the Grössencharacter associated to a CM elliptic curve
Let be a quadratic imaginary field and let be an elliptic curve defined over a number field (such that ), with complex multiplication by . The so-called ‘Main Theorem of Complex Multiplication’ (, Thm. 8.2) implies the existence of a Grössencharacter of , associated to the curve satisfying several interesting properties which we collect in the following statement.
Theorem (, Thm. 9.1, Prop. 10.4, Cor. 10.4.1).
In particular, if then is defined over (actually, it may be defined over ), is a generator of (by part (2), and the explicit generator can be pinned down using part (4)). Thus, if is the number of roots of unity in , then where is any generator of . Also, by part (5), .
- 1 J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York.
- 2 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
|Title||the Grössencharacter associated to a CM elliptic curve|
|Date of creation||2013-03-22 15:45:29|
|Last modified on||2013-03-22 15:45:29|
|Last modified by||alozano (2414)|
|Defines||grossencharacter associated to an elliptic curve|