Christophe Ritzenthaler

Saiho-ji garden

Papers

Programs

Talks

Links

You can also find most of my documents directly on Arxiv. I simply add a link to my PhD thesis (2003) and to my Habilitation (2009), both in French.
  1. Lercier, R., & Ritzenthaler, C. (2024). Siegel modular forms of degree three and invariants of ternary quartics. Proceedings of the American Mathematical Society, 152(6), 2267–2282. DOI: 10.1090/proc/14940
  2. Ritzenthaler, C. (2024). Expanding our horizons: research with developing countries. European Mathematical Society Magazine, 131, 32–35. DOI: 10.4171/MAG/166
  3. Bergström, J., Howe, E. W., Lorenzo García, E., & Ritzenthaler, C. (2024). Lower bounds on the maximal number of rational points on curves over finite fields. Mathematical Proceedings of the Cambridge Philosophical Society, 176(1), 213–238. DOI: 10.1017/S0305004123000476
  4. García, E. L., Ritzenthaler, C., & Villegas, F. R. (2024). An arithmetic intersection for squares of elliptic curves with complex multiplication. Preprint, arXiv:2412.08738 [math.NT]. arXiv:2412.08738
  5. Lombardo, D., García, E. L., Ritzenthaler, C., & Sijsling, J. (2023). Decomposing Jacobians via Galois covers. Experimental Mathematics, 32(1), 218–240. DOI: 10.1080/10586458.2021.1926008
  6. Bergström, J., Howe, E. W., García, E. L., & Ritzenthaler, C. (2023). Refinements of Katz-Sarnak theory for the number of points on curves over finite fields. Preprint, arXiv:2303.17825 [math.NT]. arXiv:2303.17825
  7. Kirschmer, M., Narbonne, F., Ritzenthaler, C., & Robert, D. (2022). Spanning the isogeny class of a power of an elliptic curve. Mathematics of Computation, 91(333), 401–449. DOI: 10.1090/mcom/3672
  8. Lercier, R., García, E. L., & Ritzenthaler, C. (2021). Stable models of plane quartics with hyperelliptic reduction. In Arithmetic, geometry, cryptography and coding theory (pp. 223–237). DOI: 10.1090/conm/770/15437
  9. Ghorpade, S. R., Ritzenthaler, C., Rodier, F., & Tsfasman, M. A. (2021). Arithmetic, geometry, and coding theory: homage to Gilles Lachaud. In Arithmetic, geometry, cryptography and coding theory (pp. 131–150). DOI: 10.1090/conm/770/15433
  10. Lercier, R., Liu, Q., García, E. L., & Ritzenthaler, C. (2021). Reduction type of smooth plane quartics. Algebra & Number Theory, 15(6), 1429–1468. DOI: 10.2140/ant.2021.15.1429
  11. Lercier, R., Sijsling, J., & Ritzenthaler, C. (2021). Functionalities for genus 2 and 3 curves. Preprint, arXiv:2102.04372 [math.AG]. arXiv:2102.04372
  12. Bassa, A., & Ritzenthaler, C. (2020). Good recursive towers over prime fields exist. Mathematische Annalen, 378(1-2), 599–604. DOI: 10.1007/s00208-020-02039-9
  13. Lercier, R., Ritzenthaler, C., & Sijsling, J. (2020). Reconstructing plane quartics from their invariants. Discrete & Computational Geometry, 63(1), 73–113. DOI: 10.1007/s00454-018-0047-4
  14. Kirschmer, M., Narbonne, F., Ritzenthaler, C., & Robert, D. (2020). Spanning the isogeny class of a power of an ordinary elliptic curve over a finite field. Application to the number of rational points of curves of genus ≤ 4. Preprint, arXiv:2004.08315 [math.NT]. arXiv:2004.08315
  15. Gélin, A., Howe, E., & Ritzenthaler, C. (2019). Principally polarized squares of elliptic curves with field of moduli equal to ℚ. In ANTS XIII (pp. 257–274). DOI: 10.2140/obs.2019.2.257
  16. Lercier, R., Ritzenthaler, C., Rovetta, F., Sijsling, J., & Smith, B. (2019). Distributions of traces of Frobenius for smooth plane curves over finite fields. Experimental Mathematics, 28(1), 39–48. DOI: 10.1080/10586458.2017.1328321
