This one day workshop is to discuss new results in number theory. The organiser is Brendan Creutz. The workshop is free and registration is not required. All talks will be in Jack Erskine 235 at the University of Canterbury.

Program

Titles and Abstracts

• Daniel Delbourgo "Selmer groups for GL(2) x GL(2)"
  For an elliptic curve E/Q and a two-dimensional Artin representation ρ, the order of vanishing of the L-function L(E,ρ,s) at the point s=1 should equal the corank of the Selmer group associated to E twisted by ρ. This is the BSD conjecture (with ρ-twists). A related conjecture (IMC) predicts that the Selmer group for E twisted by ρ over the cyclotomic extension of Q should be related to the special values L(E,ρ⊗χ,1) as χ ranges over Dirichlet characters of p-power conductor. In this talk, we'll discuss why this second prediction IMC depends only on the residual Galois representations associated to E and ρ.
  
  
• Steven Galbraith "Climbing and descending tall volcanos"
  We revisit the question of relating the elliptic curve discrete logarithm problem (ECDLP) between ordinary elliptic curves over finite fields with the same number of points. This problem has been considered in 1999 by Galbraith and in 2005 by Jao, Miller, and Venkatesan. We apply recent results from isogeny cryptography and cryptanalysis, especially the Kani construction, to this problem. We improve the worst case bound in Galbraith's 1999 paper from $\tilde{O}( q^{1.5} )$ to (heuristically) $\tilde{O}( q^{0.4} )$ operations. We stress that this paper is motivated by pre-quantum elliptic curve cryptography using ordinary elliptic curves.
  
  
• Victor Lu "16-Descent on Elliptic Curves (In progress)"
  The method of descent for Diophantine equations involves deriving auxiliary equations which may contain information on the solutions of the original equation. In modern number theory descent is used to find potential rational points of Abelian varieties. For elliptic curves over number fields K, algorithms for explicit p^n-descent for p = 2 and n = 1,2,3 has been developed. We aim to extend it to n = 4 case, which would allow us to find better bounds for the group of rational points E(K) when the Tate-Shafarevich group of E/K has nontrivial 2-primary part. In this talk we describe the general theory behind descent, and some of the work we've done so far for 16-descent on elliptic curves over number fields. Please note that this research is still ongoing.
  
  
• Jesse Pajwani "Galois invariants of obstruction sets"
  Let $k$ be a global field and let $X$ be a variety over $k$. We are often interested in how the set of rational points $X(k)$ sits inside the set of adelic points $X(\mathbb{A}_k)$, and we often develop obstruction sets $X(\mathbb{A}_k)^{obs}$ such that $X(k) \subseteq X\mathbb{A}_k)^{obs} \subseteq X(\mathbb{A}_k)$. Examples of these obstruction sets include the topological closure of the $k$ points, the Brauer-Manin obstruction, or the finite étale descent obstruction.
  The sets $X(k)$ and $X(\mathbb{A}_k)$ have an additional structure coming from Galois theory. If $L/k$ is a finite Galois extension, then $X(k) = X_L(L)^{{Gal}(L/k)}$, and a similar statement holds for $X(\mathbb{A}_k)$. It is a reasonable question to ask whether these obstruction sets also have this property: i.e., is it true that $X(\mathbb{A}_k)^{obs} = (X_L(\mathbb{A}_L)^{obs})^{{Gal}(L/k)}$?
  In this talk, I'll give a survey of joint work with Creutz and Voloch that shows that for many natural obstructions, the above principle holds when X is a subvariety of an abelian variety. I'll also show how we can construct varieties such that the above principle fails for these natural obstructions. Finally, I'll discuss how the above work allows us to reduce the problem of finding rational points on isotrivial curves over global function fields to the case where our isotrivial curve is constant.
  
  
• Felipe Voloch "Mordell's conjecture in positive characteristic and descent obstructions"
  I will give a brief historical overview of the Mordell conjecture over function fields of positive characteristic (mainly). Then, I'll discuss its relationship with descent obstructions for rational points and present a recent result obtained with B. Creutz, which completes my earlier work with B. Poonen, showing that finite descent obstructions fully characterize the rational points on non-isotrivial curves over function fields.