---
title: "Bibliography"
canonical: "https://docs.magma-maths.org/ArithmeticGeometry/HyperellipticCurves/bibliography.html"
source: "https://docs.magma-maths.org/ArithmeticGeometry/HyperellipticCurves/bibliography.md"
part: "Arithmetic Geometry"
chapter: "Hyperelliptic Curves"
breadcrumbs:
  - {"title": "Magma Handbook", "source": "../../index.md"}
  - {"title": "Arithmetic Geometry", "source": "../index-arithmetic-geometry.md"}
  - {"title": "Hyperelliptic Curves", "source": "index-hyperelliptic-curves.md"}
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next: {"title": "Hypergeometric Motives", "source": "../HypergeometricMotives/index-hypergeometric-motives.md"}
---

# Bibliography

<a id="cite-basson15"></a>
- **[Bas15]** Romain Basson. *Arithmétique des espaces de modules des courbes hyperelliptiques de genre $3$ en caractéristique positive*. PhD thesis, Université de Rennes 1, Rennes, 2015.
<a id="cite-bruin-phdthesis"></a>
- **[Bru02]** N. R. Bruin. *Chabauty methods and covering techniques applied to generalized Fermat equations*. Volume 133 of CWI Tract. Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 2002. ISBN 90-6196-508-X. Dissertation, University of Leiden, Leiden, 1999.
<a id="cite-brudoe-44split"></a>
- **[BD09]** Nils Bruin and Kevin Doerksen. The arithmetic of genus two curves with (4,4)-split Jacobians. ArXiv preprint. http://arxiv.org/abs/0902.3480, 2009.
<a id="cite-brusto-twocovdesc"></a>
- **[BS09]** Nils Bruin and Michael Stoll. Two-cover descent on hyperelliptic curves. *Math. Comp.*, 78:2347–2370, 2009.
<a id="cite-card-ng2"></a>
- **[Car03a]** G. Cardona. On the number of curves of genus 2 over a finite field. *Finite Fields and Their Applications*, 9(4):505–526, 2003.
<a id="cite-cana07"></a>
- **[CN07]** G. Cardona and E. Nart. Zeta function and cryptographic exponent of supersingular curves of genus 2. In *Pairing-based cryptography—Pairing 2007*, volume 4575 of Lecture Notes in Comput. Sci., pages 132–151. Springer, Berlin, 2007.
<a id="cite-canapu-g2"></a>
- **[CNP05a]** G. Cardona, E. Nart, and J. Pujolas. Curves of genus two over fields of even characteristic. *Mathematische Zeitschrift*, 250:177–201, 2005.
<a id="cite-cardona"></a>
- **[CQ05b]** Gabriel Cardona and Jordi Quer. Field of moduli and field of definition for curves of genus 2. *Lecture Notes Ser. Comput.*, 13:71–83, 2005.
<a id="cite-cassels-flynn"></a>
- **[CF96]** J.W.S. Cassels and E.V. Flynn. *Prolegomena to a Middlebrow Arithmetic of Curves of Genus $2$*. Cambridge University Press, Cambridge, 1996.
<a id="cite-creutzantsx"></a>
- **[Cre12]** Brendan Creutz. Explicit descent in the Picard group of a cyclic cover of the projective line. In Everett Howe and Kiran Kedlaya, editors, *ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium*, volume 1 of OBS. Mathematics Sciences Publishers, 2012.
<a id="cite-gaudry-har"></a>
- **[GH]** P. Gaudry, R. Harley. “Counting Points on Hyperelliptic Curves over Finite Fields.” 313–332.
<a id="cite-gottschling"></a>
- **[Got59]** E. Gottschling. Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades. *Math. Annalen*, 138:103–124, 1959.
<a id="cite-holmes10"></a>
- **[Hol06]** David Holmes. Canonical heights on hyperelliptic curves. Preprint http://arxiv.org/abs/1004.4503, 2006.
<a id="cite-hubrechts-pc"></a>
- **[Hub06]** Hendrik Hubrechts. Point counting in families of hyperelliptic curves. Preprint http://arxiv.org/abs/math.NT/0601438, 2006.
<a id="cite-igusag2"></a>
- **[Igu60a]** J. Igusa. “Arithmetic variety of moduli for genus two.” 72, 612–649. 1960.
<a id="cite-kedlaya1"></a>
- **[Ked01]** Kiran S. Kedlaya. Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology. *J. Ramanujan Math. Soc.*, 16:323 – 338, 2001.
<a id="cite-kllrss17"></a>
- **[KLL+18]** Pnar Klıçer, Hugo Labrande, Reynald Lercier, Christophe Ritzenthaler, Jeroen Sijsling, and Marco Streng. Plane quartics over $\Q$ with complex multiplication. *Acta Arith.*, 185(2):127–156, 2018.
<a id="cite-ler-lub1"></a>
- **[LL06]** R. Lercier and D. Lubicz. A Quasi-Quadratic Time Algorithm for Hyperelliptic Curve Point Counting. *to appear*, 2006.
<a id="cite-lrs12"></a>
- **[LRS12]** R. Lercier, C. Ritzenthaler, and J. Sijsling. Fast computation of isomorphisms of hyperelliptic curves and explicit descent. In E. W. Howe and K. S. Kedlaya, editors, *Proceedings of the Tenth Algorithmic Number Theory Symposium*, 463–486. Mathematical Sciences Publishers, 2012.
