# Conference "Arithmetic Geometry: Explicit Methods and Applications"

## December 7 - 11, 2015, Moscow, Russia

**Nurdagül Anbar (Kgs. Lyngby) : On the limits of a cubic tower**

Abstract in pdf

**Paloma Bengoechea (York) : Badly approximable numbers in Twisted Diophantine Approximation**

T. B. A.

**Wouter Castryck (Ghent) : Point counting on curves using a gonality preserving lift**

Tuitman recently developed an efficient algorithm for point counting on curves over finite fields of small characteristic p > 2 using p-adic cohomology. The algorithm potentially applies to a very large class of curves: for instance it specializes to Kedlaya's algorithm when applied to hyperelliptic (i.e. 2-gonal) curves. The remaining bottleneck is to find a "good" lift of the input curve to characteristic zero, which is easy for hyperelliptic curves but hard in general. In this talk we will show how such a lift can be found for 3-gonal and 4-gonal curves, and discuss some resulting point counting timings. This is joint (ongoing) work with Jan Tuitman.

**Alain Couvreur (Paris) : From geometry to codes and conversely**

I will start a survey on the main constructions of codes from algebraic varieties, in particular in dimension larger than or equal to 2. Then I'll present several open problems concerning either the combinatorial aspects of algebraic geometry codes (minimum distance and higher Hamming weights) or the algorithmic aspects (decoding problems).

**Julie Desjardins (Paris) : Density of rational points on elliptic surfaces and del Pezzo surfaces of degree 1**

A study of the variation of the root number of the fibers of an elliptic surface over**P**^{1}_{Q}can lead to interesting results such as the density of rational points on del Pezzo surfaces of degree 1. When the elliptic surface is isotrivial, it can happen that the root number of the fibers is constant. However, a result of Helfgott, conditional to many conjectures (parity conjecture, squarefree conjecture and Chowla conjecture), predicts the density of rational points on non-isotrivial elliptic surfaces. In this talk, we present a work in progress in this direction. For non-isotrivial elliptic surface satisfying some technical conditions, we show the density of rational points without assuming the squarefree conjecture.

**Tony Ezome (Franceville) : Computing functions on Jacobians and their quotients**

In this talk we show how to efficiently evaluate functions on Jacobians and their quotients. We deduce an algorithm to compute isogenies between Jacobians of genus two curves.

We will start from the genus one case (with elliptic curves).

**Elisa Lorenzo Garcia (Leiden) : On CM genus 3 curves**

We will first review the Gross-Zagier formula for singular moduli and some bounds for the denominators of the coefficients of class polynomials for genus 1 and 2 curves. Finally, we will discuss some recent progress on their generalizations to genus 3 curves. In particular, we will show some results on bad reduction of CM genus 3 curves and we will explain the main differences and difficulties of these generalizations. As examples, we will focus on two special families of genus 3 curves: hyperelliptic and Picard curves.

**Richard Griffon (Paris) : Analogues of the Brauer-Siegel theorem for elliptic curves over function fields**

The Brauer-Siegel theorem describes the asymptotic behaviour of the product of the regulator and the class number of a number field in terms of its discriminant. It can be seen as an measure of the "arithmetic complexity" of this number field. Now consider an elliptic curve E defined over a global field, assuming its Tate-Shafarevich group is finite, one can form the product of the order of this group and of the Néron-Tate regulator of E. Heuristically, this product measures the complexity of computing the Mordell-Weil group of E. This prompts the question of giving upper and lower bounds of this quantity in terms of simpler invariants of E, e.g. its height. In this talk, I will explain how such bounds can be estimated unconditionally in the setting of elliptic curves over function fields in positive characteristic. The two families of elliptic curves we study provide new examples where an analogue of the classical Brauer-Siegel theorem holds.

**Stefan Hellbusch (Oldenburg) : On the computation of the period matrix of a Riemann surface**

Let X be a compact and connected Riemann surface, defined by a polynomial f in**Q**[x,y]. I will give an overview of our work (joint with Christian Neurohr) on algorithms for Riemann surfaces, including the numerical computation of the period matrix of X and related topics such as the computation of the fundamental and monodromy groups. The focus of the talk will be choosing a computationally good basis of the fundamental group, which, as we will see, plays a big role in the complexity of the whole problem.

**Florian Hess (Oldenburg) : Algorithms for Curves and Jacobians**

The first part of the talk discusses general algorithms for algebraic curves, including completion and normalisation of affine curves, effective Riemann-Roch theory and some direct applications.

The second part focuses on asymptotically efficient arithmetic in Jacobians based on linear algebra over polynomial rings, using some specialised results of the first part.

A third part, if time permits, covers additional topics, possibly isomorphism testing and automorphism group computations for algebraic curves.

**Marc Hindry (Paris): The group of rational points of an abelian variety over a global field**

Let A be an abelian variety defined over a global field K (i.e. a number field of the function field of a curve C over a finite field). The Mordell-Weil theorem, combined with results of Lang-Néron, states that the group of rational points A(K) is finitely generated. We plan to survey qualitative and quantitative results and open questions around A(K), such as:

(1) structure and size of the torsion part, when K grows or when K is fixed but A varies. Computability of this finite group.

