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’ ([2], 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 ([2], Thm. 9.1, Prop. 10.4, Cor. 10.4.1).
Let be a prime of of good reduction for
, i.e. the reduction of modulo is
smooth. There exists a Grössencharacter of ,
, such that:
-
1.
is unramified at a prime of if and only if has good reduction at ;
-
2.
belongs to , thus multiplication by is a well defined endomorphism
of . Moreover ;
-
3.
the following diagram is commutative
where be the -power Frobenius map
and the vertical maps are reduction mod ;
-
4.
let be the number of points in over the finite field
and put . Then
-
5.
(due to Deuring) let be the -function associated to the elliptic curve . If then . If , and , then .
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),
.
References
- 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 |
---|---|
Canonical name | TheGrossencharacterAssociatedToACMEllipticCurve |
Date of creation | 2013-03-22 15:45:29 |
Last modified on | 2013-03-22 15:45:29 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 4 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 11G05 |
Related topic | Grossencharacter |
Related topic | EllipticCurve |
Defines | grossencharacter associated to an elliptic curve |