NZ Number Theory Workshop 2019

We will host the NZ Number Theory Workshop 2019 at the University of Canterbury on August 26th 2019.
The organisers are Brendan Creutz and Felipe Voloch.
Talks will be at the School of Mathematics and Statistics, Jack Erskine Bldg., Room 446. The workshop is open to everyone and no registration is necessary.

Programme

  9:30-10:20   Geertrui van de Voorde (UC)
  10:30-11:00   Coffee Break (Room 447)
  11:00-11:50   Lukas Zobernig (UofA)
  12:00-1:45   Lunch Break (no catering provided)
  1:45-2:35   Igor Shparlinski (UNSW)
  2:45-3:35   Jeremy Booher (UC)
  3:45-4:10   Break (Room 447)
  4:10-5:00   Problem session/general discussion

We have informal plans to go to the UC Staff Club for drinks at 5pm and go for dinner (place TBD) afterwards.

Titles and abstracts

Geertrui van de Voorde (UC): On products of elements with prescribed trace

A question from finite geometry brought us to the following problem.
Is it possible to write every element of a (finite) field as the product of two elements with prescribed trace?
More generally, let L be a (not necessarily finite) field with subfield K and let TrL/K denote the trace map from L to K. Consider a, b ∈ K. For which elements α ∈ L can we find x, y ∈ L such that
x ยท y = α
TrL/K(x) = a
TrL/K(y) = b.
This problem is related to various other subjects, such as PN-functions and polynomials with prescribed coefficients. In this talk, we will see how these problems are interrelated and how some techniques from number theory allowed us to answer this problem in particular cases, e.g. for finite fields GF(qn) where n ≥ 5. (joint work with J. Sheekey and J.F. Voloch )

Lukas Zobernig (UofA): Higher Genus Isogeny Graphs and Cryptography

In this talk, we will revisit what is known about the isogeny graph structure of higher genus abelian varieties.
We are interested in the connection between isogeny graphs of different genera, such as products of elliptic curves appearing in genus-2 isogeny graphs for example. We are further interested in abelian varieties over fields of positive characteristic. For these we may define the "p-rank" and "a-number". The p-rank and a-number lead to the notion of supersingular and superspecial abelian varieties which sit in certain connected components of the isogeny graph.
Finally, we will motivate our interests by exploring possible cryptographic applications of genus-2 isogeny graphs.

Igor Shparlinski (UNSW): Effective Hilbert's Nullstellensatz and Finite Fields

We give an overview of recent applications of effective versions of Hilbert's Nullstellensatz to various problems in the theory of finite fields. In particular, we present some results about the size of the set generated by s-fold products of some rational fractions in a finite field. This result has some algorithmic applications.
We also show that almost all points on algebraic varieties over finite fields avoid Cartesian products of small order groups. This result is a step towards Poonen's conjecture.
We finish with an outline of some open problems.

Jeremy Booher (UC): a-numbers of Curves in Artin-Schreier Covers

Let f : Y -> X be a branched Z/pZ-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic p>0. We investigate the relationship between the a-numbers of Y and X and the ramification of the map f. This is analogous to the relationship between the genus (respectively p-rank) of Y and X given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the a-number of Y is not determined by the a-number of X and the ramification of the cover, so we instead give bounds on the a-number of Y. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator. This is joint work with Bryden Cais.

Participants