---
title: "Bibliography"
canonical: "https://docs.magma-maths.org/ArithmeticGeometry/EllipticCurvesOverQAndNumberFields/bibliography.html"
source: "https://docs.magma-maths.org/ArithmeticGeometry/EllipticCurvesOverQAndNumberFields/bibliography.md"
part: "Arithmetic Geometry"
chapter: "Elliptic Curves over ${\\mathbb{Q}}$ and Number Fields"
breadcrumbs:
  - {"title": "Magma Handbook", "source": "../../index.md"}
  - {"title": "Arithmetic Geometry", "source": "../index-arithmetic-geometry.md"}
  - {"title": "Elliptic Curves over ${\\mathbb{Q}}$ and Number Fields", "source": "index-elliptic-curves-over-qand-number-fields.md"}
prev: {"title": "Mordell–Weil Groups and Descent Methods", "source": "mordell-weil.md"}
next: {"title": "Elliptic Curves over Function Fields", "source": "../EllipticCurvesOverFunctionFields/index-elliptic-curves-over-function-fields.md"}
---

# Bibliography

<a id="cite-bremnercassels"></a>
- **[BC84]** A. Bremner and J. W. S. Cassels. On the equation $Y^{2}=X(X^{2}+p)$. *Math. Comp.*, 42(165):257–264, 1984.
<a id="cite-bruin-phdthesis"></a>
- **[Bru02]** N. R. Bruin. *Chabauty methods and covering techniques applied to generalized Fermat equations*. Volume 133 of CWI Tract. Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 2002. ISBN 90-6196-508-X. Dissertation, University of Leiden, Leiden, 1999.
<a id="cite-bruin-magmabook"></a>
- **[Bru]** N. Bruin. “Some ternary Diophantine equations of signature $(n,n,2)$.”
<a id="cite-bruin-crelle"></a>
- **[Bru03]** Nils Bruin. Chabauty methods using elliptic curves. *J. reine angew. Math.*, 562:27–49, 2003.
<a id="cite-brusto-mws"></a>
- **[BS10]** Nils Bruin and Michael Stoll. The Mordell-Weil sieve: Proving non-existence of rational points on curves. *LMS J. Comput. Math.*, 13:272–306, 2010.
<a id="cite-cassels66"></a>
- **[Cas66]** J. W. S. Cassels. Diophantine equations with special reference to elliptic curves. *J. London Math. Soc.*, 41:150–158, 1966.
<a id="cite-cohen1"></a>
- **[Coh93]** Henri Cohen. *A Course in Computational Algebraic Number Theory*. Volume 138 of Graduate Texts in Mathematics. Springer, Berlin–Heidelberg–New York, 1993.
<a id="cite-five-authors-descenti"></a>
- **[CFO+08]** 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.*, 615:121–155, 2008.
<a id="cite-five-authors-descentiii"></a>
- **[CFO+]** J.E. Cremona, T.A Fisher, C. O'Neil, D. Simon, and M Stoll. Explicit $n$-Descent On Elliptic Curves, III. Algorithms. ArXiv preprint. http://arxiv.org/abs/1107.3516.
<a id="cite-five-authors-descentii"></a>
- **[CFO+09]** J.E. Cremona, T.A Fisher, C. O'Neil, D. Simon, and M Stoll. Explicit $n$-Descent On Elliptic Curves, II. Geometry. *J. reine angew. Math.*, 632:63–84, 2009.
<a id="cite-cremona-fisher-stoll"></a>
- **[CFS10]** J.E. Cremona, T.A Fisher, and M Stoll. Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves. *Algebra & Number Theory*, 4(6):763–820, 2010.
<a id="cite-crem-forms"></a>
- **[Cre99]** John Cremona. Reduction of binary cubic and quartic forms. *LMS JCM*, 2:62–92, 1999.
