Past Seminars- Number Theory

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  • Zev Klagsbrun
    Thu May 31, 2018
    3:00 pm
    We determine the average size of the Φ-Selmer group in any quadratic twist family of abelian varieties having an isogeny Φ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely...
  • Vadim Ponomarenko
    Thu May 24, 2018
    3:00 pm
    Since Fermat characterized (without proof) those integers represented by the quadratic form x^2+y^2, number theorists have been extending these results.  Recently a paper appeared in Journal of Number Theory answering the question for x^2 ± xy ± y^2.  It turns out that this was not news (although JNT refuses to...
  • Daqing Wan
    Thu May 17, 2018
    3:00 pm
    Given a global function field K of characteristic p>0, the fundamental arithmetic invariants include the genus, the class number, the p-rank and more generally the slope sequence of the zeta function of K. In this expository lecture, we explore possible stability of these invariants in a p-adic Lie tower of K. Strong stability is expected when...
  • Weiyan Chen
    Tue Apr 17, 2018
    3:00 pm
    It is a classical topic dating back to Maclaurin (1698–1746) to study certain special points on smooth cubic plane curves, such as the 9 inflection points (Maclaurin and Hesse), the 27 sextatic points (Cayley), and the 72 points "of type 9" (Gattazzo). Motivated by these algebro-geometric constructions, we ask the following...
  • Bryden Cais
    Thu Apr 12, 2018
    3:00 pm
    Let Y --> X be a branched G-covering of curves over a field k.  The genus of X and the genus of Y are related by the famous Hurwitz genus formula.  When k is perfect of characteristic p and G is a p-group, one also has the Deuring-Shafarevich formula which relates the p-rank of X to that of Y.  In this talk, we will discuss...
  • Lê Thái Hoàng
    Thu Mar 15, 2018
    3:00 pm
    The Möbius randomness principle states that the Möbius function μ does not correlate with simple or low complexity sequences F(n), that is, we have non-trivial bounds for sums ∑ μ(n) F(n). By analogy between the integers and the ring F_q[t] of polynomials over a finite field F_q, we study this principle in the latter...
  • Danny Nguyen
    Thu Mar 1, 2018
    3:00 pm
    Short generating functions were first introduced by Barvinok to enumerate integer points in polyhedra. Adding in Boolean operations and projection, they form a whole complexity hierarchy with interesting structure. We study them in the computational complexity point of view. Assuming standard complexity assumption, we show that these functions...