[ David Gruenewald ]

[ Research | Contact | Teaching ]

Research

Interests

I am a postdoctoral researcher (on the job market). In 2012 I completed a postdoc working with John Boxall in the Laboratoire de Mathématiques Nicolas Oresme at the Université de Caen, supported by the ANR's PACE project (Pairings and Advances in Cryptology for E-cash). Before that, I spent 3 months in Department of Mathematics at Radboud Universiteit Nijmegen, supported by the DIAMANT cluster working with Wieb Bosma on complex continued fractions. Before that I was a postdoc at eRISCS at the Université d'Aix-Marseille (in Marseille). Before that, I was a doctoral student in the Number Theory group of the School of Mathematics and Statistics at the University of Sydney, where my supervisor was David R. Kohel.

I am interested in computing with modular forms and their associated moduli spaces. My attention is currently focused on genus 2, where there are practical applications to hyperelliptic curve cryptography.

I was awarded my PhD in December 2009. In my thesis entitled "Explicit Algorithms for Humbert Surfaces", I find explicit practical models for moduli spaces of Abelian surfaces, in particular Humbert surfaces and Shimura curves. The equations can be found here.

In July 2008 I went to Microsoft Research for the northern summer, working as a research intern under the guidance of Kristin Lauter and Reinier Bröker. We implemented an improved version of the CRT algorithm in Magma which makes use of (3,3)-isogeny relations I had previously computed.

I enjoy all computational aspects of number theory. At the beginning of 2006 I did some computations for Alf van der Poorten on width 6 Somos sequences arising from continued fraction expansions of genus 2 curves. I primarily use Magma for my computations, but have also worked with Sage and Mathematica.

Publications

Abstract: We discuss heuristic asymptotic formulae for the number of isogeny classes of pairing-friendly abelian varieties of fixed dimension g ≥ 2 over prime finite fields. In each formula, the embedding degree k ≥ 2 is fixed and the rho-value is bounded above by a fixed real ρ0 > 1. The first formula involves families of ordinary abelian varieties whose endomorphism ring contains an order in a fixed CM-field K of degree g and generalizes previous work of the first author when g=1. It suggests that, when ρ0 < g, there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when ρ0 > g. The second formula involves families whose endomorphism ring contains an order in a fixed totally real field K0+of degree g. It suggests that, when ρ0 > 2g/(g+2) (and in particular when ρ0 > 1 if g = 2), there are infinitely many isogeny classes of g-dimensional abelian varieties over prime fields whose endomorphism ring contains an order of K0+. We also discuss the impact that polynomial families of pairing-friendly abelian varieties has on our heuristics, and review the known cases where they are expected to provide more isogeny classes than predicted by our heuristic formulae.
Abstract: Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing for every even degree d algebraic numbers of degree d that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the generalization of the nearest integer continued fraction expansion for complex numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent.
Abstract: For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CM field K, the Igusa invariants j1(A), j2(A), j3(A) generate an unramified abelian extension of the reflex field of K. In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on j1(A), j2(A), j3(A). Our description can be expressed by maps between various Siegel modular varieties, and we can explicitly compute the action for ideals of small norm. We use the Galois action to modify the CRT method for computing Igusa class polynomials, and our run time analysis shows that this yields a significant improvement. Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby implying that the ‘isogeny volcano’ algorithm to compute endomorphism rings of ordinary elliptic curves over finite fields does not have a straightforward generalization to computing endomorphism rings of abelian surfaces over finite fields.

Talks

(Invited talks are highlighted in red)

Conference and Workshop participation

 


Contact Details

Email:

davidg at maths . usyd . edu . au


Teaching

All MATHxxxx tutorials mentioned below correspond to lecture courses given at the University of Sydney.

All MATHyyy courses (after 2015) correspond to lecture courses given at ACU Strathfield.

= lectures and tutorials given at ACU.


Does this web page look familiar? I borrowed the template from Ben Smith (I’ll return it or the favour one day!)


Last modified on: 27th November 2016