Abstract
In Part 1 we explain how to construct families of elliptic curves with the same mod 3, 4, or 5 representation as that of a given elliptic curve over Q. In §4 we give equations for the families in the mod 4 case. The mod 3 and mod 5 cases were given in [9] (see also [8]). The results remain true (with the same proofs) with the field of rational numbers replaced by any field whose characteristic does not divide the level.
Keywords
- Elliptic Curve
- Elliptic Curf
- Number Field
- Prime Divisor
- Cyclic Subgroup
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References
J. E. Cremona, Algorithms for modular elliptic curves, Cambridge Univ. Press, Cambridge, 1992.
F. Diamond, On deformation rings and Hecke rings,preprint.
F. Diamond, K. Kramer, Modularity of a family of elliptic curves, Math. Research Letters 2 (1995), 299–305.
A. Grothendieck, Modèles de Néron et monodromie,in Groupes de monodromie en géometrie algébrique, SGA7 I, A. Grothendieck, ed., Lecture Notes in Math. 288, Springer, Berlin-Heidelberg-New York, 1972, pp. 313–523.
D. S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (1976), 193–237.
B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES 47 (1977), 133–186.
K. Rubin, Modularity of mod 5 representations,this volume.
K. Rubin, A. Silverberg, A report on Wiles’ Cambridge lectures, Bull. Amer. Math. Soc. (N. S.) 31, no. 1 (1994), 15–38.
K. Rubin, A. Silverberg, Families of elliptic curves with constant mod p representations, in Conference on Elliptic Curves and Modular Forms, Hong Kong, December 18–21, 1993, Intl. Press, Cambridge, Massachusetts, 1995, pp. 148–161.
J-P. Serre, Cohornologie galoisienne, Lecture Notes in Mathematics 5, Springer-Verlag, Berlin-New York, 1965.
J-P. Serre, J. Tate, Good reduction of abelian varieties,Ann. of Math. 88 (1968), 492–517.
G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, 1971.
T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59.
T. Shioda, On rational points of the generic elliptic curve with level N structure over the field of modular functions of level N, J. Math. Soc. Japan 25 (1973), 144–157.
A. Silverberg, Yu. G. Zarhin, Semistable reduction and torsion subgroups of abelian varieties, Ann. Inst. Fourier 45 (1995), 403–420.
A. Silverberg, Yu. G. Zarhin, Variations on a theme of Minkowski and Serre, J. Pure and Applied Algebra 111 (1996), 285–302.
J. Silverman, The Arithmetic of Elliptic Curves,Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1986.
R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553–572.
A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995), 443–551.
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Silverberg, A. (1997). Explicit Families of Elliptic Curves with Prescribed Mod N Representations. In: Cornell, G., Silverman, J.H., Stevens, G. (eds) Modular Forms and Fermat’s Last Theorem. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1974-3_15
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DOI: https://doi.org/10.1007/978-1-4612-1974-3_15
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