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Fisher, Tom

Some examples of 5 and 7 descent for elliptic curves over \(\mathbb{Q}\). (English) Zbl 1007.11031

J. Eur. Math. Soc. (JEMS) 3, No. 2, 169-201 (2001).
Let \(E/K\) be an elliptic curve defined over a number field \(K\). Although there is no known algorithm to compute the Mordell-Weil group \(E(K)\) of \(E/K\), the proof of the Mordell-Weil theorem gives an effective upper bound for the order of \(E(K)/nE(K)\) for an integer \(n\geq 2\). This yields an upper bound on the rank of \(E(K)\), which is an overestimate precisely whenever \(\text{ Ш}(E/K)\) contains nontrivial \(n\)-torsion. This type of calculation is known as \(n\)-descent. A great deal of work has already been done for \(2\) and \(3\)-descents. In this paper the author addresses the question of \(5\) and \(7\)-descents for special types of elliptic curves. Namely, those which admit an isogeny \(E\to E'\) of degree 5 or 7 and also such that \(E'\) admits a \(K\)-rational point of order 5 or 7. A result of J.-P. Serre [Invent. Math. 15, 259-331 (1972; Zbl 0235.14012)] shows that this latter condition is always satisfied when \(E\) and \(E'\) are semi-stable.
The author starts by giving an explicit example of a point of order 5 on \(\text{ Ш}(E/K)\) and then makes a 2-descent, using J. E. Cremona’s program mwrank [mwrank, a program for 2-descent elliptic curves over \(\mathbb Q\), http://www.maths.nottingham.ac.uk/personal/jec/ftp/progs], in order to show that \(E'(\mathbb Q)\cong\mathbb Z/5\mathbb Z\). He then performs similar calculations for all pairs of elliptic curves \(E\) and \(E'\) appearing in J. E. Cremona’s tables [Modular elliptic curve data for conductor up to 5300, http://www.maths.nottingham.ac.uk/personal/jec/ftp/data] of conductor \(\leq 5300\). He explicitly obtains and lists examples for curves for which \(\text{ Ш}(E/\mathbb Q)\cong(\mathbb Z/5\mathbb Z)^2\). Analogous calculations are done for \(n=7\).
Reviewer: Amílcar Pacheco (Rio de Janeiro)

Cited in 5 Reviews
Cited in 28 Documents

MSC:

11G05 Elliptic curves over global fields
14G05 Rational points

Keywords:

elliptic curves; descent; Tate-Shafarevich group

Citations:

Zbl 0235.14012

Software:

ecdata; mwrank
PDFBibTeX XMLCite
\textit{T. Fisher}, J. Eur. Math. Soc. (JEMS) 3, No. 2, 169--201 (2001; Zbl 1007.11031)
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References:

