Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-27T15:58:46.627Z Has data issue: false hasContentIssue false

Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields

Published online by Cambridge University Press:  01 August 2014

Reynald Lercier
Affiliation:
DGA MI, La Roche Marguerite, 35174 Bruz , France Institut de recherche mathématique, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France email reynald.lercier@m4x.org
Christophe Ritzenthaler
Affiliation:
Institut de recherche mathématique, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France email christophe.ritzenthaler@univ-rennes1.fr
Florent Rovetta
Affiliation:
Institut de Mathématiques de Luminy, UMR 6206 du CNRS, Luminy, Case 907, 13288 Marseille, France email florent.rovetta@univ-amu.fr
Jeroen Sijsling
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom email sijsling@gmail.com

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

Type
Research Article
Copyright
© The Author(s) 2014 

References

Abramovich, D. and Oort, F., ‘Alterations and resolution of singularities’, Resolution of singularities (Obergurgl, 1997), Progress in Mathematics 181 (Birkhäuser, Basel, 2000) 39108.CrossRefGoogle Scholar
Artebani, M. and Quispe, S., ‘Fields of moduli and fields of definition of odd signature curves’, Arch. Math. 99 (2012) no. 4, 333344.CrossRefGoogle Scholar
Bars, F., ‘Automorphism groups of genus 3 curves’. Notes del seminari Corbes de Gèneres 3, 2006.Google Scholar
Bergström, J., PhD Thesis, Kungl. Tekniska Högskolan, Stockholm, 2001.Google Scholar
Bergström, J., ‘Cohomology of moduli spaces of curves of genus three via point counts’, J. reine angew. Math. 622 (2008) 155187.Google Scholar
Cardona, G., ‘On the number of curves of genus 2 over a finite field’, Finite Fields Appl. 9 (2003) no. 4, 505526.CrossRefGoogle Scholar
Dixmier, J., ‘On the projective invariants of quartic plane curves’, Adv. Math. 64 (1987) 279304.Google Scholar
Dolgachev, I. V., Classical algebraic geometry: a modern view (Cambridge University Press, Cambridge, 2012).CrossRefGoogle Scholar
Freeman, D. M. and Satoh, T., ‘Constructing pairing-friendly hyperelliptic curves using Weil restriction’, J. Number Theory 131 (2011) no. 5, 959983.CrossRefGoogle Scholar
Girard, M. and Kohel, D. R., ‘Classification of genus 3 curves in special strata of the moduli space’, Algorithmic number theory. 7th International Symposium, ANTS-VII, Berlin, Germany, July 23–28, 2006, Lecture Notes in Computer Science 4076 (ed. Hess, Florian et al. ; Springer, Berlin, 2006) 346360.Google Scholar
Glasby, S. P. and Howlett, R. B., ‘Writing representations over minimal fields’, Comm. Algebra 25 (1997) no. 6, 17031711.CrossRefGoogle Scholar
Gorchinskiy, S. and Viviani, F., ‘Picard group of moduli of hyperelliptic curves’, Math. Z. 258 (2008) no. 2, 319331.CrossRefGoogle Scholar
Gross, B. H. and Harris, J., ‘On some geometric constructions related to theta characteristics’, Contributions to automorphic forms, geometry, and number theory (Johns Hopkins University Press, Baltimore, MD, 2004) 279311.Google Scholar
Guillevic, A. and Vergnaud, D., ‘Genus 2 hyperelliptic curve families with explicit jacobian order evaluation and pairing-friendly constructions’, Pairing-based cryptography — Pairing 2012, Lecture Notes in Computer Science 7708 (Springer, Berlin, 2012) 234253.CrossRefGoogle Scholar
Harris, J. and Mumford, D., ‘On the Kodaira dimension of the moduli space of curves’, Invent. Math. 67 (1982) no. 1, 2388; with an appendix by William Fulton.CrossRefGoogle Scholar
Henn, P.-G., ‘Die Automorphismengruppen der algebraischen Funktionenkorper vom Geschlecht 3’, PhD Thesis, Heidelberg, 1976.Google Scholar
Homma, M., ‘Automorphisms of prime order of curves’, Manuscripta Math. 33 (1980) no. 1, 99109.CrossRefGoogle Scholar
