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Elliptic curves of conductor $ 11$


Authors: M. K. Agrawal, J. H. Coates, D. C. Hunt and A. J. van der Poorten
Journal: Math. Comp. 35 (1980), 991-1002
MSC: Primary 10D12; Secondary 10B10, 14K07
DOI: https://doi.org/10.1090/S0025-5718-1980-0572871-5
MathSciNet review: 572871
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Abstract: We determine all elliptic curves defined over Q of conductor 11. Firstly, we reduce the problem to one of solving a diophantine equation, namely a certain Thue-Mahler equation. Then we apply recent sharp inequalities for linear forms in the logarithms of algebraic numbers to bound solutions of that equation. Finally, some straightforward computations yield all solutions of the diophantine equation. Our results are in accordance with the conjecture of Taniyama-Weil for conductor 11.


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  • [1] I. O. ANGELL, "A table of complex cubic fields," Bull. London Math. Soc., v. 5, 1973, pp. 37-38; "A table of totally real cubic fields," Math. Comp., v. 30, 1976, pp. 184-187. MR 0318099 (47:6648)
  • [2] A. BAKER & H. DAVENPORT, "The equations $ 3{x^2} - 2 = {y^2}$ and $ 8{x^2} - 7 = {z^2}$," Quart J. Math. Oxford Ser. (2), v. 20, 1969, pp. 129-137. MR 0248079 (40:1333)
  • [3] A. BAKER & D. W. MASSER (Eds.), Transcendence Theory: Advances and Applications, Academic Press, New York, 1977. MR 0457365 (56:15573)
  • [4] B. J. BIRCH & W. KUYK (Eds.), Modular Functions of One Variable. IV, Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin and New York, 1975. MR 0376533 (51:12708)
  • [5] REINHARD BÖLLING, "Elliptische Kurven mit Primzahl führer," Math. Nachr., v. 80, 1977, pp. 253-278. MR 0498609 (58:16698)
  • [6] P. BUNDSCHUH, "Fractions continues et indépendance algébrique en p-adique. Journées arithmétiques de Caen," Astérisque, 41-42, 1977, pp. 179-181. MR 0498412 (58:16538)
  • [7] J. COATES, "An effective p-adic analogue of a theorem of Thue. I; II: The greatest prime factor of a binary form; III: The diophantine equation $ {y^2} = {x^3} + k$," Acta Arith., v. 15, 1969, pp. 279-305; v. 16, 1970, pp. 399-412, 425-435. MR 0263741 (41:8341)
  • [8] T. W. CUSICK, "The Szekeres multidimensional continued fraction," Math. Comp., v. 31, 1977, pp. 280-317. MR 0429765 (55:2775)
  • [9] B. N. DELONE & D. K. FADDEEV, The Theory of Irrationalities of the Third Degree, Transl. Math. Monographs, vol. 10, Amer. Math. Soc., Providence, R. I., 1964. MR 0160744 (28:3955)
  • [10] W. J. ELLISON, Recipes for Solving Diophantine Problems by Baker's Method, Publications mathématiques, Bordeaux, Ann. 1, Fasc. 1, 1972. MR 0401659 (53:5486)
  • [11] R. FRICKE, Die elliptischen Funktionen und ihre Anwendungen. II, Teubner, Leipzig, 1922.
  • [12] STEPHEN GELBART, "Elliptic curves and automorphic representations," Adv. in Math., v. 21, 1976, pp. 235-292. MR 0439754 (55:12640)
  • [13] GÉRARD LIGOZAT, "Courbes modulaires de genre 1," Bull. Soc. Math. France, Mémoire 43, 1975. MR 0422158 (54:10150)
  • [14] GÉRARD LIGOZAT, Courbes Modulaires de Niveau 11. Modular Functions of One Variable. V, Lecture Notes in Math., vol. 601, Springer-Verlag, Berlin and New York, 1977. MR 0463118 (57:3079)
  • [15] J. LOXTON, M. MIGNOTTE, A. J. VAN DER POORTEN & M. WALDSCHMIDT, "Linear forms in logarithms with rational coefficients." (In preparation.)
  • [16] KURT MAHLER, "On a geometrical representation of p-adic numbers," Ann. of Math. (2), v. 41, 1940, pp. 8-56. MR 0001772 (1:295b)
