**Christophe Ritzenthaler**

My research mainly deals with explicit geometric and arithmetic properties of curves and moduli spaces sometimes with a view to crytographic or coding theory applications.

I'm local coordinator of the ANR PEACE (Parameter spaces for Efficient Arithmetic and Curve security Evaluation).

**Publications **

Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects, collaboration with R. Lercier, J. Algebra 372 (2012), 595-636. See also the programming section.

Genus bounds for curves with fixed frobenius eigenvalues, collaboration with N. Elkies and E. Howe, to appear in the Proceedings of the American Mathematical Society.

Complete addition laws on abelian varieties, collaboration with C. Arene and D. Kohel, LMS Journal of Computation and Mathematics 15 (2012) 308-316.

Fast computation of isomorphisms of hyperelliptic curves and explicit descent, collaboration with R. Lercier and J. Sijsling, to appear in the Proceedings of ANTS 2012. See also the programming section.

Optimal curves of genus 1, 2 and 3, Publications Mathematiques de Besancon, 99-117, 2011.

A geometric approach of Serre’s obstruction for genus 3 curve, collaboration with A. Beauville, Math. Ann. 350: 793-799, 2011.

Faster
computation of Tate pairing, collaboration with C. Arene, T. Lange
and M. Naehring, Journal of Number Theory, 131 :842–857, 2011. For a video summary of the paper.

On rationality of the intersection points of a line with a plane
quartic,
collaboration with R. Oyono. M. Anwar (ed.) et al., Arithmetic of
finite fields. Third international workshop, WAIFI 2010, Istanbul,
Turkey, June 27–30, 2010 Proceedings. Berlin : Springer. Lecture Notes
in Computer Science 6087, 224-237 (2010), 2010. See also the programming section.

Explicit
computations of Serre’s obstruction for genus 3 curves and application
to optimal curves, LMS J. Comput. Math., 13 :192–207, 2010.

А.
И. Зыкин, Ж. Лашо, К. Ритценталер. "Якобианы и абелевы многообразия
размерности 3: формула Клейна и вопрос Серра", 2010, том 431, 3, с.
313–315. Russian version of G. Lachaud, C. Ritzenthaler, A. Zykin.
"Jacobians among abelian threefolds: a formula of Klein and a question
of Serre", Doklady Akademii Nauk, 2010, Vol. 431, No. 3, pp. 313–315.

Jacobians
among Abelian threefolds: a formula of Klein and a question of Serre,
collaboration with G. Lachaud and A. Zykin, Math. Res. Lett., 17(2), 2010.

On
the existence of dimension zero divisors in algebraic function fields
defined over F_q, collaboration with S. Ballet et R. Rolland, Acta Arithmetica, 143(4) :377–392, 2010.

Genus
3 curves with many involutions and application to maximal curves in
characteristic 2, collaboration with E. Nart, In Proceedings of AGCT-12, volume 521, pages 71–85. Contemporary Mathematics, 2010.

Jacobians
in isogeny classes of abelian surfaces over finite fields,
collaboration with E. W. Howe and E. Nart, Annales de l'institut
Fourier, **59**, 239-289, (2009).

Distortion
maps
for genus two curve, collaboration with Steven D. Galbraith,
Jordi Pujolas and Benjamin
Smith, J. Math. Cryptology, 3,
1-18, (2009).

Jacobians
in isogeny classes of supersingular abelian threefolds in
characteristic 2, collaboration with E. Nart, Finite Fields
and their applications, 14, 676-702, (2008).

On some questions of Serre on abelian threefolds, collaboration with G. Lachaud,
Algebraic Geometry and its applications, Proceedings od the First SAGA
conference, Papeete, France 2007, 1-28, (2008). The MAGMA program to
check the computations.

Fast
addition on
Jacobians of
genus 3 curves, collaboration with S. Flon et R. Oyono,
Algebraic Geometry and its applications, Proceedings od the First SAGA
conference, Papeete, France 2007, 84-115, (2008).

Principally
polarizable isogeny classes of abelian surfaces over finite fields,
collaboration with E. W. Howe, D. Maisner et E. Nart, Mathematical
Research Letters, **15, **121-127, (2008).

The
Weierstrass subgroup of a curve has maximal rank,
collaboration with Martine Girard, David R. Kohel, Bull. of London
Math. Soc., 38,
925-931, (2006).

The
p-adic
CM method
for genus 2, collaboration with P. Gaudry, T. Houtmann,
D. Kohel, A. Weng, Asiacrypt 2006, volume **4284**
of Lecture Notes in Comput. Sci., pages 114–129. Springer, Berlin,
2006. Examples which have been computed can be found here.

An explicit
formula for the arithmetic geometric mean in genus 3,
collaboration with D. Lehavi, Experimental Math., **16**,
421-440, (2007).

On
the ring of
invariants of
ordinary quartic curves in characteristic 2, collaboration
with J. Müller, J. of Algebra, 303, 530-542,
(2006). Here is the MAGMA
program to compute the invariants of conics under the
specific action of PSL_3(F_2) in characteristic 2.

Non
hyperelliptic
curves
of genus three over finite fields of characteristic two,
collaboration with E. Nart, J. of Number Theory, 116, 443-473, (2006).

Automorphism
group of C : y^3+x^4+1=0 in characteristic p, JP J. Algebra,
Number
Theory and Appl., **4**, 621-623, (2004).

Point
counting
on genus 3 non hyperelliptic curves, Algorithmic Number
Theory 6th
International Symposium, ANTS VI, University of Vermont 13-18 June
2004,
Proceedings. See also the programming section.

Automorphismes
des courbes modulaires X(N) en caracteristique p, manuscripta
math. **109**,
49-62 (2002).

**Preprints **

Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group, collaboration avec Reynald Lercier et Jeroen Sijsling. See also the programming section.

An explicit expression of the Luroth invariant, collaboration avec Romain Basson, Reynald Lercier et Jeroen Sijsling. See also the programming section.

Existence
d'une
courbe de genre 5 sur F_3 avec 13 points rationnels.

** Other **

HDR (defended the 2nd of December 2009 at
Universite Aix-Marseille).

PhD thesis (defended the 25th of Juin 2003 at Paris
VII).

Chasles'
bad
relations: historical exercise during the
wokshop History of Mathematics, Leiden (2004).

La
suite (n SIN n) est-elle dense dans R ? Hubris, pub. de
l'ENS de Cachan, (2001).

memoire
de licence (3rd year thesis) : theorie des noeuds (node
theory).

memoire
de DEA (master thesis) : Action du groupe de Mathieu M_11 sur
la courbe modulaire X(11) en caracteristique 3 (after an article of A.
Adler).