  17. Ritzenthaler, C., & Romagny, M. (2018). On the Prym variety of genus 3 covers of genus 1 curves. Épijournal de Géométrie Algébrique, 2, 8. epiga.episciences.org/4383
  18. Kılıçer, P., Labrande, H., Lercier, R., Ritzenthaler, C., Sijsling, J., & Streng, M. (2018). Plane quartics over ℚ with complex multiplication. Acta Arithmetica, 185(2), 127–156. DOI: 10.4064/aa170227-16-3
  19. Nart, E., & Ritzenthaler, C. (2017). A new proof of a Thomae-like formula for non hyperelliptic genus 3 curves. In Arithmetic, geometry, cryptography and coding theory (pp. 137–155). DOI: 10.1090/conm/686/13781
  20. Lercier, R., Ritzenthaler, C., & Sijsling, J. (2016). Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group. Mathematics of Computation, 85(300), 2011–2045. DOI: 10.1090/mcom/3032
  21. Lercier, R., Ritzenthaler, C., Rovetta, F., & Sijsling, J. (2014). Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields. LMS Journal of Computation and Mathematics, 17A, 128–147. DOI: 10.1112/S146115701400031X
  22. Elkies, N. D., Howe, E. W., & Ritzenthaler, C. (2014). Genus bounds for curves with fixed Frobenius eigenvalues. Proceedings of the American Mathematical Society, 142(1), 71–84. DOI: 10.1090/S0002-9939-2013-11839-3
  23. Basson, R., Lercier, R., Ritzenthaler, C., & Sijsling, J. (2013). An explicit expression of the Lüroth invariant. In ISSAC 2013 (pp. 31–36). DOI: 10.1145/2465506.2465507
  24. Lercier, R., Ritzenthaler, C., & Sijsling, J. (2013). Fast computation of isomorphisms of hyperelliptic curves and explicit Galois descent. In ANTS X (pp. 463–486).
  25. Arene, C., Kohel, D., & Ritzenthaler, C. (2012). Complete addition laws on abelian varieties. LMS Journal of Computation and Mathematics, 15, 308–316. DOI: 10.1112/S1461157012001027
  26. Lercier, R., & Ritzenthaler, C. (2012). Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. Journal of Algebra, 372, 595–636. DOI: 10.1016/j.jalgebra.2012.07.054
  27. Ritzenthaler, C. (2011). Optimal curves of genus 1, 2 and 3. In Actes de la conférence "Théorie des nombres et applications" (pp. 99–117). DOI: 10.5802/pmb.a-137
  28. Beauville, A., & Ritzenthaler, C. (2011). Jacobians among abelian threefolds: a geometric approach. Mathematische Annalen, 350(4), 793–799. DOI: 10.1007/s00208-010-0583-6
  29. Arène, C., Lange, T., Naehrig, M., & Ritzenthaler, C. (2011). Faster computation of the Tate pairing. Journal of Number Theory, 131(5), 842–857. DOI: 10.1016/j.jnt.2010.05.013
  30. Ritzenthaler, C. (2010). Explicit computations of Serre's obstruction for genus-3 curves and application to optimal curves. LMS Journal of Computation and Mathematics, 13, 192–207. DOI: 10.1112/S1461157009000576
  31. Lachaud, G., Ritzenthaler, C., & Zykin, A. (2010). Jacobians among Abel threefolds: a formula of Klein and a question of Serre. Mathematical Research Letters, 17(2), 323–333. DOI: 10.4310/MRL.2010.v17.n2.a11
  32. Nart, E., & Ritzenthaler, C. (2010). Genus 3 curves with many involutions and application to maximal curves in characteristic 2. In Arithmetic, geometry, cryptography and coding theory 2009 (pp. 71–85).
  33. Zykin, A. I., Lachaud, G., & Ritzenthaler, C. (2010). Jacobians among abelian threefolds: a formula of Klein and a question of Serre. Doklady Mathematics, 81(2), 233–235. DOI: 10.1134/S1064562410020183
  34. Oyono, R., & Ritzenthaler, C. (2010). On rationality of the intersection points of a line with a plane quartic. In Arithmetic of finite fields (pp. 224–237). DOI: 10.1007/978-3-642-13797-6_16
