the Grössencharacter associated to a CM elliptic curve

Let K be a quadratic imaginary field and let A/F be an elliptic curveMathworldPlanetmath defined over a number fieldMathworldPlanetmath F (such that KF), with complex multiplicationMathworldPlanetmath by K. The so-called ‘Main Theorem of Complex Multiplication’ ([2], Thm. 8.2) implies the existence of a Grössencharacter of F, ψA/F:𝒜F associated to the curve A/F 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 F of good reduction for A/F, i.e. the reductionPlanetmathPlanetmath A~/F of A modulo is smooth. There exists a Grössencharacter of F, ψA/F:AFC, such that:

  1. 1.

    ψA/F is unramified at a prime 𝔔 of F if and only if A/F has good reduction at 𝔔;

  2. 2.

    ψA/F() belongs to 𝒪K, thus multiplication by [ψA/F()] is a well defined endomorphismPlanetmathPlanetmath of A/F. Moreover NF()=NK(ψA/F());

  3. 3.

    the following diagram is commutativePlanetmathPlanetmathPlanetmath


    where ϕ:A~A~ be the NF()-power Frobenius mapPlanetmathPlanetmath and the vertical maps are reduction mod ;

  4. 4.

    let |A~(𝒪F/)| be the number of points in A~ over the finite fieldMathworldPlanetmath 𝒪F/ and put a=NF()+1-|A~(𝒪F/)|. Then

  5. 5.

    (due to Deuring) let L(A/F,s) be the L-function associated to the elliptic curve A/F. If KF then L(A/F,s)=L(ψA/F,s)L(ψA/F¯,s). If KF, and F=FK, then L(E/F,s)=L(ψA/F,s).

In particular, if hK=1 then A is defined over K (actually, it may be defined over ), ψA/K() is a generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of (by part (2), and the explicit generator can be pinned down using part (4)). Thus, if e is the number of roots of unityMathworldPlanetmath in K, then ψA/Kk()=αk where α is any generator of . Also, by part (5), L(A/,s)=L(ψA/K,s).


  • 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