Number Theory Workshop, Auckland, November 23, 2023
This two day meeting is to discuss new results in number theory.
The organiser is Steven Galbraith.
The meeting will take place in room 303310 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

Thursday November 2 
10:0011:00 
Nils Bruin 
11:3012:30 
Daniel Delbourgo 
2:003:00 
Victor Flynn 
3:154:15 
Brendan Creutz 
4:305:30 
Student talks: Raiza Corpuz and tba 

Friday November 3 
9:3010:30 
Jesse Pajwani 
10:4511:45 
Tim Trudgian 
11:5012:50 
Valeriia Starichkova 
2:003:00 
tba 
3:154:15 
tba 
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 CMextension $K/F$. 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 padic 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 padic field can be determined by looking at the special fiber of a nice enough model of the variety over the ring of padic integers. In the case of curves over padic 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 χregularity of primes"
Following the work of EllenbergJainVenkatesh over Q, for a
Dirichlet character χ of any order we study the proportion and
distribution of χregular 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 λinvariant of Lp(s,χ) at such
characters χ, which agrees remarkably closely with the numerical data.
We employ padic 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 ptorsion in TateShafarevich 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 TateShafarevich 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 TateShafarevich
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 padic section conjecture is a long standing conjecture of Grothendieck about curves of high genus over padic fields, linking the padic 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) "Zerodensity results: a thirdbest 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 zetafunction 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 zetafunction have real parts close to 1.
I shall give an overview of zerodensity estimates, which is the thirdbest 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.