[1] R. Bölling: Die Ordnung der Schafarewitsch-Tate-Gruppe kann beliebig gross wer- den. Math. Nachr. 67 , 157-179 (1975) · Zbl 0314.14008 · doi:10.1002/mana.19750671003
[2] J.W.S. Cassels: Arithmetic on curves of genus 1, VI. The Tate-Shafarevich group can be arbitrarily large. J. Reine Angew. Math. 214/215 , 65-70 (1964) · Zbl 0236.14012 · doi:10.1515/crll.1964.214-215.65
[3] J.W.S. Cassels: Arithmetic on curves of genus 1, VIII. On conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math. 217 , 180-199 (1965) · Zbl 0241.14017 · doi:10.1515/crll.1965.217.180
[4] J.W.S. Cassels: Lectures on elliptic curves. LMSST 24 : Cambridge University Press 1991 · Zbl 0752.14033
[5] J.W.S. Cassels, A. Fröhlich (eds.): Algebraic number theory. Academic Press 1967 · Zbl 0153.07403
[6] I. Connell: Calculating root numbers of elliptic curves over Q . Manuscripta Math. 82 , 93-104 (1994) · Zbl 0805.14017 · doi:10.1007/BF02567689
[7] J.E. Cremona, B. Mazur: Visualizing elements in the Shafarevich-Tate group. Exp. Math. 9 (1), 13-28 (2000) · Zbl 0972.11049 · doi:10.1080/10586458.2000.10504633
[8] J.E. Cremona: Algorithms for modular elliptic curves (second edition). Cambridge University Press 1997
[9] J.E. Cremona: mwrank, a program for 2-descent on elliptic curves over Q . See http://www.maths.nottingham.ac.uk/personal/jec/ftp/progs
[10] J.E. Cremona: Modular elliptic curve data for conductors up to 5300. See http://www.maths.nottingham.ac.uk/personal/jec/ftp/data
[11] T.A. Fisher: On 5 and 7 descents for elliptic curves. Cambridge PhD Thesis (2000)
[12] H. Halberstam, H.-E. Richert: Sieve methods. LMS Monographs 4 : Academic Press 1974 · Zbl 0298.10026
[13] K. Hulek: Projective geometry of elliptic curves. Soc. Math. de France, Astérisque 137 (1986) · Zbl 0602.14024
[14] F. Klein: Über die Transformation siebenter Ordnung der elliptischen Funktionen. Math. Ann. 14 , 428-471 (1879). English translation in [Le] 287-331 · JFM 11.0297.01
[15] K. Kramer: A family of semistable elliptic curves with large Tate-Shafarevich groups. Proc. Am. Math. Soc. 89 , 379-386 (1983) · Zbl 0567.14018 · doi:10.2307/2045480
[16] S. Levy (ed.): The eightfold way, the beauty of Klein’s quartic curve. MSRI Publi- cations 35 , Cambridge University Press 1999 · Zbl 0941.00006
[17] M. DeLong: A formula for the selmer group of a rational three-isogeny (1999) · Zbl 1009.11037 · doi:10.4064/aa105-2-2
[18] B. Mazur: Arithmetic on curves. Bull. Am. Math. Soc. 14 , 207-259 (1986) · Zbl 0593.14021 · doi:10.1090/S0273-0979-1986-15430-3
[19] B. Mazur: On the passage from local to global in number theory. Bull. Am. Math. Soc. 29 , 14-50 (1993) · Zbl 0813.14016 · doi:10.1090/S0273-0979-1993-00414-2
[20] J.S. Milne: Arithmetic duality theorems. Persp. in Math. 1 . Academic Press 1986 · Zbl 0613.14019
[21] P. Monsky: Generalizing the Birch-Stephens theorem. Math. Z. 221 , 415-420 (1996) · Zbl 0853.11048 · doi:10.1007/PL00004518
[22] D. Mumford: On the equations defining abelian varieties. I. Invent. math. 1 , 287-354 (1966) · Zbl 0219.14024 · doi:10.1007/BF01389737
[23] D. Mumford: Abelian varieties. TIFR Studies in Math. 5 : Oxford University Press 1970 · Zbl 0223.14022
[24] C.H. O’Neil: Jacobians of curves of genus one. Harvard PhD thesis (1999)
[25] D.E. Rohrlich: Variation of the root number in families of elliptic curves. Compositio Math. 87 , 119-151 (1993) · Zbl 0791.11026
[26] J.-P. Serre: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. math. 15 , 259-331 (1972) · Zbl 0235.14012 · doi:10.1007/BF01405086
[27] J.-P. Serre: Local fields. GTM 67 . Springer 1979 201
[28] J.H. Silverman: The arithmetic of elliptic curves. GTM 106 . Springer 1986 · Zbl 0585.14026
[29] J.H. Silverman: Advanced topics in the arithmetic of elliptic curves. GTM 151 . Springer 1994 · Zbl 0911.14015
[30] J. Tate: Duality theorems in Galois cohomology over number fields. Proc. Intern. Congress Math. Stockholm (1962), 288-295 · Zbl 0126.07002
[31] J. Tate: The arithmetic of elliptic curves. Invent. math. 23 , 179-206 (1974) · Zbl 0296.14018 · doi:10.1007/BF01389745
[32] J. Vélu: Isogénies entre courbes elliptiques. C. R. Acad. Sci. Paris 273 , 238-241 (1971) · Zbl 0225.14014
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