The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like.
This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying the group structure, arithmetic structure of the number of points (primality, smoothness, etc.) and certain divisibility conditions.
These questions are related to such celebrated problems as Lang-Trotter and Sato-Tate conjectures. More recently the interest to these questions was re-fueled by the needs of pairing based cryptography.
In a series of talks we will describe the state of art in some of these directions, demonstrate the richness of underlying mathematics and pose some open questions.
CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université[-]

We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like.
This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying the group structure, arithmetic structure of the number of points (primality, smoothness, etc.) and certain divisibility conditions.
These questions are related to such celebrated problems as Lang-Trotter and Sato-Tate conjectures. More recently the interest to these questions was re-fueled by the needs of pairing based cryptography.
In a series of talks we will describe the state of art in some of these directions, demonstrate the richness of underlying mathematics and pose some open questions.
CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université[-]

We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like.
This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying the group structure, arithmetic structure of the number of points (primality, smoothness, etc.) and certain divisibility conditions.
These questions are related to such celebrated problems as Lang-Trotter and Sato-Tate conjectures. More recently the interest to these questions was re-fueled by the needs of pairing based cryptography.
In a series of talks we will describe the state of art in some of these directions, demonstrate the richness of underlying mathematics and pose some open questions.
CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université[-]

The modular curve $Y^1(N)$ parametrises pairs $(E,P)$, where $E$ is an elliptic curve and $P$ is a point of order $N$ on $E$, up to isomorphism. A unit on the affine curve $Y^1(N)$ is a holomorphic function that is nowhere zero and I will mention some applications of the group of units in the talk.
The main result is a way of generating generators (sic) of this group using a recurrence relation. The generators are essentially the defining equations of $Y^1(N)$ for $n < (N + 3)/2$. This result proves a conjecture of Maarten Derickx and Mark van Hoeij.[-]

The absolute Galois group of the rational numbers acts on the various flavours (profinite, prounipotent, pro-$\ell$) of the fundamental group of a smooth projective curve over the rationals. The image of the corresponding homomorphism normalizes the image of the profinite mapping class group in the automorphism group of the geometric fundamental group of the curve. The image of the Galois action modulo these “geometric automorphisms” is independent of the curve. A basic problem is to determine this image. This talk is a report on a joint project with Francis Brown whose goal is to understand the image mod geometric automorphisms in the prounipotent case. Standard arguments reduce the problem to one in genus 1, where one can approach the problem by studying the periods of iterated integrals of modular forms and their relation to multiple zeta values.[-]

Après avoir expliqué la notion de Z-invariance pour les modèles de mécanique statistique, nous introduisons une famille à un paramètre (dépendant du module elliptique) de Laplaciens massiques Z-invariants définis sur les graphes isoradiaux. Nous démontrons une formule explicite pour son inverse, la fonction de Green massique, qui a la propriété remarquable de ne dépendre que de la géométrie locale du graphe. Nous expliquerons les conséquences de ce résultat pour le modèle des forêts couvrantes, en particulier la preuve d'une transition de phase d'ordre 2 avec le modèle des arbre couvrants critiques sur les graphes isoradiaux, introduit par Kenyon. Finalement, nous considérons la courbe spectrale de ce Laplacien massique et montrons qu'il s'agit d'une courbe de Harnack de genre 1.
Il s'agit d'un travail en collaboration avec Cédric Boutillier et Kilian Raschel.[-]