Xenia Dimitrakopoulou and Elvira Lupoian.Link opens in a new window
202223 Term 3
24th April 
Steven Groen (Warwick) Supersingular EkedahlOort strata of unitary Shimura varieties (including an explanation of what that means) Let us study Abelian varieties in characteristic $p$ that have complex multiplication by an imaginary quadratic field of signature $(q2,2)$. This talk concerns the interaction between two stratifications of the moduli space of these Abelian varieties, called a unitary Shimura variety. The EkedahlOort stratification is based on the isomorphism class of the $p$torsion group scheme $A[p]$, while the Newton stratification is based on the isogeny class of the $p$divisible group $A[p^\infty ]$. Our goal is to determine which EkedahlOort strata intersect the smallest Newton stratum, called the super singular locus. I will present two new methods that give a partial answer to this question: one based on products of smaller Abelian varieties and one based on the Siegel modular variety. This is joint work with Deewang Bhamidipati, Maria Fox, Heidi Goodson, Sandra Nair and Emerald Stacy. 
Tuesday 2nd May in MB0.08 
James Rawson (Warwick) Some obstructions to solvable points on higher genus curves Falting's theorem tells us that for curves of genus at least 2, the number of points defined over any number field is finite. At the other extreme, there are infinitely many points on any curve defined over the algebraic closure of $\mathbb{Q}$. A natural question is to ask about points defined over some family of number fields, a classically interesting example is the solvable extensions. For low genera ( <5), infinite families of such points can be constructed, by methods of Pal and Wiles and Ciperani. However, for curves with genus at least 5, little is known. I will discuss a geometric approach to this problem and some obstruction to the existence of solvable points. 
Tuesday 9th May in MB0.08 
Anja Meyer (Manchester) Towards the cohomology of $SL_{2}\left( \mathbb{Z} / p^{n} \mathbb{Z} \right)$ The cohomology of matrix groups with entries in fields is well studied. However, for matrices with entries in finite rings commonly used methods fail. In this talk I will present a method of finding the mod $p$ cohomology $SL_{2} \left( \mathbb{Z} / p^{n} \mathbb{Z} \right)$, $n >1$, where as we will see, the cohomology of congruence subgroups $\Gamma \left( n \right) $ play a central role. I will further present the computation for the cohomology of a Sylow3 of $SL_{2}\left( \mathbb{Z} / 3^{n} \mathbb{Z} \right) $ as a worked example. 
15th May 
High Moments of Dirichlet Character sums Calculating moments of families of Lfunctions is a ubiquitous problem in analytic number theory, and has seen great progress in recent years. The aim of this talk is to explain how one can obtain sharp upper bounds on high real moments of Dirichlet character sums assuming the Generalised Riemann Hypothesis. The main ingredient is an upper bound on shifting moments of Dirichlet Lfunctions (assuming GRH), the proof of which uses techniques on joint values of Dirichlet polynomials developed by Harper. 
22nd May 
Xenia Dimitrakopoulou (Warwick) Constructing $p$adic $L$functions with a view towards unitary groups You've heard of $p$adic numbers and of $L$functions; get ready for $p$adic $L$functions! In this talk, we will give a brief introduction to $p$adic $L$functions and automorphic representations. We will describe how we can use these objects to interpolate critical values of $L$functions and focus on the case of the product unitary groups $U_{n+1} \times U_{n}$. 
29th May 
Bank Holiday (no seminar) 
5th June 
Alexandru Pascadi (Oxford) Kloostermania and primes in arithmetic progression to large moduli Many problems in analytic number theory seminar reduce to the task of bonding various exponential sums, often involving the notorious Kloosterman sums. We'll outline hoe "Kloostermnia" some up in proving equidistribution results about sequences from multiplicative number theory (like primes, smooth numbers, or divisor functions), in arithmetic progressions with large moduli. If time allows, we'll also discuss some of the ways to bound the merging sums of Kloosterman sums, using deep results from algebraic geometry and the spectral theory of automorphic forms. 
12th June 
Alvaro Gonzalez Hernandez (Warwick) An arithmetic expedition into K3 surfaces This talk will be a 50min tour of the theory of K3 surfaces, which are the (second) most natural analogue of elliptic curves in dimension two. We will start with a gentle talk through the geometry of K3 surfaces, exploring their most singular features. After that, I will focus on more arithmetic aspects by explaining how to construct elliptic fibrations in a surface and, together, we will conquer our first K3 and find a nice view of its lattice. Finally, we will rappel down the wild cliffs of positive characteristic geometry, but hopefully, we will all make a safe return to the fields of characteristic zero. 
19th June 
Mod $p$ Hilbert Modular Forms Serre's conjecture tells us that an odd, irreducible mod $p$ Galois representation arises from a mod $p$ modular form. A similar phenomenon for totally real fields and Hilbert modular forms has also been conjectured. In this talk, we'll introduce the definition of Hilbert modular forms as sections of automorphic line bundles on Hilbert modular varieties. We'll also discuss the lift ability from characteristic $p$ to characteristic $0$, which will help us to give an algorithm in computing mod $p$ modular forms. 
26th June 
When does an equation have a solution in the primes? When looking at a homogeneous Diophantine equation, the Hasse principle tells us that we expect there to be a nontrivial solution in the integers provided that a certain local solubility condition is satisfied. Many results exist about when this principle holds. But what if we want to know when a homogeneous Diophantine equation has a solution in the primes? In this case, it turns out that something similar to the Hasse principle holds. We discuss how one can come up with such a principle, as well as some probabilistic results about when it holds. We will see that it holds for almost all equations in certain families, and outline how this can be shown using the theory of lattices. 
202223 Term 2
9th January 
Muhammad ManjiLink opens in a new window (Warwick) Main conjectures in Iwasawa theory  past, present and future The classical Iwasawa main conjecture is a statement relating the padic zeta function to ideal class groups in a tower of Galois extensions, proven by Mazur and Wiles in 1984. This lets us relate the mysterious Riemann zeta function to class field theory. The framework of this was later generalised to a modular forms analogue; relating the padic Lfunction of a modular form with a "Selmer group" coming from its cohomology  proved in 2006 by Skinner and Urban. This result is strongly related to the BirchSwinnertonDyer conjecture. Is there a more general pattern here? We will focus on the two cases above and time permitting discuss what more general conjectures look like. 

