Number Theory Seminar
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Thu, 24/04/2008 16:00 |
Ronald van Luijk (Warwick) |
Number Theory Seminar  |
L3 |
| It is a wide open question whether the set of rational points on a smooth quartic surface in projective three-space can be nonempty, yet finite. In this talk I will treat the case of diagonal quartics V, which are given by: a x^4 + b y^4 + c z^4 + d w^4 = 0 for some nonzero rational a,b,c,d. I will assume that the product abcd is a square and that V contains at least one rational point P. I will prove that if none of the coordinates of P is zero, and P is not contained in one of the 48 lines on V, then the set of rational points on V is dense. This is based on joint work with Adam Logan and David McKinnon. | |||
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Thu, 01/05/2008 16:00 |
Martin Taylor (Manchester) |
Number Theory Seminar |
L3 |
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Thu, 15/05/2008 16:00 |
Bill Hart (Warwick) |
Number Theory Seminar |
L3 |
| We recall that an elliptic curve is a curve of genus one with a rational point on it. Certain algorithms for determining the structure of the group of rational points on an elliptic curve produce a whole set of curves of genus one and then require that we determine which of these curves has a rational point. Unfortunately no algorithm which has been proved to terminate is known for doing this. Such an algorithm or proof would probably have profound implications for the study of elliptic curves and may shed light on the Birch and Swinnerton-Dyer conjecture. This talk will be about joint work with Samir Siksek (Warwick) on the development of a new algorithmic criterion for determining that a given curve of genus one has no rational points. Both the theory behind the criterion and recent attempts to make the criterion computationally practical, will be detailed. | |||
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Thu, 22/05/2008 16:00 |
Lillian Pierce (Princeton) |
Number Theory Seminar |
L3 |
| Recently there has been increasing interest in discrete analogues of classical operators in harmonic analysis. Often the difficulties one encounters in the discrete setting require completely new approaches; the most successful current approaches are motivated by ideas from classical analytic number theory. This talk will describe a menagerie of new results for discrete analogues of operators ranging from twisted singular Radon transforms to fractional integral operators both on R^n and on the Heisenberg group H^n. Although these are genuinely analytic results, key aspects of the methods come from number theory, and this talk will highlight the roles played by theta functions, Waring's problem, the Hypothesis K* of Hardy and Littlewood, and the circle method. | |||
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Thu, 29/05/2008 16:00 |
Antal Balog (Budapest) |
Number Theory Seminar |
L3 |
| Let E be an elliptic curve over the rationals. To get an asymptotic to the number of primes p <= x such that the reduction of E mod p has prime order seems to be a hard question. Not so statistically if we let E varies over a large class of elliptic curves. We describe such a statistical result with emphasis on the analytic aspects of the proof. This is a joint work with Chantal David and Alina Cojocaru. | |||
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Thu, 05/06/2008 16:00 |
Nils Bruin (Vancouver) |
Number Theory Seminar |
L3 |
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Thu, 12/06/2008 16:00 |
Bjorn Poonen (Berkeley) |
Logic Seminar Number Theory Seminar |
L3 |
| Refining Julia Robinson's 1949 work on the undecidability of the first order theory of Q, we prove that Z is definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points. | |||
