Email this article Versal families of elliptic curves with rational 3-torsion
- Authors: Bekker B.M.1, Zarhin Y.G.2
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Affiliations:
- St. Petersburg State University, Mathematics and Mechanics Faculty
- Department of Mathematics, Pennsylvania State University
- Issue: Vol 212, No 3 (2021)
- Pages: 6-19
- Section: Articles
- URL: https://journal-vniispk.ru/0368-8666/article/view/142349
- DOI: https://doi.org/10.4213/sm9429
- ID: 142349
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Abstract
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
About the authors
Boris Mikhailovich Bekker
St. Petersburg State University, Mathematics and Mechanics Faculty
Email: bekker@pdmi.ras.ru
Candidate of physico-mathematical sciences, Associate professor
Yuri Gennad'evich Zarhin
Department of Mathematics, Pennsylvania State University
Email: zarhin@math.psu.edu
Doctor of physico-mathematical sciences, Professor
References
- Б. М. Беккер, Ю. Г. Зархин, “Деление на $2$ рациональных точек на эллиптических кривых”, Алгебра и анализ, 29:4 (2017), 196–239
- 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
- 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
- D. S. Kubert, “Universal bounds on the torsion of elliptic curves”, Proc. London Math. Soc. (3), 33:2 (1976), 193–237
- 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
- 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
- K. Rubin, A. Silverberg, “Mod $2$ representations of elliptic curves”, Proc. Amer. Math. Soc., 129:1 (2001), 53–57
- 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
- J. H. Silverman, J. Tate, Rational points on elliptic curves, Undergrad. Texts Math., Springer-Verlag, New York, 1992, x+281 pp.
- A. Wiles, “Modular elliptic curves and Fermat's last theorem”, Ann. of Math. (2), 141:3 (1995), 443–551
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