<a id="cite-lerrit12"></a>
- **[LR12]** Reynald Lercier and Christophe Ritzenthaler. Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. *J. Algebra*, 372:595–636, 2012. http://dx.doi.org/10.1016/j.jalgebra.2012.07.054.
<a id="cite-lrs"></a>
- **[LRS21]** Reynald Lercier, Christophe Ritzenthaler, and Jeroen Sijsling. Functionalities for genus $2$ and $3$ curves. https://arxiv.org/abs/2102.04372, 2021. [arXiv:2102.04372](https://arxiv.org/abs/2102.04372).
<a id="cite-mull-stoll-16"></a>
- **[MS16]** J.S. Müller and M. Stoll. Canonical heights on genus two Jacobians. *Algebra & Number Theory*, 10:2153–2234, 2016. https://doi.org/10.2140/ant.2016.10.2153. [doi:10.2140/ant.2016.10.2153](https://doi.org/10.2140/ant.2016.10.2153).
<a id="cite-mae90"></a>
- **[Mae90a]** T. Maeda. On the invariant fields of binary octavics. *Hiroshima Math. J.*, 20:619–632, 1990.
<a id="cite-mssv02"></a>
- **[MSSV02]** K. Magaard, T. Shaska, S. Shpectorov, and H. Völklein. The locus of curves with prescribed automorphism group. *Sūrikaisekikenkyūsho Kōkyūroku*, pages 112–141, 2002. Communications in arithmetic fundamental groups (Kyoto, 1999/2001).
<a id="cite-mestre-inv"></a>
- **[Mes91]** J.-F. Mestre. Construction de courbes de genre $2$ à partir de leurs modules. In T. Mora and C. Traverso, editors, *Effective methods in algebraic geometry*, volume 94 of Progr. Math., 313–334. Birkhäuser, 1991. Proc. Congress in Livorno, Italy, April 17–21, 1990.
<a id="cite-mumford2"></a>
- **[Mum84]** David Mumford. *Tata Lectures on Theta II*. Volume 43 of Progress in Mathematics. Birkhäuser, 1984.
<a id="cite-mueller-thesis"></a>
- **[Mul10a]** Jan Steffen Müller. *Computing canonical heights on Jacobians*. PhD Thesis, Universität Bayreuth, 2010. available at http://www.math.uni-hamburg.de/home/js.mueller/.
<a id="cite-mueller10"></a>
- **[Mul10b]** Jan Steffen Müller. Explicit Kummer surface formulas for arbitrary characteristic. *LMS J. Comput. Math.*, 4:47–64, 2010.
<a id="cite-poonen-schaefer"></a>
- **[PS97b]** Bjorn Poonen and Ed Schaefer. Explicit descent for Jacobians of cyclic covers of the projective line. *J. Reine Angew. Math.*, 488:141–188, 1997.
<a id="cite-poonen-stoll"></a>
- **[PS99]** Bjorn Poonen and Michael Stoll. The Cassels-Tate pairing on polarized abelian varieties. *Ann. of Math. (2)*, 150(3):1109–1149, 1999.
<a id="cite-shaska1"></a>
- **[SV01]** Tony Shaska and Völklein. *Elliptic subfields and automorphisms of genus $2$ function fields*. Springer, 2001, http://au.arxiv.org/abs/math.AG/0107142.
<a id="cite-shioda67"></a>
- **[Shi67]** T. Shioda. On the Graded Ring of Invariants of Binary Octavics. *Am. Jour. Math.*, 89(4):1022–1046, 1967.
<a id="cite-smart1"></a>
- **[Sma97]** N. P. Smart. $S$-unit equations, binary forms and curves of genus $2$. *Proc. London Math. Soc. (3)*, 75(2):271–307, 1997.
<a id="cite-smith-phd"></a>
- **[Smi05]** Benjamin A. Smith. *Explicit endomorphisms and correspondences*. PhD thesis, University of Sydney, 2005. http://hdl.handle.net/2123/1066.
<a id="cite-stoll-hc-1999"></a>
- **[Sto99b]** M. Stoll. On the height constant for curves of genus two. *Acta Arithmetica*, 90:183–201, 1999. https://doi.org/10.4064/aa-90-2-183-201. [doi:10.4064/aa-90-2-183-201](https://doi.org/10.4064/aa-90-2-183-201).
<a id="cite-stoll01"></a>
- **[Sto01]** Michael Stoll. Implementing 2-descent for Jacobians of hyperelliptic curves. *Acta Arith.*, 98(3):245–277, 2001.
<a id="cite-verc-ked"></a>
- **[Ver02]** F. Vercauteren. Computing zeta functions of hyperelliptic curves over finite fields of characteristic 2. In *Advances in cryptology—CRYPTO 2002*, volume 2442 of LNCS, pages 369–384. Springer, Berlin, 2002.
<a id="cite-wam-red"></a>
- **[Wam99]** P. Van Wamelen. Examples of Genus Two CM Curves Defined over the Rationals. *Mathematics of Computation*, 68:307–320, 1999.
<a id="cite-weth97th"></a>
- **[Wet97]** J. L. Wetherell. *Bounding the number of rational points on certain curves of high rank*. PhD thesis, U.C. Berkeley, 1997.