Twists of elliptic curves

In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic and quartic twists. The curve and its twists have the same j-invariant.

Applications of twists include cryptography,[1] the solution of Diophantine equations,[2][3] and when generalized to hyperelliptic curves, the study of the Sato–Tate conjecture.[4]

  1. ^ Bos, Joppe W.; Halderman, J. Alex; Heninger, Nadia; Moore, Jonathan; Naehrig, Michael; Wustrow, Eric (2014). "Elliptic Curve Cryptography in Practice". In Christin, Nicolas; Safavi-Naini, Reihaneh (eds.). Financial Cryptography and Data Security. Lecture Notes in Computer Science. Vol. 8437. Berlin, Heidelberg: Springer. pp. 157–175. doi:10.1007/978-3-662-45472-5_11. ISBN 978-3-662-45471-8. Retrieved 2022-04-10.
  2. ^ Mazur, B.; Rubin, K. (September 2010). "Ranks of twists of elliptic curves and Hilbert's tenth problem". Inventiones Mathematicae. 181 (3): 541–575. arXiv:0904.3709. Bibcode:2010InMat.181..541M. doi:10.1007/s00222-010-0252-0. ISSN 0020-9910. S2CID 3394387.
  3. ^ Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (2007-03-15). "Twists of X(7) and primitive solutions to x2+y3=z7". Duke Mathematical Journal. 137 (1). arXiv:math/0508174. doi:10.1215/S0012-7094-07-13714-1. ISSN 0012-7094. S2CID 2326034.
  4. ^ Lombardo, Davide; Lorenzo García, Elisa (February 2019). "Computing twists of hyperelliptic curves". Journal of Algebra. 519: 474–490. arXiv:1611.04856. Bibcode:2016arXiv161104856L. doi:10.1016/j.jalgebra.2018.08.035. S2CID 119143097.