(2) How does the rank (of the infinite part of A(K)) varies? Is it bounded when, say, K and the dimension of A are fixed? More modestly, how does the parity of the rank varies in families? Is the rank computable?

(3) How large can the size of generators of (the infinite part of) the group of rational points be? Is a set of generators computable?

These questions lead naturally to the introduction of other objects : canonical heights, moduli spaces, Shafarevich-Tate group, L-functions, etc.

**Matthias Junge (Oldenburg) : Asymptotic fast arithmetic in the divisor class group of function fields with large genus**

Abstract in pdf

**Kamal Khuri-Makdisi (Beirut) : Algorithmic representation of a curve and its Jacobian**

I will discuss a way to represent an algebraic curve C of genus g, and divisor classes on C, in a way that allows efficient addition on in the divisor class group Pic^{0}(C). The algorithms are based on linear algebra between subspaces of certain Riemann-Roch spaces and a multiplication map that, once computed for C, allow one to add two divisor classes in the time it takes to do linear algebra on spaces of dimension O(g^{1+ε}); this is asymptotically O(g^{2.373}) with the best known techniques. The representation of C in terms of a multiplication map is very well suited for modular curves, and serves as a way to represent large genus curves compactly without explicit equations.

**Philippe Lebacque (Besançon) : The K(pi,1)-property and Tsfasman-Vladuts invariants for global fields**

T. B. A.

**Gilles Lachaud (Marseille) : The distribution of the trace in the compact group of type G**_{2}and applications

We study the distribution of the trace on the seven-dimensional representation of the compact semi-simple Lie group of type G_{2}. We recall its relevance to the equidistribution of the family of exponential sums

with parameter t in**F**_{p}, where (x/p) is the quadratic character, and p is any prime other than 2 and 7. We describe the fundamental simplex of G_{2}(the space of conjugacy classes) and its image by the fundamental map. With the help of Weyl's integration formula, we give an exact formula for the distribution of the trace in terms of special functions. This answers a question raised by J.-P. Serre and N. M. Katz.

**Julia Pieltant (Paris) : Non-special divisors of degree g-1 and tensor rank of multiplication in the extensions of****F**_{2}and**F**_{3}

In this talk we will be interested in the existence of zero-dimensional divisors in algebraic function fields, and especially of non-special divisors of degree g-1, which form the borderline case. We will explain how this question arise from the study of tensor rank of multiplication in finite fields, namely when one try to design good Chudnovsky-Chudnovsky-type multiplication algorithms. When a function field is defined over a finite field with at least 4 elements then the existence of non-special divisors of degree g-1 has already been established, thus we will focused on the cases were the ground field has 2 or 3 elements.

**Matthieu Rambaud (Paris) : How can we find good multiplication algorithms using interpolation ?**

The Chudnovsky and Chudnovsky method provides today's best known bounds for the bilinear complexity of the multiplication in finite fields. It is grounded on interpolation on algebraic curves. When a curve is fixed, one can predict that the interpolation algorithms from it, can reach at most a certain complexity (with exceptions). This fixed-curve complexity seems often reached in practice. We will illustrate an explicit method to achieve it.

**Hugues Randriambololona (Paris) : Geometric aspects of products and powers of codes**

We will present an overview of the theory of products and powers of linear codes, briefly mentionning some applications as a motivation, but quickly shifting to the underlying geometric aspects. Depending on time, the following topics will be touched:- automorphism group of powers
- dimension of the product of random codes
- analogues of the Kneser and Vosper theorems (Mirandola-Zémor)
- upper bounds on the joint parameters
- lower bounds on the joint parameters
- powers of hyperplane subcodes

**Sergey Rybakov (Moscow) : Zeta functions of minimal cubic surfaces over finite fields**

We construct minimal cubic surfaces with given zeta functions over many finite fields. This is a joint work with A. Trepalin.

**Bart de Smit (Leiden): Groups of points arising from elliptic curves**

From an elliptic curve E defined over a number field K, we obtain groups of points defined over various fields related to K. One can for instance think of K itself, its finite extensions, its archimedean and non-archimedean completions, and the residue fields at the primes of K. I will go into the problem of determining the "distribution" of such groups when K is fixed and E varies. There are many open questions, but also several proven results. I will certainly include my recent theorem (with my student Angelakis) on the average structure of the*adelic*point group of an elliptic curve. It is based on the study of the Galois representation attached to elliptic curves.

**Bart de Smit (Leiden) : Characterizing global fields with L-functions**

It is well known that two number fields can have the same zeta-function without being isomorphic. On the other hand every number field has an abelian character whose L-function is only occurs over that number field. For function fields the situation at present is much less clear.

**Pavel Solomatin (Leiden) : L-functions of genus two abelian coverings of elliptic curves over finite fields**

Initially motivated by the relations between Anabelian Geometry and Artin's L-functions of the associated Galois-representations, we introduce and study the list of zeta-functions of genus two abelian coverings of elliptic curves over finite fields. Our goal is to provide a complete description of such a list.