Christophe Ritzenthaler

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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).