16th January 
Jenny RobertsLink opens in a new window (Bristol) Newform Eisenstein Congruences of Local Origin The theory of Eisenstein congruences dates back to Ramanujan’s surprising discovery that the Fourier coefficients of the discriminant function are congruent to the 11^{th} power divisor sum modulo 691. This observation can be explained via the congruence of two modular forms of weight 12 and level 1; the discriminant function and the Eisenstein series, E_{12}. We explore a generalisation of this result to newforms of weight k > 2, squarefree level and nontrivial character. Time permitting, we will discuss what these congruences tell us about the BlochKato conjecture. 

23rd January 
Maleeha Khawaja (Sheffield) The Fermat equation over real biquadratic fields Let K be a totally real number field. A key ingredient in the modular approach to resolving the Fermat equation x^n+y^n=z^n over K is modularity of the Frey curve over K. Thanks to the work of Box, we know that elliptic curves over totally real quartic fields not containing sqrt(5) are modular. For real biquadratic fields not containing sqrt(5), with modularity of the Frey curve in hand, we can then study the Fermat equation using the modular approach. In this talk we will look at the other obstacles that arise, using Q(sqrt2, sqrt3) as an illustrating example. 

30th January 
Andrew Scoones (York) On the abc conjecture in Algebraic Number Fields While the abc conjecture remains open, much work has been done on weaker versions, 

6th February 
Wissam Ghantous (Oxford) Collisions in supersingular isogeny graphs In this talk we will study the graph structure of supersingular isogeny graphs. These graphs are known to have very few loops and multiedges. We formalize this idea by studying and finding bounds for their number of loops and multiedges. We also find conditions under which these graphs are simple. To do so, we introduce a method of counting the total number of collisions (which are special endomorphisms) based on a trace formula of Gross and a known formula of Kronecker, Gierster and Hurwitz. The method presented in this talk can be used to study many kinds of collisions in supersingular isogeny graphs. As an application, we will see how this method was used to estimate a certain number of collisions and then show that isogeny graphs do not satisfy a certain cryptographic property that was falsely believed (and proven!) to hold. 

13th February 
Mar CurcoIranzo (Utrecht) Generalised Jacobians of Modular Curves and their Qrational torsion The Jacobian J0(N) of the modular curve X0(N) has received much attention within arithmetic geometry for its relation with cusp forms and elliptic curves. In particular, the group of Qrational points on X0(N) controls the cyclic Nisogenies of elliptic curves. A conjecture of Ogg predicted that, for N prime, the torsion of this group comes all form the cusps. The statement was proved by Mazur and later generalised to arbitrary level N into what we call generalised Ogg's conjecture. Consider now the generalised Jacobian J0(N)m with respect to a modulus m. This algebraic group also seems to be related to the arithmetic of X0(N) through the theory of modular forms. In the talk we present new results that compute the Qrational torsion of J0(N) for N an odd integer with respect to a cuspidal modulus m. These generalise previous results of Yamazaki, Yang and Weil. In doing so, we will also discuss how our results relate to generalised Ogg's conjecture.


