London Junior Number Theory Seminar

Abstract: Studying cohomology of a variety with an action of a finite group is a classical and well-researched topic. However, most of the results consider only the tame ramification case or concern the image of cohomology in the K-theory. During this talk I will focus on the case of a curve over a field of characteristic p with an action of a finite p-group.

My previous research suggests that the de Rham cohomology decomposes as a sum of certain 'local' and 'global' part. This is true under certain mild assumptions. During the talk I mention also related open problems, concerning crystalline cohomology and Harbater-Katz-Gabber covers.

Abstract: Given a smooth variety over a non-archimedean valued field K, one can construct a p-adic analytification as a space of geometric valuations following Berkovich. The result is a nice locally compact Hausdorff topological space that is an inductive limit of combinatorial shadows of the variety. These combinatorial shadows, also called skeleta or tropicalisations, can be realised as dual intersection complexes of degenerations. So these non-archimedean analytifications contain all information related to the reductions of the variety.

Now given a wild morphism of smooth K-varieties, I will explain how to compare certain canonical skeleta in terms of piecewise linear functions on the analytifications related to logarithmic differentials. Time depending I will explain an application to curves with potentially good ordinary reduction.

Abstract (PDF version): We present the formulation of a new Main Conjecture in the non-commutative Iwasawa theory of number fields. This conjecture predicts a relation between leading coefficients at 0 of Artin L-functions and the refined Euler characteristic of a classical complex of Iwasawa modules over the Galois group of a rank-one p-adic Lie extension L_∞/K. As opposed to existing conjectures, this is done without assuming L_∞ to be totally real and, crucially, under no requirement that Gal(L_∞/K) be abelian. If time permits, the dependence of the conjecture on its parameters and the connection to existing conjectures will be discussed.

Abstract: Gross and Zagier proved that the derivative of the Hasse-Weil L-function of an elliptic curve is non-zero precisely when its Heegner point has infinite order. A few years later, Kolyvagin showed that this Heegner point lies at the bottom of a tower of points satisfying certain Euler system relations, and used these to derive cohomology classes that bound the Selmer group and hence the Mordell-Weil rank. I will describe his main argument and how this proves the Birch and Swinnerton-Dyer conjecture for a significant proportion of elliptic curves.

Abstract: Modularity lifting theorems first appeared in the 90s with the work of Wiles and Taylor as an essential tool to prove that all (semistable) elliptic curves over the rationals are modular, and since then have been a key ingredient in extending these results. In this talk I will give an overview of the general techniques used, focusing on the classical case of GL2_Q, with a view towards explaining the Taylor-Wiles method in a non-technical way.

Abstract: It has long been known that there is an analogy between the theory of number fields, function fields and Riemann surfaces. This led Arakelov to develop an arithmetic intersection theory which parallels the intersection theory on a smooth projective surface. I will give an introduction to these ideas. If time permits (most likely it does not) I will discuss some of the relations between arithmetic intersections and heights.

Abstract: In the 1920s, Mordell and Weil asserted that the rational points on an elliptic curve form a finitely generated abelian group. The structure of this group is determined by the rank and torsion subgroup; we will discuss the first (and more mysterious) of these pieces of data. We will focus on determining whether the rank is odd or even, beginning with an overview of known results. We will then look at practical methods to compute this parity, which only involve the arithmetic of curves over local fields (the reals and p-adics). As an application, we will show that the famous Birch and Swinnerton—Dyer conjecture correctly predicts the parity of the rank in a new case. This is all joint work with Céline Maistret.

Abstract: Euler systems are a collection of compatible Galois cohomology classes attached to a Galois representation that play a crucial role in the study of special values of L-functions; for instance, they have been used to prove the Iwasawa main conjecture, cases of the Birch--Swinnerton-Dyer conjecture and have recently been used to prove cases of the Bloch--Kato conjecture. A fundamental technique in these recent advances is to show that Euler systems vary in p-adic families. In this talk, we will first give a general introduction to the theme of p-adic variation in number theory, then introduce the necessary background from the theory of Euler systems and finally discuss the idea and importance of p-adically varying Euler systems.

