Number Theory Workshop, Auckland, November 2-3, 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 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
|
Thursday November 2 |
10:00-11:00 |
Nils Bruin |
11:30-12:30 |
Daniel Delbourgo |
2:00-3:00 |
Victor Flynn |
3:15-4:15 |
Brendan Creutz |
4:30-5:30 |
Student talks: Raiza Corpuz and tba |
|
Friday November 3 |
9:30-10:30 |
Jesse Pajwani |
10:45-11:45 |
Tim Trudgian |
11:50-12:50 |
Valeriia Starichkova |
2:00-3:00 |
tba |
3:15-4:15 |
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 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.