Getaltheorie in het Vlakke land

Arithmétique en plat pays

APP 2012

Lundi 21 mai 2012

Journée printanière




Lieu : Département de mathématiques



  • 11h-12h Olivier Ramaré (Lille)

  • On the quantitative equivalence of some results on prime numbers

    There are many explicit asymptotic evaluations concerning of the number π(x) of primes up to x but much less concerning the summatory function of the Moebius function. Both problems are known to be equivalent, but a strong quantitative form of this equivalence is missing. We will present some recent work in this direction, as well as some consequences regarding

    Σn≤ x μ(n), Σn≤ x μ(n)/n,   Σn≤ x,(n,r)=1 μ(n)/n,   Σn≤ x,(n,r)=1 μ(n)Log(x/n)/n,   Σn≤ x Λ(n)/n.

    All these results will improve on earlier ones. In the last mentioned case, we will even get an extremely sharp estimate.


  • 14h-15h Daniel Loughran (Paris)

  • Rational points on certain intersections of two quadrics

    In this talk we consider the problem of counting the number of rational points of bounded height on certain intersections of two quadrics in five variables. These are del Pezzo surfaces of degree four, and we focus on the case where the surface has a conic bundle structure.

  • 15h-16h Claudia Degroote (Gand)
  • Recursively enumerable sets of polynomials over certain algebraic extensions of Q are diophantine

    Hilbert's tenth problem asked if there exists an algorithm that, given a polynomial over the integers in any number of variables, decides whether this polynomial has an integer solution. In 1970, Matiyasevich proved, following earlier work of Davis, Putnam and Robinson, that recursively enumerable subsets are diophantine for the integers. From this DPRM-theorem follows the negative answer to Hilbert's tenth problem, namely that diophantine equations over the integers are undecidable. Undecidability of diophantine equations has been proved for various other rings and fields. On the other hand, much less is known about generalizations of the stronger DPRM-theorem. In this talk, I will first give an overview of some known results. Then I will turn my attention to recursive, algebraic extensions L of the rationals, with some extra conditions that I will explain about. To finish, I will present a new result, namely that subsets of the polynomial ring L[X] that are recursively enumerable for every recursive presentation are diophantine. This is joint work with Jeroen Demeyer (University of Ghent).

  • 16h30-17h30 Johannes Nicaise (Leuven)
  • Neron component groups of wildly ramified abelian varieties

    I will present some results on the behaviour of Neron component groups of abelian varieties under ramified base change, with special emphasis on wildly ramified abelian varieties. The main idea is that Neron models behave well under base change if the extension is "sufficiently orthogonal" to the minimal extension where the abelian variety acquires semi-abelian reduction, but it is a considerable challenge to define this orthogonality property in a suitable way. I will explain what it means for Jacobians and abelian varieties with potential multiplicative reduction. The hardest case is that of potential good reduction; I will discuss some open problems and possible approaches in that setting. This is a joint project with Lars Halvard Halle (Oslo).


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