Abstract: Developed by Fontaine and Ilusie as a tool to understand p-adic Galois representations associated to varieties with bad reduction, Logarithmic geometry has proved to be a magical tool in other areas of Mathematics. One of the main applications is to the theory of moduli spaces (e.g. Shimura varieties) where log geometry gives us the 'correct' compactification. In this talk, I will give an example based introduction to the field building up to the result that the Deligne-Mumford compactification of genus g curves with n marked points has a natural description in log geometry.

Abstract: The Gross-Zagier formula relates the height of the Heegner points on the modular curve X_0(N) and the first derivative of certain L-series. In this talk I will give the precise statement of this formula, and introduce its application to BSD conjecture. I will also give a rough sketch of the proof.

Abstract: In this talk I shall talk about a converse theorem for a family of L functions of degree 2 with gamma factor coming from a holomorphic cuspform. These L functions coincide with either those coming from a newform or a product of L functions arising from Dirichlet characters.

Our converse theorem requires some weak analytic data on the Euler factors, naturally appearing in the Selberg class. We also suppose that the twisted L functions satisfy expected functional equations of the right conductor and the Gamma factors are unchanged under twists.

Abstract: Two elliptic curves over the rational numbers are isogenous if and only if they have the same number of points modulo every prime. I will explain a proof of this fact (a special case of the Tate conjecture) found by the Chudnovsky brothers, relying on a criterion for algebraic dependence of power series. Time permitting, I will survey further Diophantine applications of algebraicity theorems.

Abstract: Absolute Galois groups are incredibly interesting and studied by so many number theorists, but what can they actually tell us about the underlying field? Some things are easy, eg, if Gal_k is trivial, then k is separably closed. In certain cases, it turns out you can recover the field entirely from it's Galois group. I'll run through the proof of this for global fields, and then talk about the techniques that you need in order to do this for higher dimensional fields. I'll also hint at how this becomes harder for "étale fundamental groups of varieties", but no knowledge of this will be required.

Abstract: The Siegel-Faltings theorems state that curves with non-abelian fundamental groups have finitely many S-integral points; however, explicit computation of the set of such points is still a major generically-open problem.

In this talk, I'll show how local Tate duality provides a symplectic form on the cotangent bundle of certain moduli spaces associated to any variety. I'll then relate the S-integral points to the intersection of some nice (i.e. Lagrangian) subspaces of these cotangent bundles and state how these intersections can be explicitly described as the critical locus of a regular function. This is joint work with Minhyong Kim.

Abstract/Answer: It's part of a building! In the theory of reductive groups over non-archimedean fields, Bruhat and Tits' work on buildings is essential. Among other things, they examine certain subgroups of G(Q_p) called Iwahori (resp. parahoric) subgroups. These resemble Borel (resp. parabolic) subgroups of G, but also take the valuation into account. Naturally, they appear when studying Shimura varieties and their level structures. I will focus on the cases of SL_3 and GL_2 to make things more concrete.

Abstract: The Mordell-Weil rank of an elliptic curve E over a number field K remains mysterious and, in general, it can be hard to find out whether your curve has even one K-rational point of infinite order. The parity conjecture gives us a way of predicting the parity of its rank over K, using something called the global root number, which is calculated using relatively easily computable local factors at each place of K, known as local root numbers. We will look at some of the history of parity related questions, examples of parity phenomena, what the state of the art is and generalisations to abelian varieties and twists thereof.

