Explicit arithmetic intersection theory and computation of Néron-Tate heights

van Bommel, Raymond; Holmes, David; Müller, J. Steffen

doi:10.1090/mcom/3441

Explicit arithmetic intersection theory and computation of Néron-Tate heights
HTML articles powered by AMS MathViewer

by Raymond van Bommel, David Holmes and J. Steffen Müller HTML | PDF

Math. Comp. 89 (2020), 395-410 Request permission

Abstract:

We describe a general algorithm for computing intersection pairings on arithmetic surfaces. We have implemented our algorithm for curves over $\mathbb {Q}$, and we show how to use it to compute regulators for a number of Jacobians of smooth plane quartics, and to numerically verify the conjecture of Birch and Swinnerton-Dyer for the Jacobian of the split Cartan curve of level 13, up to squares.

References

Jennifer S. Balakrishnan and Amnon Besser, Computing local $p$-adic height pairings on hyperelliptic curves, Int. Math. Res. Not. IMRN 11 (2012), 2405–2444. MR 2926986, DOI 10.1093/imrn/rnr111
Jennifer S. Balakrishnan, Netan Dogra, J. Steffen Müller, Jan Tuitman, and Jan Vonk, Explicit Chabauty-Kim for the split Cartan modular curve of level 13, Ann. of Math. (2) 189 (2019), no. 3, 885-944.
Burcu Baran, An exceptional isomorphism between modular curves of level 13, J. Number Theory 145 (2014), 273–300. MR 3253304, DOI 10.1016/j.jnt.2014.05.017
Burcu Baran, An exceptional isomorphism between level 13 modular curves via Torelli’s theorem, Math. Res. Lett. 21 (2014), no. 5, 919–936. MR 3294556, DOI 10.4310/MRL.2014.v21.n5.a1
Raymond van Bommel, Numerical verification of the Birch and Swinnerton-Dyer conjecture for hyperelliptic curves of higher genus over $\mathbb {Q}$ up to squares, to appear in Exp. Math., DOI 10.1080/10586458.2019.1592035
Nils Bruin, Bjorn Poonen, and Michael Stoll, Generalized explicit descent and its application to curves of genus 3, Forum Math. Sigma 4 (2016), Paper No. e6, 80. MR 3482281, DOI 10.1017/fms.2016.1
Robert F. Coleman and Benedict H. Gross, $p$-adic heights on curves, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 73–81. MR 1097610, DOI 10.2969/aspm/01710073
Tim Dokchitser, Models of curves over DVRs, arXiv e-prints (2018), available at 1807.00025.
Gerd Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387–424. MR 740897, DOI 10.2307/2007043
E. Victor Flynn, Franck Leprévost, Edward F. Schaefer, William A. Stein, Michael Stoll, and Joseph L. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70 (2001), no. 236, 1675–1697. MR 1836926, DOI 10.1090/S0025-5718-01-01320-5
E. V. Flynn and N. P. Smart, Canonical heights on the Jacobians of curves of genus $2$ and the infinite descent, Acta Arith. 79 (1997), no. 4, 333–352. MR 1450916, DOI 10.4064/aa-79-4-333-352
Benedict H. Gross, Local heights on curves, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 327–339. MR 861983
Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
F. Hess, Computing Riemann-Roch spaces in algebraic function fields and related topics, J. Symbolic Comput. 33 (2002), no. 4, 425–445. MR 1890579, DOI 10.1006/jsco.2001.0513
Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
David Holmes, Computing Néron-Tate heights of points on hyperelliptic Jacobians, J. Number Theory 132 (2012), no. 6, 1295–1305. MR 2899805, DOI 10.1016/j.jnt.2012.01.002
David Holmes, An Arakelov-theoretic approach to naïve heights on hyperelliptic Jacobians, New York J. Math. 20 (2014), 927–957. MR 3272917
Paul Hriljac, Heights and Arakelov’s intersection theory, Amer. J. Math. 107 (1985), no. 1, 23–38. MR 778087, DOI 10.2307/2374455
Fredrik Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput. 66 (2017), no. 8, 1281–1292. MR 3681746, DOI 10.1109/TC.2017.2690633
V. A. Kolyvagin and D. Yu. Logachëv, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, 171–196 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1229–1253.
Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. MR 969124, DOI 10.1007/978-1-4612-1031-3
Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
Pascal Molin and Christian Neurohr, Computing period matrices and the Abel-Jacobi map of superelliptic curves, Math. Comp. 88 (2019), no. 316, 847–888. MR 3882287, DOI 10.1090/mcom/3351
Jan Steffen Müller, Computing canonical heights using arithmetic intersection theory, Math. Comp. 83 (2014), no. 285, 311–336. MR 3120591, DOI 10.1090/S0025-5718-2013-02719-6
J. Steffen Müller and Michael Stoll, Computing canonical heights on elliptic curves in quasi-linear time, LMS J. Comput. Math. 19 (2016), no. suppl. A, 391–405. MR 3540967, DOI 10.1112/S1461157016000139
Jan Steffen Müller and Michael Stoll, Canonical heights on genus-2 Jacobians, Algebra Number Theory 10 (2016), no. 10, 2153–2234. MR 3582017, DOI 10.2140/ant.2016.10.2153
David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651, DOI 10.1007/978-1-4899-2843-6
A. Néron, Quasi-fonctions et hauteurs sur les variétés abéliennes, Ann. of Math. (2) 82 (1965), 249–331 (French). MR 179173, DOI 10.2307/1970644
Christian Neurohr, Efficient integration on Riemann surfaces & applications, PhD thesis, Carl von Ossietzky Universität Oldenburg, 2018. http://oops.uni-oldenburg.de/3607/1/neueff18.pdf.
Bjorn Poonen and Edward F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math. 488 (1997), 141–188. MR 1465369
Bjorn Poonen and Michael Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149. MR 1740984, DOI 10.2307/121064
Samir Siksek, Infinite descent on elliptic curves, Rocky Mountain J. Math. 25 (1995), no. 4, 1501–1538. MR 1371352, DOI 10.1216/rmjm/1181072159
Joseph H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), no. 183, 339–358. MR 942161, DOI 10.1090/S0025-5718-1988-0942161-4
Michael Stoll, On the height constant for curves of genus two. II, Acta Arith. 104 (2002), no. 2, 165–182. MR 1914251, DOI 10.4064/aa104-2-6
Michael Stoll, An explicit theory of heights for hyperelliptic Jacobians of genus three, Algorithmic and experimental methods in algebra, geometry, and number theory, Springer, Cham, 2017, pp. 665–715. MR 3792747