20th February 
PedroJose Cazorla Garcia (Manchester) The generalised LebesgueRamanujanNagell equation The LebesgueRamanujanNagell equation x^2 + D = y^n has been student extensively by number theorists during the last century, both by its intrinsic interest and as a generalisation to Catalan's conjecture. With the advent of the modular methodology developed bu Wiles, Ribet and others and employment in the proof of Fermat's last theorem, along with the evolution of computational number theory techniques, it is now feasible to consider the generalised LebesgueRamanujanNagell equation C1x^2 + C2 = y^n . In this talk, we will discuss a variety of techniques which allow us to solve the aforementioned equation in the range 1< 1 C1, C2 < 20, involving the modularity of Galois representations and Thue equations.


27th February 
Robin Visser (Warwick) Computing abelian varieties with good reduction outside a finite set of primes Let $K$ be a number field, $d$ a positive integer, and $S$ a finite set of primes in $K$. In the 1960s, Shafarevich conjectured that there are only finitely many isomorphism classes of dimension $d$ abelian varieties $A/K$ with good reduction outside $S$. Whilst this was eventually proved by Faultings in 1983, his proof did not produce an effective algorithm to explicitly calculate all such abelian varieties for a given $K$,$d$ and $S$. In this talk, I shall give a brief survey of some known methods to effectively compute abelian varieties in certain cases, as well as comment on several existing problems in this area. 

6th March 
Katerina Santicola (Warwick) Reverse Engineering Polynomials Number theorists love to solve Diophantine equations. We start with an equation, and we try to find all the integer solutions. In this talk, we're going the other direction. Starting with a finite set of perfect powers, can we find a polynomial whose image hits these perfect powers, and only these ? We will go over how to do this for integral and rational perfect powers, and then discuss some applications to constructing hyperelliptic curves with prescribed rational points. 

13th March 
Margherita Pagano (Leiden)
BrauerManin obstruction to weak approximation coming from primes of good reduction
A way to study rational points on a variety is by looking at their image in the padic points. The BrauerManin obstruction can be used to study the density of rational points inside the product of the padic points. In this talk I will discuss the role that primes of good reduction on a variety might play in the BrauerManin obstruction to weak approximation.

202223 Term 1
3rd October 
Elvira LupoianLink opens in a new window (Warwick) TwoTorsion Subgroups of Genus 5 Curves and the Generalized Ogg Conjecture Let be a smooth, projective and nonhyperelliptic curve of genus 5 over and let be its Jacobian. Recall that is a 5dimension abelian variety whose points can be identified with elements of the zero Picard group of . The MordellWeil theorem states that for any number field , is a finitely generated group; that is, , where is a finite group, the torsion subgroup, and is the rank. In this talk I will present a method of computing the 2torsion subgroup of ; that is the group and hence the 2torsion over any number field . This method was used to verify the Generalized Ogg conjecture for with . 

10th October 
Diana MocanuLink opens in a new window (Warwick) The modularity approach over totally real fields I will use this occasion to introduce the modular method for solving Diophantine Equations, famously used to prove the Fermat Last Theorem. Then, we will see how to generalize it for a totally real number field and a Fermattype equation over $K$. We call the triple of exponents the signature of the equation. We will see various results concerning the solutions to the Fermat equation with signature and using image of inertia comparison and the study of $S$unit equations. This is a completed project under the supervision of Samir Siksek. If time permits, I will show some ongoing work on how one can approach the more difficult family of equations of signatures (fixed r). 

17th October 
Havard DammJohnsenLink opens in a new window (Oxford) Computing GrossStark units using Hilbert modular forms Recent work of DarmonPozziVonk shows that logarithms of certain units of number fields, socalled GrossStark units, arise as constant terms of overconvergent modular forms. These are obtained by doing a suitable weightspace deformation of the restriction of an Eisenstein series on the Hilbert modular group. In this talk I will explain what some of these words mean, describe an algorithm for computing the GrossStark units efficiently, and show some very concrete examples. Time permitting, I will also describe some ongoing work on extending these results. 

