On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields. (English) Zbl 0225.14015

Shimura, Goro

Nagoya Math. J. 43, 199-208 (1971).

As 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 \(\mathbb Q\).

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)

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 · doi:10.2307/1970507
[4]	DOI: 10.2969/jmsj/01010001 · Zbl 0081.07603 · doi:10.2969/jmsj/01010001
[5]	DOI: 10.1007/BF01361551 · Zbl 0158.08601 · doi:10.1007/BF01361551
[6]	DOI: 10.1007/BF01181192 · JFM 54.0405.01 · doi:10.1007/BF01181192
[7]	Math. Ann. 97 pp 210– (1926)
[8]	pp 85– (1953)
[9]	DOI: 10.1007/BF01594160 · Zbl 0015.40202 · doi:10.1007/BF01594160