This two day meeting is to discuss new results in number theory. The organiser is Steven Galbraith.

The meeting will take place in room 303-310 at the University of Auckland. The workshop is free and registration is not required. Provisionally the meeting will run 10am to 5:30pm on Thursday and 9:30am to 4:15pm on Friday.

Program

Titles and Abstracts

• Nils Bruin (SFU) "tba"
  
  
  
• Raiza Corpuz (Waikato) "Equivalences of the Iwasawa main conjecture"
  Let $p$ be an odd prime, and suppose that $E_1$ and $E_2$ are two elliptic curves which are congruent modulo $p$. Fix an Artin representation $\tau: G_F \to \text{\rm GL}_2(\mathbb{C})$ over a totally real field $F$, induced from a Hecke character over a CM-extension $K/F$. We compute the variation of the $\mu$- and $\lambda$-invariants of the Iwasawa Main Conjecture, as one switches between $\tau$-twists of $E_1$ and $E_2$, thereby establishing an analogue of Greenberg and Vatsal's result. Moreover, we show that provided an Euler system exists, IMC$(E_1, \tau)$ is true if and only if IMC$(E_2, \tau)$ is true. This is joint work with Daniel Delbourgo from University of Waikato.
  
  
• Brendan Creutz (Canterbury) "Degrees of points on varieties over p-adic fields"
  Let X/k be a variety over a field k and let P be a point on X with coordinates in the algebraic closure of k. The degree of P is the degree of the field extension of k generated by the coordinates of P. For example, the point (0,i) on the conic x^2 + y^2 = -1 has degree 2. In this talk I will explain how the set of all degrees of points on a variety over a p-adic field can be determined by looking at the special fiber of a nice enough model of the variety over the ring of p-adic integers. In the case of curves over p-adic fields this gives an algorithm to compute the degree set, which yields some surprising possibilities. This is joint work with Bianca Viray.
  
  
• Daniel Delbourgo (Waikato/Auckland) "Random matrix theory and the χ-regularity of primes"
  Following the work of Ellenberg-Jain-Venkatesh over Q, for a Dirichlet character χ of any order we study the proportion and distribution of χ-regular primes p, which seems to depend on how the ideal (p) splits inside the field Q(χ). We make a similar prediction for the behaviour of the cyclotomic λ-invariant of Lp(s,χ) at such characters χ, which agrees remarkably closely with the numerical data. We employ p-adic random matrix models for both GL(n) and GSp(2n) as n approaches infinity, and get identical predictions. This is joint work with Prof. Heiko Knospe from Cologne.
  
  
• Victor Flynn (Oxford) "Arbitrarily Large p-torsion in Tate-Shafarevich Groups of Absolutely Simple Abelian Varieties over~$\Q$"
  Joint work with Ari Shnidman. We consider Question$_p$ : do there exist absolutely simple abelian varieties defined over~$\Q$ with arbitrarily large $p$-part of the Tate-Shafarevich group? Previously this was only known for $p = 2,3,5,7,13$. We shall show that the answer is yes for all primes~$p$ using an approach for demonstrating arbitrarily large Tate-Shafarevich groups, which only requires the existence of $\Q$-rational $p$-torsion of rank~$2$, and does not require an explicit model of any isogenous variety.
  
  
• Jesse Pajwani (Canterbury) "The valuative section conjecture and étale homotopy"
  The p-adic section conjecture is a long standing conjecture of Grothendieck about curves of high genus over p-adic fields, linking the p-adic points of a curve to sections of a short exact sequence of étale fundamental groups. A powerful way of interpreting the section conjecture is as a fixed point statement, and this interpretation makes the statement look like many other theorems in algebraic topology. For this talk, we'll first introduce the framing of the section conjecture as a fixed point statement, and then show this interpretation allows us to give an alternate proof of part of a result of Pop and Stix towards the section conjecture. This new proof generalises to other fields, and the new fields allow us to extend the original result to a larger class of varieties.
  
  
• Valeriia Starichkova (UNSW) "tba"
  
  
  
• Tim Trudgian (UNSW Canberra at ADFA) "Zero-density results: a third-best result towards the Riemann hypothesis"
  Ideally, prove the Riemann hypothesis. But, if this is too hard, then a good second best is: show that no zeroes of the Riemann zeta-function have real parts close to 1. If \textit{this} is too hard, then a good third best is: show that \textit{not too many} zeroes of the Riemann zeta-function have real parts close to 1.
  I shall give an overview of zero-density estimates, which is the third-best attempt mentioned above. This will include some recent work by colleagues at UNSW Canberra, one excellent idea I had that fails, and another idea for future research.