Show Summary Details
More options …

Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

Online
ISSN
1435-5345
See all formats and pricing
More options …
Volume 2016, Issue 720

Volume 1826 (1826)

Jennifer S. Balakrishnan
• Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom of Great Britain and Northern Ireland
• Email
• Other articles by this author:
/ Amnon Besser
/ J. Steffen Müller
Published Online: 2014-07-08 | DOI: https://doi.org/10.1515/crelle-2014-0048

Abstract

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell–Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell–Weil group tensored with $ℚ$.

Dedicated to the memory of Robert F. Coleman (1954–2014)

References

• [1]

Balakrishnan J. S., Iterated Coleman integration for hyperelliptic curves, ANTS-X: Proceedings of the tenth algorithmic number theory symposium, Open Book Series 1, Mathematical Sciences Publishers (2013), 41–61. Google Scholar

• [2]

Balakrishnan J. S. and Besser A., Computing local p-adic height pairings on hyperelliptic curves, Int. Math. Res. Not. IMRN (2012), no. 11, 2405–2444. Google Scholar

• [3]

Balakrishnan J. S. and Besser A., Coleman–Gross height pairings and the p-adic sigma function, J. reine angew. Math. (2013), 10.1515/crelle-2012-0095. Google Scholar

• [4]

Balakrishnan J. S., Bradshaw R. W. and Kedlaya K., Explicit Coleman integration for hyperelliptic curves, Algorithmic number theory, Lecture Notes in Comput. Sci. 6197, Springer, Berlin (2010), 16–31. Google Scholar

• [5]

Balakrishnan J. S., Dan-Cohen I., Kim M. and Wewers S., A non-abelian conjecture of Birch and Swinnerton–Dyer type for hyperbolic curves, preprint 2012, http://arxiv.org/abs/1209.0640.

• [6]

Balakrishnan J. S., Kedlaya K. S. and Kim M., Appendix and erratum to “Massey products for elliptic curves of rank 1”, J. Amer. Math. Soc. 24 (2011), no. 1, 281–291. Google Scholar

• [7]

Balakrishnan J. S., Müller J. S. and Stein W., A p-adic analogue of the conjecture of Birch and Swinnerton–Dyer for modular abelian varieties, preprint 2012, http://arxiv.org/abs/1210.2739.

• [8]

Besser A., p-adic Arakelov theory, J. Number Theory 111 (2005), no. 2, 318–371. Google Scholar

• [9]

Besser A. and Furusho H., The double shuffle relations for p-adic multiple zeta values, Primes and knots, Contemp. Math. 416, American Mathematical Society, Providence (2006), 9–29. Google Scholar

• [10]

Besser A. and de Jeu R., The syntomic regulator for ${K}_{4}$ of curves, Pacific J. Math. 260 (2012), no. 2, 305–380. Google Scholar

• [11]

Besser A. and Zerbes S., Vologodsky integration on semi-stable curves, in preparation. Google Scholar

• [12]

Bilu Y. F. and Hanrot G., Solving superelliptic Diophantine equations by Baker’s method, Compositio Math. 112 (1998), no. 3, 273–312. Google Scholar

• [13]

Bosma W., Cannon J. and Playoust C., The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3–4, 235–265. Google Scholar

• [14]

Bugeaud Y., Mignotte M., Siksek S., Stoll M. and Tengely S., Integral points on hyperelliptic curves, Algebra Number Theory 2 (2008), no. 8, 859–885. Google Scholar

• [15]

Chabauty C., Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885. Google Scholar

• [16]

Coates J. and Kim M., Selmer varieties for curves with CM Jacobians, Kyoto J. Math. 50 (2010), no. 4, 827–852. Google Scholar

• [17]

Coleman R., Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. Google Scholar

• [18]

Coleman R., Torsion points on curves and p-adic abelian integrals, Annals of Math. 121 (1985), 111–168. Google Scholar

• [19]

Coleman R., Duality for the de Rham cohomology of an abelian scheme, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 5, 1379–1393. Google Scholar

• [20]

Coleman R. and Gross B., p-adic heights on curves, Algebraic number theory, Adv. Stud. Pure Math. 17, Academic Press, Boston (1989), 73–81. Google Scholar

• [21]

Curilla C. and Kühn U., On the arithmetic self-intersection number of the dualizing sheaf for Fermat curves of prime exponent, preprint 2009, http://arxiv.org/abs/0906.3891.

