## $p$-adic heights of Heegner points and $\Lambda$-adic regulators

HTML articles powered by AMS MathViewer

- by Jennifer S. Balakrishnan, Mirela Çiperiani and William Stein PDF
- Math. Comp.
**84**(2015), 923-954 Request permission

## Abstract:

Let $E$ be an elliptic curve defined over $\mathbb {Q}$. The aim of this paper is to make it possible to compute Heegner $L$-functions and anticyclotomic $\Lambda$-adic regulators of $E$, which were studied by Mazur-Rubin and Howard.

We generalize results of Cohen and Watkins and thereby compute Heegner points of non-fundamental discriminant. We then prove a relationship between the denominator of a point of $E$ defined over a number field and the leading coefficient of the minimal polynomial of its $x$-coordinate. Using this relationship, we recast earlier work of Mazur, Stein, and Tate to produce effective algorithms to compute $p$-adic heights of points of $E$ defined over number fields. These methods enable us to give the first explicit examples of Heegner $L$-functions and anticyclotomic $\Lambda$-adic regulators.

## References

- J. S. Balakrishnan,
*On $3$-adic heights on elliptic curves*, Preprint (2012), 1–8, http://www.math.harvard.edu/~jen/three_adic_heights.pdf. - Massimo Bertolini,
*Selmer groups and Heegner points in anticyclotomic $\mathbf Z_p$-extensions*, Compositio Math.**99**(1995), no. 2, 153–182. MR**1351834** - Henri Cohen,
*A course in computational algebraic number theory*, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR**1228206**, DOI 10.1007/978-3-662-02945-9 - Henri Cohen,
*Number theory. Vol. I. Tools and Diophantine equations*, Graduate Texts in Mathematics, vol. 239, Springer, New York, 2007. - Christophe Cornut,
*Mazur’s conjecture on higher Heegner points*, Invent. Math.**148**(2002), no. 3, 495–523. MR**1908058**, DOI 10.1007/s002220100199 - J. E. Cremona,
*Elliptic curve data*, http://www.warwick.ac.uk/~masgaj/ftp/data/. - 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 - Benedict H. Gross,
*Heegner points on $X_0(N)$*, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 87–105. MR**803364** - Benedict H. Gross,
*Kolyvagin’s work on modular elliptic curves*, $L$-functions and arithmetic (Durham, 1989), Cambridge Univ. Press, Cambridge, 1991, pp. 235–256. - David Harvey,
*Efficient computation of $p$-adic heights*, LMS J. Comput. Math.**11**(2008), 40–59. MR**2395362**, DOI 10.1112/S1461157000000528 - Benjamin Howard,
*The Iwasawa theoretic Gross-Zagier theorem*, Compos. Math.**141**(2005), no. 4, 811–846. MR**2148200**, DOI 10.1112/S0010437X0500134X - Dimitar Jetchev,
*Global divisibility of Heegner points and Tamagawa numbers*, Compos. Math.**144**(2008), no. 4, 811–826. MR**2441246**, DOI 10.1112/S0010437X08003497 - Anthony W. Knapp,
*Elliptic curves*, Mathematical Notes, vol. 40, Princeton University Press, Princeton, NJ, 1992. MR**1193029** - Barry Mazur and Karl Rubin,
*Elliptic curves and class field theory*, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 185–195. MR**1957032** - Barry Mazur, William Stein, and John Tate,
*Computation of $p$-adic heights and log convergence*, Doc. Math.**Extra Vol.**(2006), 577–614. MR**2290599** - B. Mazur and J. Tate,
*The $p$-adic sigma function*, Duke Math. J.**62**(1991), no. 3, 663–688. MR**1104813**, DOI 10.1215/S0012-7094-91-06229-0 - Bernadette Perrin-Riou,
*Fonctions $L$ $p$-adiques, théorie d’Iwasawa et points de Heegner*, Bull. Soc. Math. France**115**(1987), no. 4, 399–456 (French, with English summary). MR**928018** - W. Stein,
*Algebraic number theory, a computational approach*, 2007, http://wstein.org/books/ant/. - W. A. Stein et al.,
*Sage Mathematics Software (Version 5.10)*, The Sage Development Team, 2013, http://www.sagemath.org. - The PARI Group, Bordeaux,
*PARI/GP, version 2.5.0*, 2011, available from http://pari.math.u-bordeaux.fr/. - V. Vatsal,
*Special values of anticyclotomic $L$-functions*, Duke Math. J.**116**(2003), no. 2, 219–261. MR**1953292**, DOI 10.1215/S0012-7094-03-11622-1 - M. Watkins,
*Some remarks on Heegner point computations*, Preprint (2006), http://arxiv.org/abs/math/0506325. - Christian Wuthrich,
*On $p$-adic heights in families of elliptic curves*, J. London Math. Soc. (2)**70**(2004), no. 1, 23–40. MR**2064750**, DOI 10.1112/S0024610704005277

## Additional Information

**Jennifer S. Balakrishnan**- Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
- MR Author ID: 910890
- Email: jen@math.harvard.edu
**Mirela Çiperiani**- Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712
- MR Author ID: 838646
- Email: mirela@math.utexas.edu
**William Stein**- Affiliation: Department of Mathematics, University of Washington, Box 354350 Seattle, Washington 98195
- MR Author ID: 679996
- Email: wstein@uw.edu
- Received by editor(s): May 21, 2013
- Received by editor(s) in revised form: August 2, 2013
- Published electronically: September 11, 2014
- Additional Notes: The first author was supported by NSF grant DMS-1103831

The second author was supported by NSA grant H98230-12-1-0208

The third author was supported by NSF Grants DMS-1161226 and DMS-1147802. - © Copyright 2014 American Mathematical Society
- Journal: Math. Comp.
**84**(2015), 923-954 - MSC (2010): Primary 11Y40, 11G50, 11G05
- DOI: https://doi.org/10.1090/S0025-5718-2014-02876-7
- MathSciNet review: 3290969