On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields.

*(English)*Zbl 0225.14015As Hecke showed, every \(L\)-function of an imaginary quadratic field \(K\) with a Größencharacter \(\lambda\) is the Mellin transform of a cusp form \(f(z)\) belonging to a certain congruence subgroup \(\Gamma\) of \(\mathrm{SL}_2(\mathbb Z)\). After a suitable normalization of \(\lambda\), \(\Gamma\) is given by

\[ \Gamma = \left\{ \begin{pmatrix}{a &b\\ c &d}\end{pmatrix} \in \mathrm{SL}_2(\mathbb Z)\mid a\equiv d\equiv 1,\;c\equiv 0\pmod{D\cdotN(c)}\right\}, \]

where \(-D\) is the discriminant of \(K\) and \(c\) is the conductor of \(\lambda\). If the weight of \(f(z)\) is \(2\), \(f(z)\,dz\) is a differential form of the first kind on the compactification \((H/\Gamma)^*\) of the quotient \(H/\Gamma\), where \(H\) denotes the upper half complex plane. Denote by \(\mathrm{Jac}(H/\Gamma)\) the Jacobian variety of \((H/\Gamma)^*\), and identify the tangent space of \(\mathrm{Jac}(H/\Gamma)\) at the origin with the space of all differential forms of the first kind on \((H/\Gamma)^*\). Let \(A\) be the smallest abelian subvariety of \(\mathrm{Jac}(H/\Gamma)\) that has \(f(z)\,dz\) as a tangent at the origin. The author proves the following result:

The abelian variety \(A\) is a product of copies of an elliptic curve whose endomorphism algebra is isomorphic to \(K\).

Let \(E\) be an elliptic curve defined over \(\mathbb Q\) such that \(\mathrm{End}_{\mathbb Q}(E)\) is isomorphic to \(K\). As Deuring showed, the zeta-function of \(E\) over \(\mathbb Q\) is exactly the \(L\)-function of a certain Größencharacter \(\lambda\) of \(K\). The author obtains an abelian variety \(A\) by a procedure described by the following scheme:

elliptic curve \(E\mapsto\) zeta-function with a Größencharacter \(\lambda\mapsto\) cusp form \(f(z)\mapsto\) abelian subvariety \(A\) of \(\mathrm{Jac}(H/\Gamma')\),

where the group \(\Gamma'\) is of the form:

\[ \Gamma' = \left\{ \begin{pmatrix}{a &b\\ c &d}\end{pmatrix} \in \mathrm{SL}_2(\mathbb Z)\mid c\equiv 0\pmod{D\cdotN(c)}\right\}, \]

and proves the following result:

\(A\) is an elliptic curve isogenous to \(E\) over \(\mathbb Q\).

\[ \Gamma = \left\{ \begin{pmatrix}{a &b\\ c &d}\end{pmatrix} \in \mathrm{SL}_2(\mathbb Z)\mid a\equiv d\equiv 1,\;c\equiv 0\pmod{D\cdotN(c)}\right\}, \]

where \(-D\) is the discriminant of \(K\) and \(c\) is the conductor of \(\lambda\). If the weight of \(f(z)\) is \(2\), \(f(z)\,dz\) is a differential form of the first kind on the compactification \((H/\Gamma)^*\) of the quotient \(H/\Gamma\), where \(H\) denotes the upper half complex plane. Denote by \(\mathrm{Jac}(H/\Gamma)\) the Jacobian variety of \((H/\Gamma)^*\), and identify the tangent space of \(\mathrm{Jac}(H/\Gamma)\) at the origin with the space of all differential forms of the first kind on \((H/\Gamma)^*\). Let \(A\) be the smallest abelian subvariety of \(\mathrm{Jac}(H/\Gamma)\) that has \(f(z)\,dz\) as a tangent at the origin. The author proves the following result:

The abelian variety \(A\) is a product of copies of an elliptic curve whose endomorphism algebra is isomorphic to \(K\).

Let \(E\) be an elliptic curve defined over \(\mathbb Q\) such that \(\mathrm{End}_{\mathbb Q}(E)\) is isomorphic to \(K\). As Deuring showed, the zeta-function of \(E\) over \(\mathbb Q\) is exactly the \(L\)-function of a certain Größencharacter \(\lambda\) of \(K\). The author obtains an abelian variety \(A\) by a procedure described by the following scheme:

elliptic curve \(E\mapsto\) zeta-function with a Größencharacter \(\lambda\mapsto\) cusp form \(f(z)\mapsto\) abelian subvariety \(A\) of \(\mathrm{Jac}(H/\Gamma')\),

where the group \(\Gamma'\) is of the form:

\[ \Gamma' = \left\{ \begin{pmatrix}{a &b\\ c &d}\end{pmatrix} \in \mathrm{SL}_2(\mathbb Z)\mid c\equiv 0\pmod{D\cdotN(c)}\right\}, \]

and proves the following result:

\(A\) is an elliptic curve isogenous to \(E\) over \(\mathbb Q\).

Reviewer: Şerban A. Basarab (Bucureşti)

##### MSC:

11F03 | Modular and automorphic functions |

14H05 | Algebraic functions and function fields in algebraic geometry |

14H52 | Elliptic curves |

11G05 | Elliptic curves over global fields |

11G10 | Abelian varieties of dimension \(> 1\) |

11G15 | Complex multiplication and moduli of abelian varieties |

14M17 | Homogeneous spaces and generalizations |

14K22 | Complex multiplication and abelian varieties |

14H40 | Jacobians, Prym varieties |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

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##### References:

[1] | On the zeta-function of an abelian variety with complex multiplication |

[2] | Publ. Math. Soc. Japan 11 (1971) |

[3] | DOI: 10.2307/1970507 · Zbl 0142.05402 |

[4] | DOI: 10.2969/jmsj/01010001 · Zbl 0081.07603 |

[5] | DOI: 10.1007/BF01361551 · Zbl 0158.08601 |

[6] | DOI: 10.1007/BF01181192 · JFM 54.0405.01 |

[7] | Math. Ann. 97 pp 210– (1926) |

[8] | pp 85– (1953) |

[9] | DOI: 10.1007/BF01594160 · Zbl 0015.40202 |

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