  • [17] KURT MAHLER, Lectures on Diophantine Approximations. Part I: p-Adic Numbers and Roth's Theorem, Univ. of Notre Dame, 1961. MR 0142509 (26:78)
  • [18] B. MAZUR &. P. SWINNERTON-DYER, "Arithmetic of Weil curves," Invent. Math., v. 25, 1974, pp. 1-61. MR 0354674 (50:7152)
  • [19] OLAF NEUMANN, "Zur Reduktion der elliptischen Kurven," Math. Nachr., v. 46, 1970, pp. 285-310. MR 0280495 (43:6215)
  • [20] OLAF NEUMANN, "Die elliptischen Kurven mit den Führern $ {3.2^m}$ und $ {9.2^m}$," Math. Nachr., v. 48, 1970, pp. 387-389. MR 0288122 (44:5320)
  • [21] OLAF NEUMANN, "Elliptische Kurven mit vorgeschriebenen Reduktionsverhalten. I," Math. Nachr., v. 49, 1971, pp. 107-123. MR 0337999 (49:2767a)
  • [22] OLAF NEUMANN, "Elliptische Kurven mit vorgeschriebenen Reduktionsverhalten. II," Math. Nachr., v. 56, 1973, pp. 269-280. MR 0338000 (49:2767b)
  • [23] A. P. OGG, "Abelian curves of 2-power conductor," Proc. Cambridge Philos. Soc., v. 62, 1966, pp. 143-148. MR 0201436 (34:1320)
  • [24] A. P. OGG, "Elliptic curves and wild ramification," Amer. J. Math., v. 89, 1967, pp. 1-21. MR 0207694 (34:7509)
  • [25] A. P. OGG, "Abelian curves of small conductor," J. Reine Angew. Math., v. 226, 1967, pp. 204-215. MR 0210706 (35:1592)
  • [26] TH. SCHNEIDER, Über p-adische Kettenbrüche, Symposia Mathematica IV, Istituto Nazionale di Alta Mathematica, 1970, pp. 181-189. MR 0272720 (42:7601)
  • [27] J.-P. SERRE, Abelian l-Adic Representations and Elliptic Curves, Benjamin, New York, 1968. MR 0263823 (41:8422)
  • [28] J.-P. SERRE, "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques," Invent. Math., v. 15, 1972, pp. 259-331. MR 0387283 (52:8126)
  • [29] BENNETT SETZER, "Elliptic curves of prime conductor," J. London Math. Soc. (2), v. 10, 1975, pp. 367-378. MR 0371904 (51:8121)
  • [30] T. N. SHOREY, A. J. VAN DER POORTEN, R. TIJDEMAN &. A. SCHINZEL, "Applications of the Gel'fond-Baker method to diophantine equations" in Transcendence Theory: Advances and Applications, Chapter 3, pp. 59-77 (Baker & Masser, Eds.), Academic Press, New York, 1977. MR 0472689 (57:12383)
  • [31] H. P. F. SWINNERTON-DYER & B. J. BIRCH, "Elliptic curves and modular functions," in Modular Functions of One Variable. IV, Chapter 2, pp. 2-32 (Birch & Kuyk, Eds.), Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin and New York, 1975. MR 0384813 (52:5685)
  • [32] G. SZEKERES, "Multidimensional continued fractions," Ann. Univ. Sci. Budapest, Eötvös Sect. Math., v. 13, 1970, pp. 113-140. MR 0313198 (47:1753)
  • [33] JOHN T. TATE, "The arithmetic of elliptic curves," Invent. Math., v. 23, 1974, pp. 179-206. MR 0419359 (54:7380)
  • [34] J. TATE, "Algorithm for determining the type of a singular fibre in an elliptic pencil," in Modular Functions of One Variable. IV, Chapter 3, pp. 33-52 (Birch & Kuyk, Eds.), Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin and New York, 1975. MR 0393039 (52:13850)
  • [35] JACQUES VELU, "Courbes elliptiques sur Q ayant bonne réduction en dehors de 11," C. R. Acad. Sci. Paris Sér. A-B, v. 273, 1971, pp. A73-A75. MR 0316457 (47:5004)
  • [36] A. WEIL, "Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen," Math. Ann., v. 168, 1967, pp. 149-156. MR 0207658 (34:7473)

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DOI: https://doi.org/10.1090/S0025-5718-1980-0572871-5
Article copyright: © Copyright 1980 American Mathematical Society

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