Versal families of elliptic curves with rational 3-torsion
Sbornik. Mathematics, Tome 212 (2021) no. 3, pp. 274-287 Cet article a éte moissonné depuis la source Math-Net.Ru

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For an arbitrary field of characteristic different from 2 and 3, we construct versal families of elliptic curves whose 3-torsion is either rational or isomorphic to $\mathbb Z/3\mathbb Z\oplus \mu_3$ as a Galois module. Bibliography: 10 titles.
Keywords: elliptic curves
Mots-clés : torsion points, Galois modules.
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B. M. Bekker; Yu. G. Zarhin. Versal families of elliptic curves with rational 3-torsion. Sbornik. Mathematics, Tome 212 (2021) no. 3, pp. 274-287. http://geodesic.mathdoc.fr/item/SM_2021_212_3_a1/

[1] B. M. Bekker, Yu. G. Zarhin, “Division by 2 of rational points on elliptic curves”, St. Petersburg Math. J., 29:4 (2018), 683–713 | DOI | MR | Zbl

[2] B. M. Bekker, Yu. G. Zarhin, “Families of elliptic curves with rational torsion points of even order”, Algebraic curves and their applications, Contemp. Math., 724, Amer. Math. Soc., Providence, RI, 2019, 1–32 | DOI | MR | Zbl

[3] B. M. Bekker, Yu. G. Zarhin, “Torsion points of order $ 2g\,{+}\,1 $ on odd degree hyperelliptic curves of genus $ g $”, Trans. Amer. Math. Soc., 373:11 (2020), 8059–8094 | DOI | MR | Zbl

[4] D. S. Kubert, “Universal bounds on the torsion of elliptic curves”, Proc. London Math. Soc. (3), 33:2 (1976), 193–237 | DOI | MR | Zbl

[5] K. Rubin, A. Silverberg, “Families of elliptic curves with constant $\operatorname{mod} p$ representations”, Elliptic curves, modular forms, Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Intl. Press, Cambridge, MA, 1995, 148–161 | MR | Zbl

[6] K. Rubin, A. Silverberg, “Mod 6 representations of elliptic curves”, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Proc. Sympos. Pure Math., 66, Part 1, Amer. Math. Soc., Providence, RI, 1999, 213–220 | MR | Zbl

[7] K. Rubin, A. Silverberg, “Mod $2$ representations of elliptic curves”, Proc. Amer. Math. Soc., 129:1 (2001), 53–57 | DOI | MR | Zbl

[8] A. Silverberg, “Explicit families of elliptic curves with prescribed $\operatorname{mod} N$ representations”, Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 1997, 447–461 | DOI | MR | Zbl

[9] J. H. Silverman, J. Tate, Rational points on elliptic curves, Undergrad. Texts Math., Springer-Verlag, New York, 1992, x+281 pp. | DOI | MR | Zbl

[10] A. Wiles, “Modular elliptic curves and Fermat's last theorem”, Ann. of Math. (2), 141:3 (1995), 443–551 | DOI | MR | Zbl