  35. Ballet, S., Ritzenthaler, C., & Rolland, R. (2010). On the existence of dimension zero divisors in algebraic function fields defined over 𝔽_q. Acta Arithmetica, 143(4), 377–392. DOI: 10.4064/aa143-4-4
  36. Galbraith, S. D., Pujolàs, J., Ritzenthaler, C., & Smith, B. (2009). Distortion maps for supersingular genus two curves. Journal of Mathematical Cryptology, 3(1), 1–18. DOI: 10.1515/JMC.2009.001
  37. Howe, E. W., Nart, E., & Ritzenthaler, C. (2009). Jacobians in isogeny classes of abelian surfaces over finite fields. Annales de l'Institut Fourier, 59(1), 239–289. DOI: 10.5802/aif.2430
  38. Lachaud, G., & Ritzenthaler, C. (2008). On some questions of Serre on abelian threefolds. In Algebraic geometry and its applications (pp. 88–115). DOI: 10.1142/9789812793430_0005
  39. Flon, S., Oyono, R., & Ritzenthaler, C. (2008). Fast addition on non-hyperelliptic genus 3 curves. In Algebraic geometry and its applications (pp. 1–28). DOI: 10.1142/9789812793430_0001
  40. Nart, E., & Ritzenthaler, C. (2008). Jacobians in isogeny classes of supersingular abelian threefolds in characteristic 2. Finite Fields and their Applications, 14(3), 676–702. DOI: 10.1016/j.ffa.2007.09.006
  41. Howe, E. W., Maisner, D., Nart, E., & Ritzenthaler, C. (2008). Principally polarizable isogeny classes of Abelian surfaces over finite fields. Mathematical Research Letters, 15(1), 121–127. DOI: 10.4310/MRL.2008.v15.n1.a11
  42. Lehavi, D., & Ritzenthaler, C. (2007). An explicit formula for the arithmetic-geometric mean in genus 3. Experimental Mathematics, 16(4), 421–440. DOI: 10.1080/10586458.2007.10129011
  43. Gaudry, P., Houtmann, T., Kohel, D., Ritzenthaler, C., & Weng, A. (2006). The 2-adic CM method for genus 2 curves with application to cryptography. In Advances in cryptology -- ASIACRYPT 2006 (pp. 114–129). DOI: 10.1007/11935230_8
  44. Girard, M., Kohel, D. R., & Ritzenthaler, C. (2006). The Weierstrass subgroup of a curve has maximal rank. Bulletin of the London Mathematical Society, 38(6), 925–931. DOI: 10.1112/S0024609306019059
  45. Müller, J., & Ritzenthaler, C. (2006). On the ring of invariants of ordinary quartic curves in characteristic 2. Journal of Algebra, 303(2), 530–542. DOI: 10.1016/j.jalgebra.2005.02.034
  46. Nart, E., & Ritzenthaler, C. (2006). Non-hyperelliptic curves of genus three over finite fields of characteristic two. Journal of Number Theory, 116(2), 443–473. DOI: 10.1016/j.jnt.2005.05.014
  47. Ritzenthaler, C. (2004). Automorphism group of C:y^3+x^4+1=0 in characteristic p. JP Journal of Algebra, Number Theory and Applications, 4(3), 621–623.
  48. Ritzenthaler, C. (2004). Point counting on genus 3 non hyperelliptic curves. In Algorithmic number theory (pp. 379–394). DOI: 10.1007/b98210
  49. Ritzenthaler, C. (2003). Methode A.G.M. pour les courbes ordinaires de genre 3 non hyperelliptiques sur F_2^N. Preprint, arXiv:math/0303072 [math.NT]. arXiv:math/0303072
  50. Ritzenthaler, C. (2003). Existence d'une courbe de genre 5 sur F_3 avec 13 points rationnels. Preprint, arXiv:math/0302147 [math.NT]. arXiv:math/0302147
  51. Ritzenthaler, C. (2002). Automorphisms of modular curves X(n) in characterictic p. Manuscripta Mathematica, 109(1), 49–62. DOI: 10.1007/s002290200286
Some packages in Magma which has been developed. They will be maintained and will eventually integrate the Magma official release. Please send comments and bugs to this email address. At the end of this section, you will find also some small programs or outputs related to articles. In my most recent articles, these data are directly stored on Arxiv as ancillary files.
Invariants and reconstruction of genus 2 hyperelliptic curves