Howe, E. W., Lauter, K. E. and Top, J., ‘Pointless curves of genus three and four’, Algebra, Geometry, and Coding Theory (AGCT 2003), Séminaires et Congrès 11 (eds Aubry, Y. and Lachaud, G.; Société Mathématique de France, Paris, 2005).Google Scholar
Huggins, B., ‘Fields of moduli and fields of definition of curves’, PhD Thesis, University of California, Berkeley, California, 2005, arXiv:math/0610247.Google Scholar
Katsylo, P., ‘Rationality of the moduli variety of curves of genus 3’, Comment. Math. Helv. 71 (1996) no. 4, 507524.CrossRefGoogle Scholar
Katz, N. M. and Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications 45 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Lercier, R. and Ritzenthaler, C., ‘Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects’, J. Algebra 372 (2012) 595636.CrossRefGoogle Scholar
Lercier, R., Ritzenthaler, C. and Sijsling, J., ‘Fast computation of isomorphisms of hyperelliptic curves and explicit descent’, Proceedings of the Tenth Algorithmic Number Theory Symposium (eds Howe, E. W. and Kedlaya, K. S.; Mathematical Sciences Publishers, 2012) 463486.Google Scholar
Lønsted, K., ‘The structure of some finite quotients and moduli for curves’, Comm. Algebra 8 (1980) no. 14, 13351370.Google Scholar
Magaard, K., Shaska, T., Shpectorov, S. and Völklein, H., ‘The locus of curves with prescribed automorphism group’, Communications in arithmetic fundamental groups, Sūrikaisekikenkyūsho Kōkyūroku 1267 (RIMS, Kyoto, 2002) 112141.Google Scholar
Meagher, S. and Top, J., ‘Twists of genus three curves over finite fields’, Finite Fields Appl. 16 (2010) no. 5, 347368.Google Scholar
Newstead, P. E., Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 51 (Tata Institute of Fundamental Research, Bombay, 1978).Google Scholar
Ohno, T., ‘The graded ring of invariants of ternary quartics I’, unpublished manuscript, 2005.Google Scholar
Ritzenthaler, C., ‘Point counting on genus 3 non hyperelliptic curves’, Algorithmic number theory, Lecture Notes in Computer Science 3076 (Springer, Berlin, 2004) 379394.CrossRefGoogle Scholar
Ritzenthaler, C., ‘Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves’, LMS J. Comput. Math. 13 (2010) 192207.CrossRefGoogle Scholar
Rökaeus, K., ‘Computer search for curves with many points among abelian covers of genus 2 curves’, Arithmetic, geometry, cryptography and coding theory, Contemporary Mathematics 574 (American Mathematical Society, Providence, RI, 2012) 145150.CrossRefGoogle Scholar
Satoh, T., ‘Generating genus two hyperelliptic curves over large characteristic finite fields’, Advances in cryptology: EUROCRYPT 2009, Lecture Notes in Computer Science 5479 (Springer, Berlin, 2009).Google Scholar
Sekiguchi, T., ‘Wild ramification of moduli spaces for curves or for abelian varieties’, Compos. Math. 54 (1985) 33372.Google Scholar
Serre, J.-P., ‘Corps locaux’, Deuxième éditionPublications de l’Université de Nancago, No. VIII (Hermann, Paris, 1968).Google Scholar
Shioda, T., ‘Plane quartics and Mordell–Weil lattices of type E 7’, Comment. Math. Univ. St. Pauli 42 (1993) no. 1, 6179.Google Scholar
Silverman, J. H., ‘The arithmetic of elliptic curves’, 2nd edn,Graduate Texts in Mathematics 106 (Springer, Dordrecht, 2009).Google Scholar
Varshavsky, Y., ‘On the characterization of complex Shimura varieties’, Selecta Math. (N.S.) 8 (2002) no. 2, 283314.CrossRefGoogle Scholar
Vermeulen, A., ‘Weierstrass points of weight two on curves of genus three’, PhD Thesis, University of Amsterdam, Amsterdam, 1983.Google Scholar
Weber, H., Theory of abelian functions of genus 3 (Theorie der Abel’schen Functionen vom Geschlecht 3), 1876.Google Scholar
Weil, A., ‘The field of definition of a variety’, Amer. J. Math. 78 (1956) 509524.CrossRefGoogle Scholar