The Group Structure On An Elliptic Curve

Date

2019-05-01

Authors

Alvarez Olson, Nathan

Journal Title

Journal ISSN

Volume Title

Publisher

Abstract

An elliptic curve is defined most generally as the solution setEpKqof a non-singularcubic polynomialfwith coefficients in a fieldK. Via the theory of elliptic functions andthe Weierstrass?-function specifically, a bijection is established between the complextorus and theC-points of an elliptic curve. This map endowsEpCqwith an abeliangroup structure, where the group law yields a nice geometric interpretation when thecurve is embedded inCP2. Mordell’s theorem, of which a special case is proved, impliesEpQqis finitely generated. Lastly, the Nagell-Lutz theorem, which places a divisibilitycondition on they-coordinates of points inGthat impliesGis finite, is proved.

Department

Description

LCSH Subject Headings

Citation