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Families of elliptic curves with the same mod 8 representations

Published online by Cambridge University Press:  19 April 2017

ZEXIANG CHEN
ZEXIANG CHEN*
Affiliation:
Nomura International Plc, One Angel Lane, London, EC4R 3AB e-mail: [email protected]
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Abstract

We compute certain twists of the classical modular curve X(8). Searching for rational points on these twists enables us to find non-trivial pairs non-isogenous elliptic curves over ℚ whose 8-torsion subgroups are isomorphic as Galois modules. We also show that there are infinitely many examples over ℚ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 1 , July 2018 , pp. 137 - 162
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[BCP] Bosma, W., Cannon, J. and Playoust, C. The Magma algebra system I: The user language. J. Symb. Comb. 24 (1997), 235–265, (See also the Magma home page at http://magma.maths.usyd.edu.au/magma/.Google Scholar
[BD] Bruin, N. and Doerken, K. The Arithmetic of genus two curves with (4, 4)-split Jacobians. http://arxiv.org/pdf/0902.3480.pdf.Google Scholar
[F1] Fisher, T.A. The Hessian of a genus one curve. Proc. Lond. Math. Soc. (3) 104 (2012), 613–648.Google Scholar
[F2] Fisher, T.A. Invariant theory for the elliptic normal quintic, I. Twists of X(5) Math. Ann. 356 (2013), no. 2, 589–616.Google Scholar
[F3] Fisher, T.A. On families of 7 and 11-congruent elliptic curves. LMS J. Comput. Math. 17 (2014), no. 1, 536–564.Google Scholar
[F4] Fisher, T.A. On families of 9-congruent elliptic curves. Acta Arith. 171 (2015), no. 4, 371–387.CrossRefGoogle Scholar
[HK] Halberstadt, E. and Kraus, A. On the modular curves YE(7). Math. Comp. 69 (2000), no. 231, 1193–1206.Google Scholar
[KS] Kani, E. and Schanz, W. Modular Diagonal Quotient Surfaces. Math. Z. 227 (1998), no. 2, 337–366.CrossRefGoogle Scholar
[KO] Kraus, A. and Oesterle, J. Sur une question de B. Mazur. Math. Ann. 293 (1992), no. 2, 259–275.Google Scholar
[M] Mazur, B. Rational isogenies of prime degree. Invent. Math. 44 (1978), 129–162.Google Scholar
[MT] Schütt, M. and Shioda, T. Elliptic surfaces. http://arxiv.org/pdf/0907.0298.pdf.Google Scholar
[PSS] Poonen, B., Schaefer, E.F. and Stoll, M. Twists of X(7) and primitive solutions to x 2 + y 3 = z 7. Duke Math. J. 137 (2007), no. 1, 103–158.Google Scholar
[RS] Rubin, K. and Silverberg, A. Families of elliptic curves with constant mod p representations. In Elliptic Curves, Modular Forms and Fermat's Last Theorem (Hong Kong, 1993), eds. Coates, John, Yau, S. T. (International Press, Cambridge, Massachusetts, 1995), 148–161; Second edition: Elliptic Curves, Modular Forms and Fermat's Last Theorem (1997), 296–309.Google Scholar
[S1] Silverberg, A. Explicit families of elliptic curves with prescribed mod N representations. In Modular Forms and Fermat's Last Theorem, eds. Cornell, Gary, Silverman, Joseph H., Stevens, Glenn (Springer–Verlag, Berlin-Heidelberg-New York 1997), 447–461.Google Scholar
[S2] Silverman, J.H. The Arithmetic of Elliptic Curves (Springer-Verlag GTM 106, 1986), Expanded 2nd Edition (2009)CrossRefGoogle Scholar