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

Molin, Pascal; Neurohr, Christian

doi:10.1090/mcom/3351

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