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7jtn         Seu Vella Lleida

Jornadas dedicadas a la memoria del profesor Javier Cilleruelo

Conferenciantes plenarios


Abstracts

  • Local densities compute isogeny classes
  • Consider an ordinary isogeny class of elliptic curves over a finite, prime field. Inspired by a random matrix heuristic (which is so strong it’s false), Gekeler defines a local factor for each rational prime. Using the analytic class number formula, he shows that the associated infinite product computes the size of the isogeny class.
    I’ll explain a transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality. Moreover, the new perspective allows a natural generalization to higher-dimensional abelian varieties. ----- This is joint work with Julia Gordon and S. Ali Altug.
    Jeffrey Achter ---------------

  • On the p-part of the Birch-Swinnerton-Dyer formula for multiplicative primes
  • Let E/ℚ be a semistable elliptic curve of analytic rank one, and let p>3 be a prime for which the Galois group of the extension ℚ(E[p])/ℚ is isomorphic to GL2(Fp). In this note, following a slight modification of the methods of Jetchev–Skinner–Wan, we use Iwasawa theory to establish the p-part of the Birch–Swinnerton-Dyer formula for E. In particular, we extend the main result of loc.cit. to primes of multiplicative reduction.
    Francesc Castellà ---------------

  • Curves of low genus and arithmetic applications
  • In 2000, Paul Vojta solved the n-squares problem under Bombieri's conjecture, by explicitly finding all the curves of geometric genus 0 or 1 on certain surfaces of general type related to this problem. In this talk I will sketch a refined and generalized version of the geometric method implicit in Vojta's work. I will also discuss new arithmetic applications conditional to Bombieri's conjecture in the case of number fields, and unconditional for function fields.
    Natalia García-Fritz ---------------

  • Fields of definition of elliptic k-curves with CM and Sato-Tate groups of abelian surfaces
  • Let A be an abelian surface over a number field k that is isogenous over the algebraic closure of k to the square of an elliptic curve E. If E does not have CM, by results of Ribet and Elkies concerning fields of definition of elliptic k-curves, E is isogenous to a curve defined over a polyquadratic extension of k. We show that one can adapt Ribet's methods to study the field of definition of E up to isogeny also in the CM case, as long as k contains the field of CM. As an application of this analysis, we provide a number field over which abelian surfaces can be found realizing each of the 52 possible Sato--Tate groups of abelian surfaces. ----- This is joint work with Francesc Fité.
    Xevi Guitart ---------------

  • Estimates for the Artin conductor
  • Let K be a number field such that K/ℚ is galois and χ the caharacter of a linear representation of Gal(K/ℚ). The Artin conductor asociated to χ (denoted by fχ) is a positive integer involved in the functional equation of the Artin L-function attached to χ.
    In 2011, we improved the lower bounds to the conductor obtained by Odlyzko in 1976. Now, we will see that the growth of the Artin conductor is at most exponential in the degree of the character.
    Amalia Pizarro-Madariaga ---------------

  • Fully maximal and fully minimal abelian varieties and curves
  • We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists over its minimal field of definition. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic. For supersingular curves of genus 3 in characteristic 2, we use a parametrization of a moduli space of such curves by Viana and Rodriguez to determine the L-polynomial and the type of each. ----- This is joint work with Valentijn Karemaker.
    Rachel Pries ---------------

  • The Apéry set of a numerical semigroup
  • Numerical semigroups are easy to define: they are just additive semigroups, contained in the non-negative integers, such that they only miss a finite amount of such integers. However, the computational problems that arise in such structures are far from easy, specially from a computational viewpoint. A rather mysterious set that seems to have the key to solve many of these problems is the so-called Apéry set, an object whose origins lie in singularity theory.
    José María Tornero ---------------

  • The mathematical legacy of Javier Cilleruelo
  • We will dedicate this talk to celebrate the Mathematics of Javier Cilleruelo. He had a special intuition for additive problems with a combinatorial flavour and his taste for Mathematics had a big impact on many of us.
    He wrote more than 80 research articles with more than 44 coauthors from all around the globe and he made.e profound contributions in a wide variety of problems. We will present a selection of his results to give a glimpse of the impact of his work.
    Ana Zumalacárregui ---------------