Welcome!
I work as a postdoctoral fellow in the Harvard Mathematics Department since September 2021, and I am a member of the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation. I focus on algorithms for abelian varieties and their moduli spaces, although I am also interested in the rest of the collaboration's work and explicit number theory and algebraic geometry in general. Other Harvard members of the collaboration include Alexander Betts, Alexander Cowan, and Professor Noam D. Elkies.
I obtained my Ph.D. in 2021 at the Institute of Mathematics in Bordeaux, under supervision of INRIA researchers Damien Robert and Aurel Page, in the home of the extraordinary Pari/GP software. Before that, I was a student at École normale supérieure (ENS) in Paris. Here is a more complete curriculum vitae.
This webpage was last updated on Nov. 1, 2021.
Publications
- Jean Kieffer. Degree and height estimates for modular equations on PEL Shimura varieties. Journal of the London Mathematical Society, to appear. arXiv.
- Jean Kieffer. Sign choices in the AGM for genus two theta constants. Publications Mathématiques de Besançon, to appear. arXiv.
- Luca De Feo, Jean Kieffer, Benjamin Smith. Towards practical key exchange from ordinary isogeny graphs. In T. Peyrin and S. Galbraith (editors), Advances in Cryptology - AsiaCrypt 2018, IACR. arXiv, SpringerLink.
Preprints and working papers
- Jean Kieffer. Evaluating modular equations in genus 2. arXiv.
- Jean Kieffer. Upper bounds on the heights of polynomials and rational fractions from their values. Submitted. arXiv.
- Jean Kieffer, Aurel Page, Damien Robert. Computing isogenies from modular equations in genus two. Submitted. arXiv.
Thesis manuscript
Jean Kieffer. Higher-dimensional modular equations, applications to isogeny computations and point counting. Ph.D. thesis, University of Bordeaux, 2021. Official TEL open archive.
Other documents
- Internship report on the implementation of the SEA algorithm for crypto-sized elliptic curves (in French): pdf.
- Master's thesis on a key exchange scheme based on CM isogeny graphs of elliptic curves over finite fields (in French): pdf. Disclaimer: this document is of lower quality than the published paper "Towards practical key exchange..." above. Slides from the defense.
Talks (selected)
- Harvard number theory seminar, Oct. 2021: Higher-dimensional modular equations and point counting on abelian surfaces.
- MIT number theory seminar, Oct. 2021: Higher-dimensional modular equations and point counting on abelian surfaces.
- Thesis defense, Bordeaux, July 2021: Slides (in French).
- AGCT, May 2021 (online): On the complexity of modular equations in genus 2. Slides.
- Geometry seminar, Bordeaux, Nov. 2020 (online): Algorithmic aspects of the moduli space of principally polarized abelian surfaces.
- C2 days, Nov. 2020 (online): Genus 2 point counting using isogenies. Slides.
- Computer algebra days (JNCF), Luminy, March 2020: Heights and interpolation of rational fractions. Slides.
- CARAMBA seminar, Nancy, Feb. 2020: Computing isogenies from modular equations in genus 2.
- Cryptography seminar, Rennes, Jan. 2020: Computing isogenies from modular equations in genus 2.
- Lambda PhD seminar, Bordeaux, Oct. 2019: Counting points on elliptic curves over finite fields.
- AGCT, Luminy, June 2019: Computing isogenies from modular equations in genus 2. Slides.
- AsiaCrypt, Brisbane, Dec. 2018: Towards practical key exchange from ordinary isogeny graphs. Slides.
- LFANT seminar, Bordeaux, May 2018: Implementing the SEA algorithm.
There will be an AMS Special Session on Explicit method for modularity during the Joint Meetings in Seattle, Jan. 7-8 2022.
Data and software
I have developed a C library called hdme (Higher-Dimensional Modular Equations) to evaluate Siegel and Hilbert modular polynomials for abelian surfaces via analytic methods. The code is publicly available on my GitHub page.
This software demonstrates that computing isogenies of degrees in the hundreds between p.p. abelian surfaces with real multiplication can be done within minutes. An important subroutine is to evaluate of genus 2 theta constants in uniform quasi-linear time on the Siegel fundamental domain: here is a little plot showing the cost of this computation (it's cheaper near the cusp). Here is another plot showing the evaluation of Hilbert modular equations at a given point as the level grows.
A follow-up project would be to access the library from Sage and use it to compute isogenies between p.p. abelian surfaces (with or without RM) there.
Useful links
Enea Milio's webpage contains examples of modular equations of Siegel and Hilbert type (discriminants 5, 8 and 12) in different coordinates. Compared to these, the elliptic modular polynomials from Andrew Sutherland's database do look tiny.
I regularly use Flint, Arb, Pari/GP, and the LMFDB database. You might want to compare performance between Flint and NTL. Here is other software that I find useful:
- PariTwine (links between Flint, Arb, Andreas Enge's other libraries and Pari),
- hcperiods (computation of period matrices of hyperelliptic curves),
- AVIsogenies (isogeny computations using algebraic theta constants).