Second 𝑝-descents on elliptic curves

Creutz, Brendan

doi:10.1090/S0025-5718-2013-02713-5

Second $p$-descents on elliptic curves
HTML articles powered by AMS MathViewer

by Brendan Creutz;

Math. Comp. 83 (2014), 365-409

DOI: https://doi.org/10.1090/S0025-5718-2013-02713-5

Published electronically: May 23, 2013

PDF | Request permission

Abstract:

Let $p$ be a prime and $C$ a genus one curve over a number field $k$ representing an element of order dividing $p$ in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of $D$ in the Shafarevich-Tate group such that $pD = C$ and obtains explicit models for these $D$ as curves in projective space. This leads to a practical algorithm for performing explicit $9$-descents on elliptic curves over $\mathbb {Q}$.

References

B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7–25. MR 146143, DOI 10.1515/crll.1963.212.7
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
A. Bremner and J. W. S. Cassels, On the equation $Y^{2}=X(X^{2}+p)$, Math. Comp. 42 (1984), no. 165, 257–264. MR 726003, DOI 10.1090/S0025-5718-1984-0726003-4
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
Nils Bruin and Michael Stoll, Two-cover descent on hyperelliptic curves, Math. Comp. 78 (2009), no. 268, 2347–2370. MR 2521292, DOI 10.1090/S0025-5718-09-02255-8
J. W. S. Cassels, Arithmetic on curves of genus $1$. IV. Proof of the Hauptvermutung, J. Reine Angew. Math. 211 (1962), 95–112. MR 163915, DOI 10.1515/crll.1962.211.95
J. W. S. Cassels, Arithmetic on curves of genus 1. V. Two counterexamples, J. London Math. Soc. 38 (1963), 244–248. MR 148664, DOI 10.1112/jlms/s1-38.1.244
J. W. S. Cassels, Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180–199. MR 179169, DOI 10.1515/crll.1965.217.180
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
J. W. S. Cassels, Second descents for elliptic curves, J. Reine Angew. Math. 494 (1998), 101–127. Dedicated to Martin Kneser on the occasion of his 70th birthday. MR 1604468, DOI 10.1515/crll.1998.001
C. Chevalley and A. Weil: Un théorème d’arithmétique sur les courbes algébriques, C.R. Acad. Sci. Paris 195 (1932), 570–572.
Pete L. Clark, The period-index problem in WC-groups. I. Elliptic curves, J. Number Theory 114 (2005), no. 1, 193–208. MR 2163913, DOI 10.1016/j.jnt.2004.10.001
Pete L. Clark and Shahed Sharif, Period, index and potential. III, Algebra Number Theory 4 (2010), no. 2, 151–174. MR 2592017, DOI 10.2140/ant.2010.4.151
J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), no. 3, 223–251. MR 463176, DOI 10.1007/BF01402975
John E. Cremona, Tom A. Fisher, and Michael Stoll, Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves, Algebra Number Theory 4 (2010), no. 6, 763–820. MR 2728489, DOI 10.2140/ant.2010.4.763
J.E. Cremona: Elliptic curves database, available online at http://www.warwick.ac.uk/staff/J.E.Cremona/ftp/data/INDEX.html
J. E. Cremona, T. A. Fisher, C. O’Neil, D. Simon, and M. Stoll, Explicit $n$-descent on elliptic curves. I. Algebra, J. Reine Angew. Math. 615 (2008), 121–155. MR 2384334, DOI 10.1515/CRELLE.2008.012
B. Creutz: Explicit second $p$-descent on elliptic curves, Ph.D. thesis, Jacobs University (2010).
Brendan Creutz and Robert L. Miller, Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula, J. Algebra 372 (2012), 673–701. MR 2990032, DOI 10.1016/j.jalgebra.2012.09.029
Z. Djabri, Edward F. Schaefer, and N. P. Smart, Computing the $p$-Selmer group of an elliptic curve, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5583–5597. MR 1694286, DOI 10.1090/S0002-9947-00-02535-6
T.A. Fisher: On $5$ and $7$ descents for elliptic curves, Ph.D. thesis, University of Cambridge (2000).
Tom Fisher, Some examples of 5 and 7 descent for elliptic curves over $\bf Q$, J. Eur. Math. Soc. (JEMS) 3 (2001), no. 2, 169–201. MR 1831874, DOI 10.1007/s100970100030
Tom Fisher, Finding rational points on elliptic curves using 6-descent and 12-descent, J. Algebra 320 (2008), no. 2, 853–884. MR 2422319, DOI 10.1016/j.jalgebra.2008.04.007
V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. MR 1106906
Serge Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983. Reprint of the 1959 original. MR 713430
Carl-Erik Lind, Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins, University of Uppsala, Uppsala, 1940 (German). Thesis. MR 22563
J. R. Merriman, S. Siksek, and N. P. Smart, Explicit $4$-descents on an elliptic curve, Acta Arith. 77 (1996), no. 4, 385–404. MR 1414518, DOI 10.4064/aa-77-4-385-404
Robert L. Miller and Michael Stoll, Explicit isogeny descent on elliptic curves, Math. Comp. 82 (2013), no. 281, 513–529. MR 2983034, DOI 10.1090/S0025-5718-2012-02619-6
L.J. Mordell: On the rational solutions of the indeterminate equations of the 3rd and 4th degrees, Proc. Camb. Phil. Soc. 21 (1922), 179–192.
Catherine O’Neil, The period-index obstruction for elliptic curves, J. Number Theory 95 (2002), no. 2, 329–339. MR 1924106, DOI 10.1016/S0022-314X(01)92770-2
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
Hans Reichardt, Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen, J. Reine Angew. Math. 184 (1942), 12–18 (German). MR 9381, DOI 10.1515/crll.1942.184.12
Karl Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68. MR 1079839, DOI 10.1007/BF01239508
Edward F. Schaefer, Class groups and Selmer groups, J. Number Theory 56 (1996), no. 1, 79–114. MR 1370197, DOI 10.1006/jnth.1996.0006
Edward F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), no. 3, 447–471. MR 1612262, DOI 10.1007/s002080050156
Edward F. Schaefer and Michael Stoll, How to do a $p$-descent on an elliptic curve, Trans. Amer. Math. Soc. 356 (2004), no. 3, 1209–1231. MR 2021618, DOI 10.1090/S0002-9947-03-03366-X
Ernst S. Selmer, The Diophantine equation $ax^3+by^3+cz^3=0$, Acta Math. 85 (1951), 203–362 (1 plate). MR 41871, DOI 10.1007/BF02395746
Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
Samir Siksek, Descent on Picard groups using functions on curves, Bull. Austral. Math. Soc. 66 (2002), no. 1, 119–124. MR 1922613, DOI 10.1017/S0004972700020736
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Denis Simon, Computing the rank of elliptic curves over number fields, LMS J. Comput. Math. 5 (2002), 7–17. MR 1916919, DOI 10.1112/S1461157000000668
S. Stamminger: Explicit $8$-descent on elliptic curves, PhD thesis, International University Bremen (2005).
Michael Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), no. 3, 245–277. MR 1829626, DOI 10.4064/aa98-3-4
A. Weil: Sur un théorème de Mordell, Bull. Sci. Math. 54 (1930), 182-191.
A.J. Wiles: Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 2 (1995), no. 3, 443–551.
T. Womack: Explicit descent on elliptic curves, PhD thesis, Nottingham (2003).
Ju. G. Zarhin, Noncommutative cohomology and Mumford groups, Mat. Zametki 15 (1974), 415–419 (Russian). MR 354612