Determination of elliptic curves with everywhere good reduction over ℚ(√37)

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1. [BW] A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19-62. Zbl0788.11026
2. [BK] A. Brumer and K. Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), 715-743. Zbl0376.14011
3. [Ca] W. Casselman, On abelian varieties with many endomorphisms, and a conjecture of Shimura's, Invent. Math. 12 (1971), 225-236. Zbl0213.47303
4. [CW] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), 223-251. Zbl0359.14009
5. [Co] S. Comalada, Elliptic curves with trivial conductor over quadratic fields, Pacific J. Math. 144 (1990), 233-258. Zbl0732.11028
6. [DR] P. Deligne et M. Rapoport, Les schémas de modules des courbes elliptiques, in: Modular Functions of One Variable II, Lecture Notes in Math. 349, Springer, 1973, 143-316. 
7. [Fr] R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, Leipzig, 1922. Zbl48.0432.01
8. [Ha1] Y. Hasegawa, ℚ-curves over quadratic fields, Manuscripta Math., to appear. 
9. [Ha2] Y. Hasegawa, Table of cuspforms on Γ₁(N) with real quadratic characters, unpublished. 
10. [HHM] Y. Hasegawa, K. Hashimoto and F. Momose, Modular conjecture for ℚ-curves and QM-curves, preprint. Zbl1110.11302
11. [He] E. Hecke, Lectures on the Theory of Algebraic Numbers, Grad. Texts in Math. 77, Springer, 1981. 
12. [Is] H. Ishii, The non-existence of elliptic curves with everywhere good reduction over certain quadratic fields, Japan. J. Math. 12 (1986), 45-52. Zbl0613.14021
13. [KT] T. Kagawa and N. Terai, Squares in Lucas sequences and some Diophantine equations, Manuscripta Math., to appear. Zbl0911.11012
14. [KM] N. Katz and B. Mazur, The Arithmetic Moduli of Elliptic Curves, Princeton Univ. Press, 1985. Zbl0576.14026
15. [Ki1] M. Kida, On a characterization of Shimura's elliptic curve over ℚ(√37), Acta Arith. 77 (1996), 157-171. 
16. [Ki2] M. Kida, private communication. 
17. [KK] M. Kida and T. Kagawa, Nonexistence of elliptic curves with good reduction everywhere over real quadratic fields, J. Number Theory 66 (1997), 201-210. Zbl0895.11023
18. [Kr] A. Kraus, Quelques remarques à propos des invariants c₄,c₆ et Δ d'une courbe elliptique, Acta Arith. 54 (1989), 75-80. 
19. [MSZ] H. H. Müller, H. Ströher and H. G. Zimmer, Torsion groups of elliptic curves with integral j-invariant over quadratic fields, J. Reine Angew. Math. 397 (1989), 100-161. Zbl0662.14020
20. [Pi1] R. G. E. Pinch, Elliptic curves over number fields, Ph.D. thesis, Oxford, 1982. 
21. [Pi2] R. G. E. Pinch, Elliptic curves with good reduction away from 3, Math. Proc. Cambridge Philos. Soc. 101 (1987), 451-459. Zbl0632.14030
22. [Ro] M. I. Rosen, Some confirming instances of the Birch-Swinnerton-Dyer conjecture over biquadratic fields, in: Number Theory, R. A. Mollin (ed.), Walter de Gruyter, 1990, 493-499. Zbl0754.14013
23. [Ru] K. Rubin, The ``main conjectures'' of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25-68. Zbl0737.11030
24. [Sa] P. Satgé, Groupes de Selmer et corps cubiques, J. Number Theory 23 (1986), 294-317. Zbl0601.14027
25. [Ser] J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. 
26. [Set] B. Setzer, Elliptic curves with good reduction everywhere over quadratic fields and having rational j-invariant, Illinois J. Math. 25 (1981), 233-245. Zbl0471.14019
27. [Shim] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Japan 11, Iwanami Shoten and Princeton Univ. Press, 1971. Zbl0221.10029
28. [Shio] K. Shiota, On the explicit models of Shimura's elliptic curves, J. Math. Soc. Japan 38 (1986), 649-659. Zbl0608.14023
29. [Sil] J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, 1986. 
30. [Str] R. J. Stroeker, Reduction of elliptic curves over imaginary quadratic number fields, Pacific J. Math. 108 (1983), 451-463. Zbl0491.14024
31. [Ta] J. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil, in: Modular Functions of One Variable IV, Lecture Notes in Math. 476, Springer, 1975, 33-52. 
32. [TdW] N. Tzanakis and B. M. M. de Weger, On the practical solution of the Thue equation, J. Number Theory 31 (1989), 99-132. Zbl0657.10014
33. [Um] A. Umegaki, A construction of everywhere good ℚ-curves with p-isogeny, preprint. 
34. [dW1] B. M. M. de Weger, A hyperelliptic diophantine equation related to imaginary quadratic number fields with class number 2, J. Reine Angew. Math. 427 (1992), 137-156; Correction: J. Reine Angew. Math. 441 (1993), 217-218. Zbl0738.11032
35. [dW2] B. M. M. de Weger, A Thue equation with quadratic integers as variables, Math. Comp. 64 (1995), 855-861. Zbl0835.11012