London Junior Number Theory Seminar (old)

Abstract: The strong form of Serre's conjecture states that every two-dimensional continuous, odd, irreducible mod p representation of the absolute Galois group of Q arises from a modular form of a specific minimal weight, level and character. In this talk we show this minimal weight is equal to two other notions of minimal weight, one inspired by work of Buzzard, Diamond and Jarvis and one coming from p-adic Hodge theory. After discussing the interplay between these three  weight characterisations in the more general setting of Galois representations over totally real fields, we investigate its consequences for generalised Serre conjectures.

Abstract: In 1999, Batyrev showed that birational Calabi-Yau varieties over \C have equal Betti numbers. The proof is a lovely journey into the world of p-adic manifolds, p-adic integration and the Weil conjectures. In this talk, we will present the proof and discuss generalisations of this result to number theory and mirror symmetry. The talk will be accessible to all.

Abstract: The section conjecture is a classical conjecture about curves of genus >=2 over p-adic fields or number fields. Roughly speaking, it says that there is a bijection between the points on the curve and sections of a certain, natural, short exact sequence. There are many ways of viewing this conjecture, but perhaps the way that it makes the most sense is as a topological statement about certain strange "spaces". In this talk, I'll introduce you to the section conjecture as it is written, then talk about how it relates to étale homotopy theory.

No knowledge of étale homotopy or any other homotopy theory is required, but the talk will be a little bit topological in parts.

Abstract:The parity conjecture predicts that the rank of an elliptic curve is related to the root number, which we can calculate relatively easily. I will explore the consequences of this conjecture, with a focus on quadratic twists. I will then discuss techniques for finding root numbers of hyperelliptic curves, and how these are connected to the elliptic case. 

Abstract:The aim of this talk is to present (at least some ideas related to) ongoing work on what I call motivic point counting. After recalling some basics regarding the classical situation of point counting from number theory/algebraic geometry, I will give an introduction to the motivic world surrounding the Grothendieck ring of varieties K0(Vark). I will define certain generalizations of relative tangent bundles, explain how they come into play and generalize them even further. Finally, I want to explain how these generalizations give rise to new tools in motivic point counting. 

Abstract: Overconvergent modular forms and completed cohomology are two important candidates for the role of p-adic modular forms. Their definitions are of very different nature and both theories have their own advantages. Therefore, it is natural to ask about the connection between the two, if there is any. In this talk we will discuss independent recent work of Lue Pan and Sean Howe which reveal a deep connection between the two sides. In both cases, the main tools are p-adic Hodge theory and the geometry of Scholze's Hodge-Tate period map. 

Abstract: In 2015, Scholze proved that torsion classes appearing in the cohomology of Bianchi groups have associated mod p Galois representations in the usual way. The aim of this talk is to connect period values of mod p classes, both torsion and not, to the ranks of suitably chosen Selmer groups, in a way comporting with the Birch and Swinnerton-Dyer conjecture for elliptic curves over imaginary quadratic fields. This follows a question posed by Calegari and Venkatesh in their book "A Torsion Jacquet-Langlands Correspondence". 

Abstract: The theory of Berkovich spaces is one of a few approaches to developing non-Archimedean geometry. This theory fixes some of the drawbacks that Tate's original approach of rigid spaces had, giving a theory which is geometrically much more intuitive. We will illustrate its use through the insights it gives to uniformization of elliptic curves over non-Archimedean fields. Familiarity with rigid spaces might be helpful, but will not be by any means necessary.

Abstract: Maass forms are complex valued functions on the upper half plane, which similar to modular forms transform under the action of a discrete subgroup G of SL(2,R). In addition, they are eigenfunctions of the hyperbolic Laplacian defined on the upper half-plane and satisfy certain growth conditions at the cusps of a fundamental domain of G. However, unlike modular forms, Maass forms need not be holomorphic. In this talk I will define Maass cusp forms for SL(2,Z) and explain an algorithm due to Hejhal for numerically computing the Laplace eigenvalues of Maass cusp forms for SL(2,Z). 

Abstract: In 1966 Chen Jingrun proved that every large even integer can be written as the sum of two primes or the sum of a prime and a semiprime. To date, this weakened version of Goldbach's conjecture is one of the most remarkable results of sieve theory. In my talk, I will outline the key concepts and developments from sieve theory that paved the way to Chen’s proof and give a sketch of his original argument. No prior familiarity with sieve theory will be assumed.

Abstract: In this talk I will sketch how p-adic geometry can help us in better understanding Shimura varieties. To do this I will introduce Shimura varieties and perfectoid spaces separately. Then we will see how they interfere. Mainly treating the Siegel case (which includes modular curves) we will see that the geometry of these Shimura varieties naturally yields perfectoid spaces. Here the ordinary locus and the Frobenius on it play a crucial role.  This connection turns out to be quite fruitful as, among other things, it tells us when certain cohomology groups of Shimura varieties vanish.

