Computing canonical heights on the projective line with no factorization
HTML articles powered by AMS MathViewer
- by Elliot Wells PDF
- Math. Comp. 86 (2017), 3019-3029 Request permission
Abstract:
We give an algorithm which requires no integer factorization for computing the canonical height of a point in $\mathbb {P}^{1}(\mathbb {Q})$ relative to a morphism $\phi : \mathbb {P}_{\mathbb {Q}}^{1} \rightarrow \mathbb {P}_{\mathbb {Q}}^{1}$ of degree $d \geq 2$.References
- Gregory S. Call and Joseph H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163–205. MR 1255693
- J. W. S. Cassels, Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991. MR 1144763, DOI 10.1017/CBO9781139172530
- The Sage Developers, Sage Mathematics Software (Version 7.0), 2016, http://www. sagemath.org.
- Thorsten Kleinjung, Kazumaro Aoki, Jens Franke, Arjen K. Lenstra, Emmanuel Thomé, Joppe W. Bos, Pierrick Gaudry, Alexander Kruppa, Peter L. Montgomery, Dag Arne Osvik, Herman te Riele, Andrey Timofeev, and Paul Zimmermann, Factorization of a 768-bit RSA modulus, Advances in cryptology—CRYPTO 2010, Lecture Notes in Comput. Sci., vol. 6223, Springer, Berlin, 2010, pp. 333–350. MR 2725602, DOI 10.1007/978-3-642-14623-7_{1}8
- Michelle Manes, $\Bbb Q$-rational cycles for degree-2 rational maps having an automorphism, Proc. Lond. Math. Soc. (3) 96 (2008), no. 3, 669–696. MR 2407816, DOI 10.1112/plms/pdm044
- J. Steffen Müller and Michael Stoll, Computing canonical heights on elliptic curves in quasi-linear time, arXiv:1509.08748v2, 2015.
- J. Steffen Müller and Michael Stoll, Canonical heights on genus two jacobians, arXiv:1603.00640v1, 2016.
- D. G. Northcott, Periodic points on an algebraic variety, Ann. of Math. (2) 51 (1950), 167–177. MR 34607, DOI 10.2307/1969504
- Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
- Joseph H. Silverman, Computing canonical heights with little (or no) factorization, Math. Comp. 66 (1997), no. 218, 787–805. MR 1388892, DOI 10.1090/S0025-5718-97-00812-0
- Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR 2316407, DOI 10.1007/978-0-387-69904-2
Additional Information
- Elliot Wells
- Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
- Email: ellwells@math.brown.edu
- Received by editor(s): March 3, 2016
- Received by editor(s) in revised form: July 12, 2016
- Published electronically: April 7, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 86 (2017), 3019-3029
- MSC (2010): Primary 37P30; Secondary 11G50, 11Y16
- DOI: https://doi.org/10.1090/mcom/3200
- MathSciNet review: 3667036