Abstract: The p-adic Langlands correspondence for GL_2(Q_p) was a very significant step in the Langlands programme and there has since been strong intuition and growing evidence of a similar result holding for GL_n(K) where n > 2 and K is a finite extension of Q_p. Let π be a GL_n(K)-representation that corresponds via the local Langlands conjecture to WD(ρ) for some regular de Rham Galois representation ρ and let r be the globalisation of ρ. Breuil conjectured that for GL_2(K) with K unramified, the lattice in the r-part of the cohomology of an appropriate Shimura curve corresponding to π is determined by the local Galois representation. Breuil’s conjecture and generalisations to GL_n(K) is a local-global compatibility result that informs our understanding of the hypothetical p-adic Langlands correspondence for GL_n(K). In this talk we will give some background, state Breuil’s conjecture and generalisations and discuss the progress that has been made so far in proving them.

Abstract: For an abelian variety over a number field, the Birch–Swinnerton-Dyer conjecture predicts the order of vanishing of the L-function at 1, and the leading coefficient of the power series expansion. Cassels proved that if the prediction holds for a given variety, it also holds for isogenous varieties. I will discuss this result, and show that it has interesting consequences independent of BSD. In particular, I will define the p-infinity Selmer rank of an elliptic curve, and show that if we have complex multiplication it is even.

Abstract: Since their introduction by Birch and Manin (1972), modular symbols have played a crucial role in Number Theory: analytic, algebraic, and even computational. In this talk we will focus on a particularly interesting aspect of them, their distribution. Motivated by its potential against the ABC conjecture as well as its close relation with special values of L-functions, the topic has received a lot of attention in the recent years. Having that as a starting point, we will attempt to narrate the evolution of the subject, including milestones such as the discovery of their Gaussian Distribution (Petridis-Risager) and the Mazur-Rubin-Stein conjectures, and reaching as far as the frontlines of current research. This talk should be approachable to anyone with minimal number theoretic background and some curiosity on how the analytic side of the field works.

Abstract: Many questions in number theory can be related to the study of number fields. Any number field is described by a single irreducible polynomial. However, this description is inconvenient in many ways. For example, many different polynomials give rise to the same field and algorithms to compute invariants like Galois groups, Frobenius elements or class groups are not very efficient. These issues have led to many other constructions of number fields. I will summarise some of them and discuss their strengths. In particular I will give the example of Galois representations arising in the cohomology of locally symmetric spaces.

Abstract: In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers \sqrt{D} is the solution to Pell’s equation for D. It is well-known that, once an integer solution to Pell’s equation exists, we can use it to generate all other solutions (u_n,v_n)_n∈Z. Our object of interest is the polynomial version of Pell’s equation, where the integers are replaced by polynomials with complex coefficients. We investigate the factors of v_n(t). In particular, we show that over the complex polynomials, there are only finitely many values of n for which v_n(t) has a repeated root. Restricting our analysis to Q[t], we give an upper bound on the number of “new” factors of v_n(t) of degree at most N. Furthermore, we show that all “new” linear rational factors of v_n(t) can be found when n ≤ 3, and all “new” quadratic rational factors when n ≤ 6.

Abstract: Class field theory can be described as an attempt at classification of abelian extensions of a number field in terms of data intrinsic to the field itself. In the talk we will explain the statements of the fundamental theorems of this theory, both in the classical language of ideals, as well as in the modern language of idèles.

Abstract: I will introduce classical Euler systems and then discuss Coleman's Conjecture (now a theorem) which gives a precise description of the set of all such systems.

Abstract: We will discuss a formulation of the Langlands reciprocity for modular forms so that it will be more clear how it is the part of the more general conjectures involving cuspidal automorphic representations. If time permits, I will sketch how the map from the automorphic side to the Galois side is constructed when the weight is at least 2.

Abstract: The main purpose of this talk is to present how one can use representation-theoretic machinery to verify classical isogenies between Jacobians of curves, and apply it to recover known theorems related to the parity conjecture. At the heart of all of this lies the study of relations between permutation representations of finite groups. This presentation will heavily rely on examples, and we will see how to establish isogenies between Jacobians of curves using such relations (i.e. p-isogeny between 2 elliptic curves), and then use them to recover the parity of the rank of these Jacobians using local data.