Abstract: In this talk we will be interested in the primes of good supersingular reduction of an elliptic curve E over the rationals. Elkies proved in the 80s that this set of primes is always infinite. We will explain his 3-page proof, which makes clever use of CM theory.

This is an expository talk and is intended to be accessible to everyone who knows what an elliptic curve is. 

Abstract: What are multiplicative functions? What are random models? In this talk, I will briefly describe both terms and introduce you to the world of random multiplicative functions (RMFs). In 1944, Wintner constructed such a function to model the Mobius function and these models have been well studied since. I will use RMFs to show the limiting distribution of sums of Dirichlet Characters and outline the associated proof. 

Abstract:  In the 1920s, Mordell and Weil asserted that the structure of the rational points on an elliptic curve, or more generally an abelian variety, is determined by two pieces of data: the rank and the torsion subgroup. In this talk we will address the first of these, looking specifically at hyperelliptic curves. We will see a *new method* for determining the parity of the rank and will discuss an approach this provides to the Parity Conjecture. 

Abstract: The operator θ = q d/dq plays an important role in the theory of elliptic modular forms modulo p. It is particularly useful for studying questions of minimality of lifts to characteristic zero and the classification of its cycles was a fundamental step in Edixhoven's proof of the weight part of Serre's modularity conjecture. Because of this, one would like to be able to define and study similar operators on more general classes of automorphic forms. To achieve such generalisations, it is important to understand the geometric nature of this operator. In this talk I will present a classical construction of theta due to Gross and, if time allows, I will also explain how to give a similar construction in the Picard case. 

Abstract: Fermat’s Last Theorem states that the equation x^n + y^n = z^n , with n at least 3, has no non-trivial solutions forintegers x, y and z. What happens if we replace the word 'integers' by 'O_K ', for K a number field? Does the same statement hold? In this talk I will give an overview of the proof of Fermat’s Last Theorem (over Q), seeing how three black boxes of Wiles, Ribet, and Mazur combine to resolve this 400-year-old problem. I will then shift setting to work over a totally real field, discuss some of the difficulties that arise, and see how they can sometimes be overcome using a variety of techniques.

Note for this talk are now available on Philippe's website, which you can get to by clicking his name above.

Abstract: The Riemann hypothesis (RH) is one of the great open problems in mathematics. It arose from the study of prime numbers in an analytic context, and—as often occurs in mathematics—developed analogies in an algebraic setting, leading to the influential Weil conjectures. RH for curves over finite fields was proven in the 1940’s by Weil using algebraic-geometric methods. In this talk, we discuss an alternate proof of this result by Stepanov (and Bombieri), using only elementary properties of polynomials. Over the decades, the proof has been whittled down to a 5 page gem! Time permitting, we also indicate connections to exponential sums and the original RH. The talk is geared to a broad mathematical audience. 

Abstract: Motivated by a series of conjectures of Mazur, Rubin and Stein, the study of the arithmetic statistics of modular symbols has received a lot of attention in recent years. In this talk, I will highlight several results about the distribution of modular symbols, including their Gaussian distribution and the residual equidistribution modulo p. I will also talk about generalisations to quadratic imaginary fields and higher dimensions. 

Abstract: The rational points of an abelian variety A form a finitely-generated group of the form A(Q) = Δ x Zr, where Δ is the finite torsion part, and r the rank. This rank is notoriously difficult to pin down in any generality. Even in the classical case of elliptic curves, one relies on the conjectural finiteness of the Tate--Shafarevich group for an algorithm which is guaranteed to terminate. In higher-dimensions, one faces the further challenge that such algorithms are largely undeveloped. We review the current approaches, theorems, and conjectures that facilitate some extraction of rank information. In particular, we look at how isogenies play an intimate role in the parity of the rank, and some general setups (not just for elliptic curves) to exploit this to get some cold, hard numbers out the other side.

Abstract: Despite the name, pseudorepresentations are not pseudoscience. I'll discuss what they are and why they're useful. 

Abstract: I will give a brief overview of the book Galois Cohomology of Elliptic Curves, by Prof J Coates and Prof R Sujatha. After a few basic results of Galois cohomology, we shall see an application of Iwasawa theory to a Z_p-extension obtained by attaching the p-power torsion points of an elliptic curve E over a number field F. Finally, if time permits, we will take a look at some specific examples, that is, Iwasawa theory for curves of conductors 11 and 294.

Abstract: Let F be a totally real field and \chi a totally odd character on the absolute Galois group of F. The Gross-Stark conjecture predicts the value of the leading term of the p-adic L-function attached to the character \chi. A key part of this value is the determinant of the Gross-Regulator matrix. In this talk I will introduce this conjecture and its relation to the Gross-Stark units. Formulas have been conjectured by Dasgupta and Spiess for the values of this determinant and the Gross-Stark units. At the end I will mention forthcoming work showing that these two formulas are compatible when F is of degree 3. 