This is a work in collaboration with R. Lercier. It has been implemented in Magma (v.2.13).

Magma Source
Invariants and reconstruction of genus 3 hyperelliptic curves

This is a work in collaboration with Reynald Lercier. It has been implemented in Magma (v.2.17).

Magma Source Paper
Fast computation of isomorphisms of hyperelliptic curves

This is a work in collaboration with Reynald Lercier and Jeroen Sijsling. It is now implemented in Magma 2.25-7.

Source Paper
Reconstructing plane quartics from their invariants

This is a work in collaboration with Reynald Lercier and Jeroen Sijsling. It is now implemented in Magma 2.25-7.

Source Paper
Decomposing Jacobians via Galois covers

This is a work in collaboration with Elisa Lorenzo García, Davide Lombardo and Jeroen Sijsling.

Source Paper
Spanning the isogeny class of the power of an elliptic curve

This is a work in collaboration with Markus Kirschmer, Fabien Narbonne and Damien Robert.

Source Paper
Functionalities for genus 2 and genus 3 curves

This is a work in collaboration with Reynald Lercier and Jeroen Sijsling. It has been implemented in Magma 2.25-7 and completes the work done before on genus 2 and 3.

Hyperelliptic Quartics Paper
  1. The Magma file to check the computation of the sign in Lemma 3.6 of A new proof of Thomae-like formula for non hyperelliptic genus 3 curves.
  2. The databases for p=11 and 13 (and the program to exploit them in Magma) and the statistics on the distributions of the trace for 7< p <59 of the article Parametrizing the moduli space of curves and application to smooth plane quartics over finite fields are contained in the following zip file.
  3. an implementation of an AGM algorithm for non-hyperelliptic curves of genus 3 has been worked out in collaboration with M. Fouquet and P. Gaudry. It has been implemented in Magma (v.2.09). This program does not work with the new version of MAGMA. You'll have to make changes in the definitions of the p-adic spaces.
  4. Here are several programs relative to the paper An explicit expression of Luroth invariant with Romain Basson, Reynald Lercier and Jeroen Sijsling. This Magma program provides a way to compute an expression of Luroth invariant and this is the final result. This Magma program generates random Luroth quartics of type L1 (with the notation of Ottaviani, Sernesi `On singular Luroth quartics') and this database contains 10,000 of them with rational coefficients. Finally this program checks that neither L1 nor L2 defines a new invariant. Note that they require the use of Echidna package available on this page.
  5. Magma programs to check the computations of Fast computation of isomorphisms of hyperelliptic curves and explicit descent with Reynald Lercier and Jeroen Sijsling: for Sections 1.5, 2.3.1 and 2.3.2: the programs, associated data and the package for genus 3 hyperelliptic curves; for Section 2.4; the general descent program, the resulting equation (8Mo) and the program to descend a particular example.
  6. Two programs related to On rationality of the intersection points of a line with a plane quartic with Roger Oyono: Computation of the correspondence curve in characteristic 2, Flexes in characteristic 3 (Maple 11).
  7. A program (collaboration with Philippe Trebuchet) in MAGMA which tests if a plane curve over a finite field is absolutely irreducible. It is based on E. Kaltofen article : Fast parallel absolute irreductibility testing, J. Symb. Comp. 1, (1985), 57-67.
  8. A program in MAGMA which computes the p-rank of the modular curves X(N).
  9. A basic program in MAGMA for computing the number of points on an elliptic curve with AGM as suggested by Jean-François Mestre.
Some slides of talks.
  1. Modular forms in small dimensions: geometry and arithmetic, Online Talk at Oberwolfach conference on moduli spaces and modular forms (Feb. 2021).
  2. Spanning the isogeny class of E^g, Online Talk at QNTAG seminar (May 2020).
  3. Modular forms in small dimensions, Talk in AGCT (June 2019).
  4. Smooth plane quartics with CM over Q, Talk in Barcelona (Feb. 2017).
  5. Reconstruction of smooth plane quartics from its invariants, Talk in Besançon (Apr. 2016).
  6. Distribution of traces of genus 3 curves over finite fields, Talk in Linz (Nov. 2013).
  7. Invariants and hyperelliptic curves: algorithmic aspects and open questions, Talk in Trento (Sep. 2012).
  8. Invariants and hyperelliptic curves: geometry, arithmetic and algorithmic aspects, Talk in Luminy (Oct. 2011).
  9. Algorithmic number theory and the allied theory of theta functions, Talk in Edinburgh (Oct. 2010).
  10. Rationality of intersection points of a line and a quartic, Talk in Istanbul (June 2010).
  11. Optimal curves of genus 1, 2 and 3, Talk in Leuven (May 2010).
  12. Completeness, Talk in Montréal (Apr. 2010).
  13. From the curve to its Jacobian and back, Talk in Montréal (Apr. 2010).
  14. Existence of dimension zero divisor, Talk at Antalya (Sept. 2009).
  15. Serre's obstruction for genus 3 curves, Talk in Guadeloupe (May 2009).
  16. Some old and new problems on genus 3 curve, Talk at ESF workshop (Mar. 2009).
  17. Quelles courbes elliptiques pour la cryptographie ? Talk at C2 (Mar. 2008).
  18. Addition law on a plane quartic, Talk in Tahiti (May 2007).
  19. AGM method for non hyperelliptic curves of genus 3, Talk in Bordeaux (Nov. 2003).
Website, working groups,etc.
  1. A database of curves with many points. It is a website in collaboration with E. Howe, K. Lauter and G. van der Geer and maintained by G. Oomens.
  2. The working group on effective theory of invariants.