Getaltheorie in het Vlakke land

Arithmétique en plat pays

APP 2015

Lundi 20 avril 2015

Journée printanière




Lieu : LPP, Salle de Réunion



  • 11h-12h Harald Helfgott (Paris)

  • The ternary Goldbach conjecture

    The ternary Goldbach conjecture (1742) asserts that every odd number greater than 5 can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant C satisfies the conjecture. In the years since then, there has been a succession of results reducing C, but only to levels much too high for a verification by computer up to C to be possible (C> 10^1300). (Work by Ramare and Tao solved the corresponding problems for six and five prime numbers instead of three.) My recent work proves the conjecture. We will go over the main ideas in the proof.

    12h-14h Restaurant Barrois, Salle 1

  • 14h-15h Maja Volkov (Mons)

  • Variétés abéliennes supersingulières à module de Tate non semisimple

    On montre l'existence de variétés abéliennes sur Qp ayant bonne réduction supersingulière et dont le module de Tate p-adique n'est pas semisimple. Ce résultat est une application de la caractérisation en termes de φ-modules filtrés, via la théorie de Hodge p-adique, des représentations p-adiques du groupe de Galois absolu de Qp provenant des schémas abéliens. On montrera comment obtenir des variétés ayant les propriétés souhaitées pour la plus petite dimension possible, à savoir des surfaces. Nos constructions se généralisent aisément en dimension supérieure.

  • 15h-16h Kohji Matsumoto (Nagoya)

  • Desingularization of multiple zeta-functions

    We introduce the method of desingularization of multiple zeta-functions of generalized Euler-Zagier type, under the motivation of finding suitable rigorous meaning of the values of multiple zeta-functions at non-positive integer points. The desingularized multiple zeta-function turns to be entire, and is written by a suitable finite linear combination of usual multiple zeta-functions. It is shown that specific combinations of Bernoulli numbers attain special values of desingularized zeta-function at non-positive integer points.
    (This is a joint work with H. Furusho, Y. Komori and H. Tsumura.)

  • 16h30-17h30 Korneel Debaene (Gand)
  • Sieving for Completely Splitting Primes

    The well-known Brun-Titchmarsh inequality gives an upper bound for the number of primes in an arithmetic progression, or in other words, primes with a certain splitting behaviour in a certain cyclotomic field. This upper bound is off by a factor 2 asymptotically, but its merit is that it is effective. When one turns ones attention to the completely splitting primes of some other families of fields, the asymptotic density is known by Cebotarev, but the question of proving an effective upper bound is certainly a non-trivial one. Let l be an odd prime, then we are concerned with primes p which are 1 mod l, for which a certain prime q is an l-th power mod p. We describe the proof of an effective Brun-Titchmarsh-style upper bound for this set of primes, whose main ingredients are a Reciprocity law, theorems on counting lattice points and the Selberg Sieve.

  • 19 h -

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    Nous voudrions que tous les participants s'inscrivent.

    Wij wensen dat alle deelnemers registreren.

    Contact: G. Bhowmik; R. Cluckers; J. Nicaise .






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