Abstract: Let V be a p-adic Galois representation which arises from geometry (i.e. as a subquotient of the etale cohomology of some variety). Then attached to V is an arithmetic object, the Bloch--Kato Selmer group, and an analytic object, the complex valued L-function. The conjecture of Bloch--Kato predicts a relation between these two objects and can be viewed as a generalisation of the Birch--Swinnerton-Dyer conjecture. In this talk I will describe this conjecture and survey some known cases. 

Abstract: In 1922, Mordell proved that the group of rational points E(Q) of an elliptic curve E/Q is a finitely generated Abelian group, written in the form E(Q) \cong E(Q)_{tors} x Z^r. While the finite part is well understood, the infinite part is much more mysterious. In this talk, we discuss ranks of quadratic twists of pairs of elliptic curves and show how to obtain quadratic twists of pairs of elliptic curves with high ranks. 

Abstract: For K a number field it is natural to look at the representations of the Galois group of a finite extension not only over K but also over its ring of integers. Taking reduction modulo a prime ideal of these integral representations we get examples of representations over fields of finite characteristic naturally occurring in number theory. For G a finite group and K a field of characteristic 0, Mashke’s theorem says that every finite dimensional KG-module is semisimple. This is true also if K is a field of characteristic p, where p is a prime not dividing the order of G. However if K is a field of prime characteristic p that divides the order of G then Mashke’s theorem no longer holds and the building blocks of KG-modules can no longer be assumed to be simple. The study of such KG- modules is called modular representation theory and in this talk we will give a whistle stop tour of some of its key introductory results. 

Abstract: Introducing topology into algebra has been an essential aspect of mathematics in the last century. This however has, among others, the drawback that categories of topological algebraic structures are not abelian (eg. the lack of cokernels). A remedy to this has been recently introduced by Clausen and Scholze, in the form of condensed mathematics which enlarges the notion of topological spaces and yields categories of algebraic objects with excellent categorical properties. I intend to introduce the basics of the theory, notably the notion of solid abelian groups and the existence of completed tensor products. If time permits, I will discuss how this theory can lead to a new (categorical) framework for functional analysis. This will be an overview and I will avoid focusing on the most technical details! 

Abstract: The deep connection between special values of L-functions and arithmetic is one of the most fascinating branches of modern number theory. For most people the way in is via the analytic class number formula or the conjecture of Birch and Swinnerton-Dyer. In this talk, I will instead use classical Galois module theory as a motivation. This will lead to (a special case of) the so-called equivariant Tamagawa Number Conjecture (eTNC). At the very end I will mention joint work with Martin Hofer that proves new cases of this conjecture. 

Abstract: Elliptic curves are a central object of study in number theory. It should be no surprise that there are many discoveries to be made when one looks at the ring of endomorphisms of an elliptic curve. For most curves, this is simply Z. But for some special curves, it is strictly larger. These are the curves with 'complex multiplication'. In this talk, we will study these curves a little bit, with an eye to a beautiful result about in the class field theory of imaginary quadratic fields. 

Abstract: We will explain the link between the Deligne-Mumford semistable reduction theorem for curves and Grothendieck’s semistable reduction theorem for abelian varieties. These results are basically equivalent and there are many known approaches. First we will sketch a explicit proof in characteristic zero of the Deligne-Mumford theorem and outline some key ideas of the Artin-Winters proof for the general case. Time permitting we will discuss applications to Néron-Ogg-Shafarevich-like inertial criteria. The talk will be an introduction to the subject and no prior knowledge other than basic algebraic geometry will be needed. 

Abstract: In this talk I want to present the well-behaved properties of reductive groups and notions that come along with them, such as maximal Tori, Borel subgroups and Parabolic subgroups. At first the definition of a reductive group over some field might seem not that intuitive and rather group theoretic. But a deeper study of this field has shown that reductive groups have a good representation theory, can be classified fairly well via root data, and also yield geometric structures via their corresponding flag varieties. I will try to touch on all of these subjects via examples such as Tori, GL_n and Sp_4.  

Abstract: We are interested in the question of whether, given a modular form, we can assign to it a family of modular forms which varies p-adic analytically over weights. In the 1980’s, Haruzo Hida gave a partial answer to this question though his theory of ordinary parts. His work has since grown and found numerous applications, notably in Wiles’ proof of the Iwasawa main conjecture for totally real fields in 1990. We will give a short introduction to his work, starting by defining the space of \Lambda-adic forms and giving an example of p-adic interpolation in the case of Eisenstein series, then defining the ordinary projector, and ending by stating some of the main results in Hida Theory. It is intended to be very accessible with no prerequisites.

Abstract: In 1976 K. Ribet discovered a new connection between modular forms and the p-part of the class group of the pth cyclotomic field. This lead to an improved form of Kummer's criterion for irregular primes and eventually was used by Mazur-Wiles in 1984 to prove the Iwasawa Main Conjecture. The protagonists of Ribet's 1976 article are modular forms, Galois representations, Bernoulli numbers and a large part of modern number theory revolves around them. The aim of the talk is to briefly introduce them and explain Ribet's proof (using a few strong results as black boxes). Hopefully it will leave you wanting more and motivate a few of the subjects in the following year.