Mod 4 Galois representations and elliptic curves

Holden, Christopher

doi:10.1090/S0002-9939-07-08899-5

Mod 4 Galois representations and elliptic curves
HTML articles powered by AMS MathViewer

by Christopher Holden PDF

Proc. Amer. Math. Soc. 136 (2008), 31-39 Request permission

Abstract:

Galois representations $\bar {\rho }: G_{\mathbb Q} \rightarrow GL_{2}(\mathbb Z/n)$ with cyclotomic determinant all arise from the $n$-torsion of elliptic curves for $n=2,3,5$. For $n=4$, we show the existence of more than a million such representations which are surjective and do not arise from any elliptic curve.

References

Clemens Adelmann, The decomposition of primes in torsion point fields, Lecture Notes in Mathematics, vol. 1761, Springer-Verlag, Berlin, 2001. MR 1836119, DOI 10.1007/b80624
Frank Calegari, Mod $p$ representations on elliptic curves, Pacific J. Math. 225 (2006), no. 1, 1–11. MR 2233722, DOI 10.2140/pjm.2006.225.1
Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier, Constructing complete tables of quartic fields using Kummer theory, Math. Comp. 72 (2003), no. 242, 941–951. MR 1954977, DOI 10.1090/S0025-5718-02-01452-7
J. E. Cremona, Classical invariants and 2-descent on elliptic curves, J. Symbolic Comput. 31 (2001), no. 1-2, 71–87. Computational algebra and number theory (Milwaukee, WI, 1996). MR 1806207, DOI 10.1006/jsco.1998.1004
Luis Dieulefait, Existence of nonelliptic mod $l$ Galois representations for every $l>5$, Experiment. Math. 13 (2004), no. 3, 327–329. MR 2103330, DOI 10.1080/10586458.2004.10504544
E. Goins, Elliptic curves and icosahedral Galois representations, Stanford University Thesis, 1999.
Karl Rubin, Modularity of mod $5$ representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 463–474. MR 1638489
N. I. Shepherd-Barron and R. Taylor, $\textrm {mod}\ 2$ and $\textrm {mod}\ 5$ icosahedral representations, J. Amer. Math. Soc. 10 (1997), no. 2, 283–298. MR 1415322, DOI 10.1090/S0894-0347-97-00226-9