This two day meeting is to discuss new results in number theory. The organiser is Steven Galbraith. The workshop is free and registration is not required (but you get what you pay for).

CHANGE OF LOCATION: We will be in room 303.102 (first floor of building 303) at the University of Auckland. This room is also called MLT2. If we get kicked out then we will go to room 303-310 on the third floor.

Program

Titles and Abstracts

• Pabasara Athukorala (Auckland) "Improving a lattice-code-based cryptosystem by Li, Ling, Xing and Yeo"
  Li, Ling, Xing, and Yeo (LLXY) have proposed a code-based encryption scheme based on factoring in finite fields. The security of the scheme relies on both the decoding problem for p-ary codes, and the bounded distance decoding problem for the \ell_1-norm in a family of lattices. We show how to apply the May-Ozerov information set decoding algorithm to attack the scheme. We also give a Niederreiter version of the scheme that has smaller ciphertexts. Compared with Classic McEliece, we obtain ciphertexts of similar size and smaller public and secret keys.
  
  
• Florian Breuer (Newcastle, Australia) "Explicit bounds on the coefficients of modular polynomials"
  For every positive integer $N$, there is a modular polynomial $\Phi_N(X,Y) \in\mathbb{Z}[X,Y]$ which vanishes exactly at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of degree $N$. The coefficients of these polynomials are notoriously large; their asymptotic growth (with implicit constants) has been known since the 1980s. Since these polynomials have applications in cryptography it is important to obtain explicit bounds on their coefficients. In this talk, I report on joint work with Fabien Pazuki and Desir\’ee Gij\’on G\’omez proving asymptotically optimal upper and lower bounds on these coefficients.
  
  
• Nils Bruin (SFU, Canada) "Genus 4 Jacobians that are (2,2)-decomposable"
  The study of decomposable Jacobians of genus 2 curves has a long and rich history. Essentially, they can be understood in terms of elliptic curves with suitably compatible n-torsion, where the surface is obtained as the quotient of the product of the curve by the graph of the identification of the n-torsion subgroups of the elliptic curves.
  We can similarly produce decomposable abelian 4-folds appropriately gluing abelian surfaces together along their n-torsion, but even for n=2 a new phenomenon arises: the Jacobians form a positive codimension sublocus in the moduli space of abelian 4-folds, so we expect an additional relation on our abelian surfaces in order to end up with a Jacobian.
  We identify this relation in terms of an explicit model of the moduli space of abelian surfaces with full 2-level structure, the Castelnuovo-Richmond-Igusa quartic. We use this to describe and classify the genus 2 curves whose Jacobian is (2,2)-isogenous to a square of a Jacobian of a genus 2 curve and identify a component of hyperelliptic genus 4 curves with this property that is of unexpectedly high dimension.
  This is joint work with Avinash Kulkarni, described in the preprint [On (2,2)-decomposable genus 4 Jacobians, arXiv:2309.01959 (2023)]
  
  
• 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, UK) "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, Australia) "Primes in short intervals"
  This talk will be inspired by my main thesis project connected to primes in short intervals, namely, we will discuss the result by Baker and Harman (1996) from their paper “On the difference between consecutive primes”, which provides the second-best result for the length of a short interval containing at least one prime number. The paper seems very interesting because it provides a combination of various ideas and techniques such as sieves, zero-density estimates, combinatorial arguments and optimisation problems which arise when one looks for the ‘good’ regions (i.e. the regions where sieves or zero-density estimates can be used). We will introduce the necessary theory on our way and provide a simplification and generalisation of the ideas introduced in the paper. This will show a connection between the zero-density estimates and primes in short intervals more explicitly. A lot of interesting questions arise on our way, which will be discussed during the talk. In particular, it is of interest to compare the methods from the paper with the current best result on primes in short intervals due to Baker, Harman, and Pintz (2001).
  
  
• Tim Trudgian (UNSW Canberra at ADFA, Australia) "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.