# Abstracts Session 1

 On some conjectures on the Mordell-Weil and the Tate-Shafarevich of an Abelian variety Andrea Surroca Following Manin's approach, we propose conjectural upper bounds for 1) the Néron-Tate height of the elements of a system of generators of the Mordell-Weil group of an Abelian variety, as well as for 2) the cardinality of its Tate-Shafarevich group. We extend Manin's approach, initially for elliptic curves over the rationals (as re-visited by Lang and Golfeld-Szpiro), to an Abelian variety of arbitrary dimension, over an arbitrary number field. The dependence of the bounds is explicit in all the parameters, and the bounds given here are not conjectured but are implied by strong but nowadays classical conjectures: Birch and Swinnerton-Dyer conjecture, Hasse-Weil conjecture. On once hand, point 1) extends in higher dimension and for arbitrary number fields a conjecture of Lang. On the other hand, assuming also Szpiro's conjectures, with point 2) we extends a theorem of Goldfeld-Szpiro to Abelian varieties of arbitrary dimension $g$ over any number field $K$, and improved it in the case $g=1$ and $K$ the field of rational numbers.
 Reductions of elliptic curves Antonella Perucca (University of Luxembourg) Let $E$ be an elliptic curve defined over a number field $K$. Fix some prime number $\ell$. If $\alpha \in E(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the reduction $(\alpha \bmod \mathfrak p)$ is well-defined and has order coprime to $\ell$. This set admits a natural density. By refining the method of R. Jones and J. Rouse (2010), we can express the density as an $\ell$-adic integral. We also prove that the density is a rational number whose denominator (up to powers of $\ell$) is uniformly bounded in a very strong sense. Finally, we describe a strategy for computing the density which covers every possible case. This is joint work with Davide Lombardo (University of Pisa).
 On cyclic extensions Bayarmagnai Gombodorj (National University of Mongolia) Let $n\geq 3$ be an integer and $\zeta_n$ be a primitive $n$-th root of unity. We will present a description of cyclic extensions of degree n in terms of $k$-rational points of the tori when the base field $k$ contains $\zeta_n + \zeta_{n}^{-1}$.
 The Prime Geodesic Theorem in the 3-dimensional hyperbolic space Dimitrios Chatzakos (CEMPI, Lille and Université de Lille) The Prime Geodesic Theorem studies the asymptotic behaviour of lengths of primitive closed geodesics on hyperbolic manifolds. For $2$-dimensional manifolds this problem was first studied by Huber and Selberg. It turns out that the lengths of these geodesics obey an asymptotic distribution analogous to the Prime Number Theorem, and the error term has been extensively studied by use of the Selberg and the Kuznetsov trace formulas. In this talk, we discuss the Prime Geodesic Theorem on $3$-dimensional hyperbolic manifolds. For the Picard manifold, we improve on the classical pointwise bound of Sarnak, using the Kuznetsov formula combined with a recent large sieve inequality of Watt. Further, for a 3-manifold of finite area, we study the second moment of the error term using the Selberg trace formula. This is a joint work in progress with Giacomo Cherubini and Niko Laaksonen.
 Geometric primality tests using curves of genus 1 and 2 Eduardo Ruiz Duarte (Rijksuniversiteit Groningen) We revisit and generalize some geometric techniques behind deterministic primality testing for some integer sequences using curves of genus 1 over finite rings. Subsequently we develop a similar primality test using the Jacobian of a genus 2 curve.
 Unlikely Intersections in families of abelian varieties Fabrizio Barroero(Universität Basel) Two varieties whose dimensions do not sum up to at least the dimension of the ambient space should not intersect. This is the guiding philosophy that led several authors to propose conjectures about subvarieties of commutative algebraic groups and of Shimura varieties. After a brief historic introduction we will talk about results for curves in families of abelian varieties, mostly obtained in collaboration with Laura Capuano.
 Expansions of quadratic numbers in a $p$-adic continued fraction Laura Capuano (University of Oxford) It goes back to Lagrange that a real quadratic irrational always has a periodic continued fraction. Starting from decades ago, several authors generalised proposed different definitions of a $p$-adic continued fraction, and the definition depends on the chosen system of residues mod $p$. It turns out that the theory of $p$-adic continued fractions has many differences with respect to the real case; in particular, no analogue of Lagrange's theorem holds, and the problem of deciding whether the continued fraction is periodic or not seemed to be not known. In recent work with F. Veneziano and U. Zannier we investigated the expansion of quadratic irrationals, for the p-adic continued fractions introduced by Ruban, giving an effective criterion to establish the possible periodicity of the expansion. This criterion, somewhat surprisingly, depends on the real value of the p-adic continued fraction.
 Explicit Small Height Bound for $\mathbb{Q} (E_\text{tor})$ Linda Frey (Universität Basel) Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We will show that there exists an explicit constant $C$ which is only dependent on the conductor and the $j-$invariant of $E$ such that the absolute logarithmic Weil height of an $\alpha \in \mathbb{Q} (E_\text{tor})^*\setminus \mu_\infty$ is always greater than $C$ where $E_\text{tor}$ denotes all the torsion points of $E$ and $\mu_\infty$ are the roots of unity.
 Counting rational points on genus one curvesManh Hung Tran (Chalmers University) The density of integral solutions of Diophantine equations can also be viewed geometrically as the density of integral points on algebraic varieties which is one of the classical problems in Diophantine geometry. We are interested in the case of projective varieties defined by homogeneous polynomials and this gives rise to the study of rational points on such varieties. In this talk we focus on smooth genus one curves which are closely related to elliptic curves. We give uniform upper bounds for the number of rational points of bounded height on smooth genus one curves in two forms: plane cubic curves and complete intersections of two quadric surfaces. The main tools to study this problem are descent and determinant methods.
 Isomorphism classes of Abelian varieties over finite fields Stefano Marseglia (Stockholms Universitet) Deligne proved that the category of ordinary abelian varieties over a finite field is equivalent to the category of free finitely generated abelian groups endowed with an endomorphism satisfying certain easy-to-state axioms. Centeleghe and Stix extended this equivalence to all isogeny classes of abelian varieties over ${\mathbb F}_p$ without real Weil numbers. Using these descriptions, under some extra assumption on the isogeny class, we obtain that in order to compute the isomorphism classes of abelian varieties we need to calculate the isomorphism classes of (non necessarily invertible) fractional ideals of some orders in certain étale algebras over ${\mathbb Q}$. We present a concrete algorithm to perform these tasks and, for the ordinary case, to compute the polarizations and also the automorphisms of the polarized abelian variety.
 New instances of the Mumford-Tate conjecture and further applications Victoria Cantoral Farfan (International Center for Theoretical Physics, Trieste) In this talk we will present new cases of the Mumford-Tate conjecture for abelian varieties defined over number fields. Moreover we will discuss some further applications in the direction of the Sato-Tate conjecture.