After reviewing old work with Teitelbaum, in which we constructed the character variety X of the additive group o_L in a finite extension L/Q_p and established the Fourier isomorphism for the distribution algebra of o_L, I will briefly report on more recent work with Berger and Xie, in which we establish the theory of (\varphi_L,\Gamma_L)-modules over X and relate it to Galois representations. Then I will discuss an ongoing project with Venjakob. Our goal is to use this theory over X for Iwasawa theory.

# Past Number Theory Seminar

I will give a gentle introduction to joint work in progress with George Boxer, Frank Calegari, and Vincent Pilloni, in which we prove that all abelian surfaces over totally real fields are potentially modular. We also prove that infinitely many abelian surfaces over Q are modular.

I will talk about some recent work with Chantal David and Matilde Lalin about the mean value of L-functions associated to cubic characters over F_q[t] when q=1 (mod 3). I will explain how to obtain an asymptotic formula with a (maybe a little surprising) main term, which relies on using results from the theory of metaplectic Eisenstein series about cancellation in averages of cubic Gauss sums over functions fields.

Serre's uniformity question concerns the possible ways the Galois group of Q can act on the p-torsion of an elliptic curve over Q. In this talk I will survey what is known about this question, and describe two recent results related to the Chabauty-Kim method. The first, which is joint work with Jennifer Balakrishnan, Steffen Muller, Jan Tuitman and Jan Vonk, completes the classification of elliptic curves over Q with split Cartan level structure. The second, which is work in progress with Samuel Le Fourn, Samir Siksek and Jan Vonk, concerns the applicability of the Chabauty-Kim method in determining the elliptic curves with non-split Cartan level structure.

Let L be a lattice in R^n and let Z in R^(m+n) a parameterized family of subsets Z_T of R^n. Starting from an old result of Davenport and using O-minimal structures, together with Martin Widmer, we proved for fairly general families Z an estimate for the number of points of L in Z_T, which is essentially best possible.

After introducing the problem and stating the result, we will present applications to counting algebraic integers of bounded height and to Manin’s Conjecture.

Mazur observed that in many cases where an elliptic curve E has a non-trivial element C in its Tate-Shafarevich group, one can find another elliptic curve E' such that ExE' admits an isogeny that kills C. For elements of order 2 and 3 one can prove that such an E' always exists. However, for order 4 this leads to a question about rational points on certain K3-surfaces. We show how to explicitly construct these surfaces and give some results on their rational points.

This is joint work with Tom Fisher.

The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.

In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, and also posed some questions: how many cusp forms of a given level are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Preston Wake, we give an answer to these questions in terms of cup products (and Massey products) in Galois cohomology. Time permitting, we may be able to indicate some partial generalisations of Mazur's results to square-free level.

We count monic quartic polynomials with prescribed Galois group, by box height. Among other things, we obtain the order of magnitude for quartics, and show that non-quartics are dominated by reducibles. Tools include the geometry of numbers, diophantine approximation, the invariant theory of binary forms, and the determinant method. Joint with Rainer Dietmann.

l-adic cohomology was built to provide an etale cohomology with coefficients in a field of characteristic 0. This, via the Grothendieck trace formula, gives a cohomological interpretation of L-functions - a fundamental tool in Deligne's theory of weights developed in Weil II. Instead of l-adic coefficients one can consider coefficients in ultra products of finite fields. I will state the fundamental theorem of Weil II for curves in this setting and explain briefly what are the difficulties to overcome to adjust Deligne's proof. I will then discuss how this ultra product variant of Weil II allows to extend to arbitrary coefficients previous results of Gabber and Hui, Tamagawa and myself for constant $\mathbb{Z}_\ell$-coefficients. For instance, it implies that, in an $E$-rational compatible system of smooth $\overline{\mathbb{Q}}_\ell$-sheaves all what is true for $\overline{\mathbb{Q}}_\ell$-coefficients (semi simplicity, irreducibility, invariant dimensions etc) is true for $\overline{\mathbb{F}}_\ell$-coefficients provided $\ell$ is large enough or that the $\overline{\mathbb{Z}}_\ell$-models are unique with torsion-free cohomology provided $\ell$ is large enough.