<a id="cite-crem-2-desc"></a>
- **[Cre01]** John Cremona. Classical invariants and 2-descent on elliptic curves. *J. Symbolic Comp.*, 31:71–87, 2001.
<a id="cite-creutzmiller"></a>
- **[CM12]** B. Creutz and R.L. Miller. Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula. *J.Algebra*, 372:673–701, 2012.
<a id="cite-creutzphd"></a>
- **[Cre10]** Brendan Creutz. *Explicit second $p$-descents on elliptic curves*. PhD Thesis, Jacobs University Bremen, 2010.
<a id="cite-elk2000"></a>
- **[Elk]** N. Elkies. “Rational Points Near Curves and Small Nonzero $|x^3 - y^2|$ via Lattice Reduction.” 33–63.
<a id="cite-fisherhigherdescent"></a>
- **[Fisb]** (no details available)
<a id="cite-fisherphd"></a>
- **[Fis00]** Tom Fisher. *On 5 and 7 descents for elliptic curves*. PhD thesis, University of Cambridge, 2000.
<a id="cite-fisherjems"></a>
- **[Fis01]** Tom Fisher. Some examples of $5$ and $7$ descent for elliptic curves over Q. *J. Eur. Math. Soc.*, 3(Issue 2):169–201, 2001.
<a id="cite-fisher6and12"></a>
- **[Fis08b]** Tom Fisher. Finding rational points on elliptic curves using 6-descent and 12-descent. *J. Algebra*, 320(2):853–884, 2008.
<a id="cite-grozag"></a>
- **[GZ86]** Gross and Zagier. Heegner Points and Derivatives of $L$-series. *Invent. Math.*, 84:225–320, 1986.
<a id="cite-harvey-padic"></a>
- **[Har08]** D. Harvey. Efficient computation of p-adic heights. *LMS J. Comput. Math.*, 11:40–59, 2008.
<a id="cite-kramer"></a>
- **[Kra81]** K. Kramer. Arithmetic of elliptic curves upon quadratic extension. *Trans. Amer. Math. Soc.*, 264(1):121–135, 1981.
<a id="cite-mazur-stein-tate"></a>
- **[MST06]** B. Mazur, W. Stein, and J. Tate. Computation of $p$-adic heights and log convergence. *Documenta Mathematica*, Extra:577–614, 2006.
<a id="cite-mazur-tate-padic"></a>
- **[MT91]** B. Mazur and J. Tate. The p-adic sigma function. *Duke Math Journal*, 62(3):663–688, 1991.
<a id="cite-merriss"></a>
- **[MSS96]** J. R. Merriman, S. Siksek, and N. P. Smart. Explicit $4$-descents on an elliptic curve. *Acta Arith.*, 77(4):385–404, 1996.
<a id="cite-schaefer-stoll"></a>
- **[SS04]** Edward F. Schaefer and Michael Stoll. How to do a $p$-descent on an elliptic curve. *Trans. Amer. Math. Soc.*, 356(3):1209–1231 (electronic), 2004.
<a id="cite-siksek"></a>
- **[Sik95]** Samir Siksek. Infinite descent on elliptic curves. *Rocky Mountain J. Math.*, 25(4):1501–1538, 1995.
<a id="cite-silverman-arith"></a>
- **[Sil86]** J. Silverman. *The arithmetic of elliptic curves*. Volume 106 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1986.
<a id="cite-stwat2002"></a>
- **[SW]** (no details available)
<a id="cite-tzanakis"></a>
- **[ST96]** R. J. Stroeker and Tzanakis T. Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms, the quartic case. *Acta Arith.*, 75:165–190, 1996.
<a id="cite-stroekertzanakis"></a>
- **[ST94]** R. J. Stroeker and N. Tzanakis. Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. *Acta Arith.*, 67:177–196, 1994.
<a id="cite-wat2003"></a>
- **[Wat02]** M. Watkins. Computing the modular degree of an elliptic curve. *Experimental Mathematics*, 11(4):487–502, 2002.