## Computing period matrices and the Abel-Jacobi map of superelliptic curves

HTML articles powered by AMS MathViewer

- by
Pascal Molin and Christian Neurohr
**HTML**| PDF - Math. Comp.
**88**(2019), 847-888 Request permission

## Abstract:

We present an algorithm for the computation of period matrices and the Abel-Jacobi map of complex superelliptic curves given by an equation $y^m=f(x)$. It relies on rigorous numerical integration of differentials between Weierstrass points, which is done using Gauss method if the curve is hyperelliptic ($m=2$) or the Double-Exponential method. The algorithm is implemented and makes it possible to reach thousands of digits accuracy even on large genus curves.## References

- Milton Abramowitz and Irene A. Stegun,
*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR**0167642** - Andrew R. Booker, Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki,
*A database of genus-2 curves over the rational numbers*, LMS J. Comput. Math.**19**(2016), no. suppl. A, 235–254. MR**3540958**, DOI 10.1112/S146115701600019X - Wieb Bosma, John Cannon, and Catherine Playoust,
*The Magma algebra system. I. The user language*, J. Symbolic Comput.**24**(1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR**1484478**, DOI 10.1006/jsco.1996.0125 - Jean-Benoît Bost and Jean-François Mestre,
*Moyenne arithmético-géométrique et périodes des courbes de genre $1$ et $2$*, Gaz. Math.**38**(1988), 36–64 (French). MR**970659** - Richard P. Brent and Paul Zimmermann,
*Modern computer arithmetic*, Cambridge Monographs on Applied and Computational Mathematics, vol. 18, Cambridge University Press, Cambridge, 2011. MR**2760886** - M. M. Chawla and M. K. Jain,
*Error estimates for Gauss quadrature formulas for analytic functions*, Math. Comp.**22**(1968), 82–90. MR**223093**, DOI 10.1090/S0025-5718-1968-0223093-3 - E. Costa, N. Mascot, J. Sijsling, and J. Voight,
*Rigorous computation of the endomorphism ring of a jacobian*.*arXiv preprint arXiv:1705.09248*, 2017. - John E. Cremona and Thotsaphon Thongjunthug,
*The complex AGM, periods of elliptic curves over $\Bbb {C}$ and complex elliptic logarithms*, J. Number Theory**133**(2013), no. 8, 2813–2841. MR**3045217**, DOI 10.1016/j.jnt.2013.02.002 - Bernard Deconinck and Mark van Hoeij,
*Computing Riemann matrices of algebraic curves*, Phys. D**152/153**(2001), 28–46. Advances in nonlinear mathematics and science. MR**1837895**, DOI 10.1016/S0167-2789(01)00156-7 - Jörg Frauendiener and Christian Klein,
*Computational approach to hyperelliptic Riemann surfaces*, Lett. Math. Phys.**105**(2015), no. 3, 379–400. MR**3312511**, DOI 10.1007/s11005-015-0743-4 - Jörg Frauendiener and Christian Klein,
*Computational approach to compact Riemann surfaces*, Nonlinearity**30**(2017), no. 1, 138–172. MR**3604607**, DOI 10.1088/1361-6544/30/1/138 - F. Johansson, Arb: a C library for ball arithmetic.
*ACM Communications in Computer Algebra***47**(2013), No. 4, 166–169. - P. Kilicer, H, Labrande, R. Lercier, C. Ritzenthaler, J. Sijsling, and M. Streng,
*Plane quartics over $\Bbb {Q}$ with complex multiplication*,*arXiv preprint arXiv:1701.06489*, 2017. - Ja Kyung Koo,
*On holomorphic differentials of some algebraic function field of one variable over $\textbf {C}$*, Bull. Austral. Math. Soc.**43**(1991), no. 3, 399–405. MR**1107394**, DOI 10.1017/S0004972700029245 - Greg Kuperberg,
*Kasteleyn cokernels*, Electron. J. Combin.**9**(2002), no. 1, Research Paper 29, 30. MR**1912810** - H. Labrande,
*Explicit computation of the Abel-Jacobi map and its inverse*. Theses, Université de Lorraine; University of Calgary, November 2016. - The LMFDB Collaboration,
*The L-functions and modular forms database*. http://www.lmfdb.org, 2013. [Online; accessed 16 September 2013]. - Nicolas Mascot,
*Computing modular Galois representations*, Rend. Circ. Mat. Palermo (2)**62**(2013), no. 3, 451–476. MR**3118315**, DOI 10.1007/s12215-013-0136-4 - Rick Miranda,
*Algebraic curves and Riemann surfaces*, Graduate Studies in Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1995. MR**1326604**, DOI 10.1090/gsm/005 - P. Molin,
*Intégration numérique et calculs de fonctions L*. PhD thesis, Université de Bordeaux I, 2010. - P. Molin and C, Neurohr,
*hcperiods: Arb and Magma packages for periods of superelliptic curves*. http://doi.org/10.5281/zenodo.1098275, July 2017. - Christopher Towse,
*Weierstrass points on cyclic covers of the projective line*, Trans. Amer. Math. Soc.**348**(1996), no. 8, 3355–3378. MR**1357406**, DOI 10.1090/S0002-9947-96-01649-2 - C. L. Tretkoff and M. D. Tretkoff,
*Combinatorial group theory, Riemann surfaces and differential equations*, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 467–519. MR**767125**, DOI 10.1090/conm/033/767125 - Paul van Wamelen,
*Equations for the Jacobian of a hyperelliptic curve*, Trans. Amer. Math. Soc.**350**(1998), no. 8, 3083–3106. MR**1432144**, DOI 10.1090/S0002-9947-98-02056-X - Paul B. van Wamelen,
*Computing with the analytic Jacobian of a genus 2 curve*, Discovering mathematics with Magma, Algorithms Comput. Math., vol. 19, Springer, Berlin, 2006, pp. 117–135. MR**2278925**, DOI 10.1007/978-3-540-37634-7_{5}

## Additional Information

**Pascal Molin**- Affiliation: IMJ-PRG & Université Paris 7, 8 place Aurélie Nemours, 75013 Paris, France
- MR Author ID: 884381
- Email: molin@math.univ-paris-diderot.fr
**Christian Neurohr**- Affiliation: Carl von Ossietzky Universität Oldenburg, Institut für Mathematik, 26129 Oldenburg, Germany
- Email: neurohrchristian@googlemail.com
- Received by editor(s): October 6, 2017
- Received by editor(s) in revised form: December 8, 2017
- Published electronically: May 30, 2018
- Additional Notes: The first author was partially supported by Partenariat Hubert Curien under grant 35487PL

The second author was partially supported by DAAD under grant 57212102. - © Copyright 2018 American Mathematical Society
- Journal: Math. Comp.
**88**(2019), 847-888 - MSC (2010): Primary 11Y16, 11Y35; Secondary 14Q05, 65D30, 11G30
- DOI: https://doi.org/10.1090/mcom/3351
- MathSciNet review: 3882287