Abstract
For any field K, a “K-curve” is an elliptic curve E defined over some finite separable extension of K such that every Galois conjugate of E is isogenous with E over a separable closure of K. A K-curve must either have complex multiplication (CM) or be isogenous to a collection of K-curves parametrized by a K-rational point on some modular curve X* (N). We give the first proof of this result, obtained in 1993 but not heretofore published. We also indicate some of the reasons for interest in K-curves, and in particular in Q-curves, and give some computational examples and open questions.
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References
Abramovic, D.: Subvarieties of Abelian Varieties and of Jacobians of Curves. Ph. D. thesis, Harvard 1991.
Abramovic, D., Harris, J.: Abelian varieties and curves in Wd(C). Compositio Math. 78 (1991) #2, 227–238.
Birch, B.J., Kuyk, W., ed.: Modular Functions of One Variable IV. Berlin: Springer 1975 (LNM 476).
Chabauty, C.: Sur les points rationnels des variétés algébriques de genre supérieur à l’unité. C. R. Acad. Sci. Paris 212 (1941), 882–885.
Elkies, N.D.: The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89 (1987) #3, 561–568.
Elkies, N.D.: Supersingular primes for elliptic curves over real number fields, Compositio Math. 72 (1989) #2, 165–172.
Elkies, N.D.: The automorphism group of the modular curve Xo(63), Compositio Math. 74 (1990) #2, 203–208.
Elkies, N.D.: Remarks on elliptic K-curves. Preprint, 1993.
Elkies, N.D.: Elliptic and modular curves over finite fields and related computational issues. Pages 21–76 in Computational Perspectives on Number Theory: Proceedings of a Conference in Honor of A.O.L. Atkin (D.A. Buell and J.T. Teitelbaum, eds.; Providence: American Mathematical Society, 1998).
Ellenberg, J.S.: Galois representations attached to Q-curves and the generalized Fermat equation A4 + B2 = CP. Amer. J. Math., to appear.
Ellenberg, J.S., Skinner, C.: On the modularity of Q-curves. Duke Math. J. 109 (2001) #1, 97–122.
Faltings, G.: Diophantine approximation on abelian varieties. Ann. of Math. (2) 133 (1991) #3, 549–576.
Flynn, V.E.: Coverings of curves of genus 2. Pages 65–84 in Algorithmic Number Theory (Leiden, 2000) (Proceedings of ANTS-IV, W. Bosma, ed.), Berlin: Springer 2000 Lect. Notes in Computer Sci. 1838.
Galbraith,S.G.: Rational points on.X,-(p).Experimental Math. 8 (1999) #4, 311318.
Galbraith, S.G.: Rational points on X,- RN) and quadratic Q-curves. J. Th. des Nombres de Bordeaux 14 (2002) #1, 205–219.
González, J., Lario, J.-C.: Rational and Elliptic Parametrizations of Q-curves. J. Number Theory 72 (1998), 13–31.
Gross, B.H.: Arithmetic on elliptic curves with complex multiplication. Berlin: Springer 1980 (LNM 776).
Jao, D.: Supersingular primes for rational points on modular curves. Ph. D. thesis, Harvard 2003.
Mazur, B., Swinnerton-Dyer, H.P.F.: Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61.
Momose, F.: Rational points on the modular curves Xsp t (p). Compositio Math. 52 (1984) #1, 115–137.
Momose, F.: Rational points on the modular curves Xó (pr). J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 33 (1986) #3, 441–466.
Momose, F.: Rational points on the modular curves Xó (N). J. Math. Soc. Japan 39 (1987) #2, 269–286.
Ribet, K.A.: Abelian varieties over Q and modular forms. CPAM preprint, Berkeley 9/92. Also pages 53–79 in 1992 Proceedings of KAIST Mathematics Workshop (Taejon: Korea Advanced Institute of Science and Technology).
Ribet, K.A.: Fields of definition of abelian varieties with real multiplication. Pages 107–118 in Arithmetic geometry (Tempe, AZ, 1993; N. Childress and J.W. Jones, eds. = Contemp. Math. 174, Amer. Math. Soc., Providence, RI, 1994).
Serre, J.P.: Répresentations l-adiques. Pages 177–193 in Algebraic number theory: papers contributed for the Kyoto International Symposium (S. Iyanaga, ed.), Tokyo: Japan Society for the Promotion of Science, 1977. [_ #112 (pages 384–400) in OEuvres III (Springer: Berlin 1986).]
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Elkies, N.D. (2004). On Elliptic K-curves. In: Cremona, J.E., Lario, JC., Quer, J., Ribet, K.A. (eds) Modular Curves and Abelian Varieties. Progress in Mathematics, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7919-4_6
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DOI: https://doi.org/10.1007/978-3-0348-7919-4_6
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