Periods of quadratic twists of elliptic curves

Pal, Vivek

doi:10.1090/S0002-9939-2011-11014-1

Periods of quadratic twists of elliptic curves
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by Vivek Pal; with an appendix by Amod Agashe

Proc. Amer. Math. Soc. 140 (2012), 1513-1525

DOI: https://doi.org/10.1090/S0002-9939-2011-11014-1

Published electronically: September 2, 2011

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Abstract:

In this paper we prove a relation between the period of an elliptic curve and the period of its real and imaginary quadratic twists. This relation is often misstated in the literature.

References

Amod Agashe, Squareness in the special $L$-value and special $L$-values of twists, Int. J. Number Theory 6 (2010), no. 5, 1091–1111. MR 2679458, DOI 10.1142/S1793042110003393
Amod Agashe, Kenneth Ribet, and William A. Stein, The Manin constant, Pure Appl. Math. Q. 2 (2006), no. 2, Special Issue: In honor of John H. Coates., 617–636. MR 2251484, DOI 10.4310/PAMQ.2006.v2.n2.a11
Amod Agashe and William Stein, Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Comp. 74 (2005), no. 249, 455–484. With an appendix by J. Cremona and B. Mazur. MR 2085902, DOI 10.1090/S0025-5718-04-01644-8
Ian Connell, Elliptic curve handbook, preprint, available at http://www.ucm.es/ BUCM/mat/doc8354.pdf.
Soonhak Kwon, Torsion subgroups of elliptic curves over quadratic extensions, J. Number Theory 62 (1997), no. 1, 144–162. MR 1430007, DOI 10.1006/jnth.1997.2036
Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. MR 1112552, DOI 10.1007/978-3-642-58227-1
Ju. I. Manin, Cyclotomic fields and modular curves, Uspehi Mat. Nauk 26 (1971), no. 6(162), 7–71 (Russian). MR 0401653
Ken Ono and Christopher Skinner, Fourier coefficients of half-integral weight modular forms modulo $l$, Ann. of Math. (2) 147 (1998), no. 2, 453–470. MR 1626761, DOI 10.2307/121015
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original. MR 1329092
Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8