24th October 
Nuno AralaLink opens in a new window (Warwick) Lines of polynomials with alternating Galois group It remains an open problem to quantify how often a degree n polynomial with integer coefficients has Galois group A_n over Q. In order to have any hope of answering this question, we need in particular to construct suitably large families of such polynomials. We focus on the question of when one can find such a family depending linearly on some parameter, since those are in some sense the largest families. We find some examples and prove some structure theorems about families with this property. 

31st October 
Philippe MichaudJacobsLink opens in a new window (Warwick) Pinpointing perfect powers: polygons plus pyramids The problem of finding squares and cubes in number sequences has long been a topic of interest to number theorists. Over the last century, this has been generalised to searching for perfect powers (numbers of the form m^k) in sequences, and Wiles' proof of Fermat's Last Theorem opened up a realm of possibilities for studying such problems. In this talk I'll discuss some of the techniques used to study perfect powers in sequences, using polygonal and pyramidal number sequences as core examples. I'll discuss how one can study Thue equations that occur in this setting by using local and modular methods combined with bounds arising from linear forms in logarithms. Part of this talk is based on joint work with Andrej Dujella, Kálmán Győry, and Ákos Pintér. My aim is for the talk to be relaxed and accessible; I won't give very many details, but I do hope to give a flavour of the techniques one can use to study a wide variety of Diophantine equations. 

7th November 
William O'ReganLink opens in a new window (Warwick) The sumproduct phenomenon The Erdős–Szemerédi theorem states that for every finite nonempty set A of integers at least one of A + A, the set of pairwise sums, or A.A, the set of pairwise products form a significantly larger set. This theorem can be viewed as an assertion that the real line cannot contain any set resembling a finite subring; it is the first example of what is know as the sumproduct phenomenon, which is now known to hold in a wide variety settings, such as rings and fields, including finite fields. In this talk, I will give a soft introduction to additive number theory and survey some of the results, including those by highly successful mathematicians such as Bourgain & Tao, displaying such phenomena. No prior knowledge will be required or expected. 

14th November 
Julio de Mello BezerraLink opens in a new window (Regensburg) Cohomological approaches to rationality of special values of Lfunctions It is known since Euler that the negative integer values of the Riemann zeta function are all rational, given by Bernoulli numbers. This rationality result has been extended to other classical zeta functions and finally to Hecke Lfunctions for totally real number fields, by two different approaches: one by Siegel and Klingen using Eisenstein series and the other due to Shintani using his Unit Theorem. In this talk we will explain these different classical approaches, starting with the Riemann zeta, and show how they give rise to more modern equivariant cohomological methods that prove these rationality statements. The stars of the show will be the generating functions whose coefficients are generalized Bernoulli numbers and whose Mellin transforms give the zeta functions. If time allows, we will also quickly mention integrality results, which are vital for the existence of padic Lfunctions and results such as the Iwasawa Main Conjecture for totally real fields.
The slides for this talk can be found here . 

21st November 
Maria CorteReal SantosLink opens in a new window (UCL) The general isogeny problem in dimension 2 The general isogeny problem is the fundamental hard problem that underlies the security of isogenybased cryptography, which asks an adversary to find the isogeny between two given (supersingular) elliptic curves over a finite field. Despite the majority of isogenybased cryptographic protocols being constructed in dimension 1, there has been a surge of interest in building protocols from two dimensional abelian varieties and the isogenies between them. In this talk, we will explore the generalisation of isogenybased cryptography to two dimensions and discuss the best attack against the general dimension 2 isogeny problem and its concrete complexity, based on joint work with Craig Costello and Sam Frengley. 

28th November 
Armando Gutierrez TerradillosLink opens in a new window (Universitat Polytecnica de Catalunya) Zeta integrals and Shalika models for $\mathrm{GU}_{2,2}$ Models of automorphic representations are intimately related to integral representations of $L$functions. The purpose of this talk is to describe a local criterion for determining the multiplicity of local Shalika models for spherical representations of $\mathrm{GU}_{2,2}$. As an application, I will explain how this model may show an integral representation of the standard $L$function of $\mathrm{GSp}_4$. 

5th December 
Lukas PraderLink opens in a new window (Regensburg) Polylogarithmic Eisenstein classes and special values of Lfunctions One striking feature of special values of Lfunctions is that they appear as constant terms of Eisenstein series. The goal of this talk is to use this fact to motivate the definition of Eisenstein cohomology classes. Further, we shall discuss how such classes may be constructed via the topological polylogarithm, and how this yields insights into special values of Lfunctions.
