On ranks of twists of elliptic curves and power-free values of binary forms

Stewart, C. L.; Top, J.

doi:10.1090/S0894-0347-1995-1290234-5

On ranks of twists of elliptic curves and power-free values of binary forms
HTML articles powered by AMS MathViewer

by C. L. Stewart and J. Top

J. Amer. Math. Soc. 8 (1995), 943-973

DOI: https://doi.org/10.1090/S0894-0347-1995-1290234-5

PDF | Request permission

References

Jan Stienstra and Frits Beukers, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic $K3$-surfaces, Math. Ann. 271 (1985), no. 2, 269–304. MR 783555, DOI 10.1007/BF01455990
B. J. Birch and N. M. Stephens, The parity of the rank of the Mordell-Weil group, Topology 5 (1966), 295–299. MR 201379, DOI 10.1016/0040-9383(66)90021-8
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
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108. MR 179168, DOI 10.1515/crll.1965.218.79
H. P. F. Swinnerton-Dyer and B. J. Birch, Elliptic curves and modular functions, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 2–32. MR 0384813
Armand Brumer, The average rank of elliptic curves. I, Invent. Math. 109 (1992), no. 3, 445–472. MR 1176198, DOI 10.1007/BF01232033
Armand Brumer and Oisín McGuinness, The behavior of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 375–382. MR 1044170, DOI 10.1090/S0273-0979-1990-15937-3
Maurice Craig, A type of class group for imaginary quadratic fields, Acta Arith. 22 (1973), 449–459. (errata insert). MR 318098, DOI 10.4064/aa-22-4-449-459
Maurice Craig, A construction for irregular discriminants, Osaka Math. J. 14 (1977), no. 2, 365–402. MR 450226

Néron models, Tate curves and Mestre’s technique of using elliptic curves to construct quadratic fields with non-trivial

-rank

On the number of integers which can be represented by a binary form

13

G. Frey, On the Selmer group of twists of elliptic curves with $\textbf {Q}$-rational torsion points, Canad. J. Math. 40 (1988), no. 3, 649–665. MR 960600, DOI 10.4153/CJM-1988-028-9
M. Fried, Constructions arising from Néron’s high rank curves, Trans. Amer. Math. Soc. 281 (1984), no. 2, 615–631. MR 722765, DOI 10.1090/S0002-9947-1984-0722765-7
Dorian Goldfeld, Conjectures on elliptic curves over quadratic fields, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 108–118. MR 564926
F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), no. 1, 1–23. MR 1080648, DOI 10.1090/S0894-0347-1991-1080648-7
George Greaves, Power-free values of binary forms, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 169, 45–65. MR 1150469, DOI 10.1093/qmath/43.1.45
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
Taira Honda, Isogenies, rational points and section points of group varieties, Jpn. J. Math. 30 (1960), 84–101. MR 155828, DOI 10.4099/jjm1924.30.0_{8}4
C. Hooley, On the power free values of polynomials, Mathematika 14 (1967), 21–26. MR 214556, DOI 10.1112/S002557930000797X
C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge Tracts in Mathematics, No. 70, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0404173
Jun-ichi Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), 612–649. MR 114819, DOI 10.2307/1970233
Kenneth Kramer, Arithmetic of elliptic curves upon quadratic extension, Trans. Amer. Math. Soc. 264 (1981), no. 1, 121–135. MR 597871, DOI 10.1090/S0002-9947-1981-0597871-8
K. Kramer and J. Tunnell, Elliptic curves and local $\varepsilon$-factors, Compositio Math. 46 (1982), no. 3, 307–352. MR 664648
J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 409–464. MR 0447191

Heights of Heegner points in a family of elliptic curves

Kurt Mahler, Zur Approximation algebraischer Zahlen. I, Math. Ann. 107 (1933), no. 1, 691–730 (German). MR 1512822, DOI 10.1007/BF01448915
Kurt Mahler, Zur Approximation algebraischer Zahlen. III, Acta Math. 62 (1933), no. 1, 91–166 (German). Über die mittlere Anzahl der Darstellungen grosser Zahlen durch binäre Formen. MR 1555382, DOI 10.1007/BF02393603
Liem Mai, The analytic rank of a family of elliptic curves, Canad. J. Math. 45 (1993), no. 4, 847–862. MR 1227663, DOI 10.4153/CJM-1993-048-9
Jean-François Mestre, Courbes elliptiques et groupes de classes d’idéaux de certains corps quadratiques, J. Reine Angew. Math. 343 (1983), 23–35 (French). MR 705875, DOI 10.1515/crll.1983.343.23
Jean-François Mestre, Rang de courbes elliptiques d’invariant donné, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 12, 919–922 (French, with English summary). MR 1168325
Jean-François Mestre, Courbes elliptiques de rang $\geq 12$ sur $\textbf {Q}(t)$, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 4, 171–174 (French, with English summary). MR 1122689
L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
Shin Nakano, Construction of pure cubic fields with large $2$-class groups, Osaka J. Math. 25 (1988), no. 1, 161–170. MR 937193
A. Néron, Propriétés arithmétiques de certaines familles de courbes algébriques, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 481–488 (French). MR 0087210
U. Schneiders and H. G. Zimmer, The rank of elliptic curves upon quadratic extension, Computational number theory (Debrecen, 1989) de Gruyter, Berlin, 1991, pp. 239–260. MR 1151868
Chad Schoen, Bounds for rational points on twists of constant hyperelliptic curves, J. Reine Angew. Math. 411 (1990), 196–204. MR 1072980, DOI 10.1515/crll.1990.411.196
R. J. Schoof, Class groups of complex quadratic fields, Math. Comp. 41 (1983), no. 163, 295–302. MR 701640, DOI 10.1090/S0025-5718-1983-0701640-0
Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. MR 1002324, DOI 10.1007/978-3-663-14060-3
Tetsuji Shioda, An infinite family of elliptic curves over $\textbf {Q}$ with large rank via Néron’s method, Invent. Math. 106 (1991), no. 1, 109–119. MR 1123376, DOI 10.1007/BF01243907
Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. MR 703488, DOI 10.1515/crll.1983.342.197
Joseph H. Silverman, Divisibility of the specialization map for families of elliptic curves, Amer. J. Math. 107 (1985), no. 3, 555–565. MR 789655, DOI 10.2307/2374369

The arithmetic of elliptic curves

C. L. Stewart, On the number of solutions of polynomial congruences and Thue equations, J. Amer. Math. Soc. 4 (1991), no. 4, 793–835. MR 1119199, DOI 10.1090/S0894-0347-1991-1119199-X
J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33–52. MR 0393039

The rank of elliptic curves

8

Néron’s proof of the existence of elliptic curves over

with rank at least 11

D. Zagier and G. Kramarz, Numerical investigations related to the $L$-series of certain elliptic curves, J. Indian Math. Soc. (N.S.) 52 (1987), 51–69 (1988). MR 989230