• [22]

Flynn E. V., A flexible method for applying Chabauty’s theorem, Compositio Math. 105 (1997), no. 1, 79–94. Google Scholar

• [23]

Gross B. H., Local heights on curves, Arithmetic geometry (Storrs 1984), Springer, New York (1986), 327–339. Google Scholar

• [24]

Harvey D., Efficient computation of p-adic heights, LMS J. Comput. Math. 11 (2008), 40–59. Google Scholar

• [25]

Holmes D., An Arakelov-theoretic approach to naive heights on hyperelliptic Jacobians, preprint 2012, http://arxiv.org/abs/1207.5948.

• [26]

Kim M., The motivic fundamental group of ${ℙ}^{1}-\left\{0,1,\mathrm{\infty }\right\}$ and the theorem of Siegel, Invent. Math. 161 (2005), no. 3, 629–656. Google Scholar

• [27]

Kim M., The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 89–133. Google Scholar

• [28]

Kim M., Massey products for elliptic curves of rank 1, J. Amer. Math. Soc. 23 (2010), no. 3, 725–747. Google Scholar

• [29]

Kim M., p-adic L-functions and Selmer varieties associated to elliptic curves with complex multiplication, Ann. of Math. (2) 172 (2010), no. 1, 751–759. Google Scholar

• [30]

Kühn U. and Müller J. S., Lower bounds on the arithmetic self-intersection number of the relative dualizing sheaf on arithmetic surfaces, preprint 2012, http://arxiv.org/abs/1205.3274.

• [31]

Lang S., Introduction to Arakelov theory, Springer, New York 1988. Google Scholar

• [32]

Liu Q., Algebraic geometry and arithmetic curves, Oxf. Grad. Texts Math. 6, Oxford University Press, Oxford 2002. Google Scholar

• [33]

Mazur B., Stein W. and Tate J., Computation of p-adic heights and log convergence, Doc. Math. (2006), 577–614. Google Scholar

• [34]

McCallum W. and Poonen B., The method of Chabauty and Coleman, Explicit methods in number theory, Panor. Synthèses 36, Société Mathématique de France, Paris (2012), 99–117. Google Scholar

• [35]

Müller J. S. and Stoll M., Canonical heights on genus two Jacobians, in preparation. Google Scholar

• [36]

Namikawa Y. and Ueno K., The complete classification of fibres in pencils of curves of genus two, Manuscripta Math. 9 (1973), 143–186. Google Scholar

• [37]

Penrose R., A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406–413. Google Scholar

• [38]

Siksek S., Explicit Chabauty over number fields, Algebra Number Theory 7 (2013), no. 4, 765–793. Google Scholar

• [39]

Smart N. P., The algorithmic resolution of Diophantine equations, London Math. Soc. Stud. Texts 41, Cambridge University Press, Cambridge 1998. Google Scholar

• [40]

Stein W. A. et al., Sage Mathematics Software (Version 5.6), The Sage Development Team, 2013, www.sagemath.org.

• [41]

Stoll M., Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), 245–277. Google Scholar

• [42]

Stoll M., On the height constant for curves of genus two. II, Acta Arith. 104 (2002), 165–182. Google Scholar

• [43]

Stoll M., An explicit theory of heights for hyperelliptic Jacobians of genus three, preprint 2014. Google Scholar

• [44]

Vologodsky V., Hodge structure on the fundamental group and its application to p-adic integration, Mosc. Math. J. 3 (2003), no. 1, 205–247. Google Scholar

Revised: 2014-04-04

Published Online: 2014-07-08

Published in Print: 2016-11-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1103831

Funding Source: European Research Council

Award identifier / Grant number: 204083

Funding Source: Engineering and Physical Sciences Research Council

Award identifier / Grant number: I020519/1

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: KU 2359/2-1

The first author was supported by NSF grant DMS-1103831. The second author’s stay at Oxford was funded by ERC grant 204083 and by EPSRC grant I020519/1. The third author was supported by DFG grant KU 2359/2-1.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 720, Pages 51–79, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

Export Citation