# **Welcome to the 4th New Zealand Number Theory Workshop 2018**

:

## We will host the 4th NZ Number Theory Workshop at Victoria University of Wellington on 16 April 2018, organized by Byoung Du (B.D.) Kim (email).

## The workshop is open to everyone, and if you would like to participate, please email the organizer above.

## This one day program will be followed by dinner, which will be somewhere in CBD, Wellington.

## **Location**

## Victoria University of Wellington, Kelburn campus

## Cotton building 3rd floor, Room 350

If you are coming from Wellington CBD (central business district), take the following buses:
Bus 20 or 22 on Lambton Quay or The Terrace (Mairangi bound)

Bus 17 on The Terrace (Victoria University of Karori bound)

Bus 18 on Ghuznee (Karori bound)

Or, you can walk. But, it will be a long walk.

## **Program**

## 10AM-11AM: Tea (We will join the department tea)

## 11AM-11:05AM: Announcements

## 11:05AM-11:50AM: Shaun Cooper

## 12:00PM-12:45PM: Brendan Creutz

## 12:55PM-1:40PM: Daniel Delbourgo

## 1:40PM-3:00PM: Lunch (during this hour, the room is used by another group)

## 3:10PM-3:55PM: Tim Trudgian

## 4:05PM-4:50PM: Felipe Voloch

## Then, we will head down to Wellington CBD for dinner (5:30-7:30).

## **Titles/Abstracts**

Shaun Cooper
Title: Elliptic integrals in Ramanujan's lost notebook

Abstract:
I will survey the elliptic integrals in Ramanujan's lost notebook.
This includes deriving their connections with modular equations,
investigating other elliptic integrals not considered by Ramanujan,
and placing all of the results within a general theory.

Brendan Creutz
Title: There are no transcendental Brauer-Manin obstructions on abelian varieties

Abstract: Since work of Manin in the 1970's it has been known that Brauer group of a variety over a number field can be used to obtain information about the rational points on said variety. The Brauer group has a subgroup of "algebraic" elements which are typically much easier to understand, and transcendental elements which are much trickier. For general varieties there can be information encoded in these transcendental elements that is not available from the algebraic subgroup. I will explain why this is not the case for (torsors under) abelian varieties. This had been known for some time, conditionally on a widely believed conjecture (finiteness of Tate-Shafarevich groups of abelian varieties). My result is unconditional.

Daniel Delbourgo
Title: A non-abelian analogue of a theorem of Emerton,
Pollack and Weston

Tim Trudgian
Title: Real zeroes of Dirichlet L-functions

Abstract: If all is right with the world then the Generalised Riemann Hypothesis is true. Done: conference over, and book an early ticket to the Fields ceremony!

The hardest nuts to crack are Dirichlet L-functions L(s, χ), where χ is a real character modulo

*q*. Harder still is the case when χ is even. I outline some work (joint with Dave Platt) on the problem.

Felipe Voloch
Title: Distinguishing curves and factoring polynomials.

Abstract: The task of factoring polynomials modulo a prime has a
satisfactory, usually efficient, solution which depends on random
choices and can be slow if we are very unlucky. An idea of Kayal
and Poonen would remedy this situation if a family of algebraic
curves is found, whose members can be distinguished by their zeta
functions. We will explain this and set it in a general context of
distinguishing algebraic curves by L-functions. In particular,
we present a family of curves that allows us to be 99.999% confident
that we have a deterministic polynomial time algorithm for factoring
polynomials modulo a prime. Joint work with A. Sutherland.

## **Participants/Speakers**

## Qin Chao The University of Waikato

## Shaun Cooper Massey University

## Brendan Creutz The University of Canterbury

## Daniel Delbourgo The University of Waikato

## Hamish Gilmore The University of Waikato

## Byoung Du Kim Victoria University of Wellington

## Timothy Trudgian UNSW Canberra at the Australian Defence Force Academy

## Felipe Voloch The University of Canterbury