Generalized \(\mathbb Q\)-curves and factors of \(J_1(N)\). (English) Zbl 1162.11350
Introduction: In [Arithmetic on elliptic curves with complex multiplication. With an appendix by B. Mazur. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0433.14032)] B. Gross introduced and studied so-called \(\mathbb Q\)-curves, which are defined as follows. Let \(K\) be an imaginary quadratic field and let \(E\) be an elliptic curve with complex multiplication by \(K\) and with \(j\)-invariant \(j(E)\), which is defined over the Hilbert class field \(H=K(j(E))\) of \(K\). Gross calls such a curve \(\mathbb Q\)-curve if it is \(H\)-isogenous to all its \(\text{Gal}(H/K)\)-conjugates \(E^\sigma\). Such curves always exist if the discriminant of \(K\) is odd. If in particular \(\text{disc}(K)= -p\) with \(p\geq 5\) a prime, Gross showed the following. The abelian variety \(R_{H/K}(E)\) obtained by Weil restriction has complex multiplication by a field \(T/K\), \([T\:K]=h=[H\:K]\). If moreover the curve \(E\) descends to the maximal real subfield \(H^+\) of \(H\), this abelian variety descends to \(\mathbb Q\) and is a \(\mathbb Q\)-isogenous factor of some \(J_0(N)\) (i.e. the Jacobian of the modular curve \(X_0(N)\)). More precisely, one may take \(N= -\text{disc}(K)N_{K/Q}({\mathfrak f})\) where \({\mathfrak f}\) denotes the conductor of the Serre-Tate character of \(R_{H/K}(E)\). It thus provides an example for the generalized Shimura-Taniyama conjecture as stated in [J.-P. Serre, Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)].
In this paper we generalize this result to a bigger class of elliptic curves. More precisely, we consider quadratic fields with arbitrary discriminants and other fields \(F\) of definition instead of \(H\). Let \(K\) be an imaginary quadratic field and \(F\) a finite extension of \(K\) together with a fixed embedding \(F\hookrightarrow\mathbb C\).
Definition. The class \({\mathcal S}= {\mathcal S}(F,K)\) consists of the elliptic curves \(E/F\) with CM by \(K\) which satisfy
(S1) The extension \(F(E_{\text{tors}})/K\) is abelian.
The subset \({\mathcal S}^+\) is defined by the descending curves in \({\mathcal S}\), i.e., those satisfying
(S2) There exists an elliptic curve \(E^+/F^+\) such that \(E=E^+\bigtimes_{F^+} F\).
In this paper we generalize this result to a bigger class of elliptic curves. More precisely, we consider quadratic fields with arbitrary discriminants and other fields \(F\) of definition instead of \(H\). Let \(K\) be an imaginary quadratic field and \(F\) a finite extension of \(K\) together with a fixed embedding \(F\hookrightarrow\mathbb C\).
Definition. The class \({\mathcal S}= {\mathcal S}(F,K)\) consists of the elliptic curves \(E/F\) with CM by \(K\) which satisfy
(S1) The extension \(F(E_{\text{tors}})/K\) is abelian.
The subset \({\mathcal S}^+\) is defined by the descending curves in \({\mathcal S}\), i.e., those satisfying
(S2) There exists an elliptic curve \(E^+/F^+\) such that \(E=E^+\bigtimes_{F^+} F\).
MSC:
| 11G05 | Elliptic curves over global fields |
| 11G15 | Complex multiplication and moduli of abelian varieties |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11R11 | Quadratic extensions |
| 14K22 | Complex multiplication and abelian varieties |
References:
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| [9] | Peter Schneider,Introduction to the Beilinson Conjectures, Beilinson’s Conjectures on Special Values ofL-functions (Michael Rapoport,Norbert Schappacher, andPeter Schneider, eds.), Perspectives in Mathematics, vol. 4, 1987. |
| [10] | Jean-Pierre Serre, Sur les représentations modulaires de degré 2 de Gal( \(\bar {\mathbb{Q}}/{\mathbb{Q}}\) ),Duke Math. J. 54 (1987), no. 1, 179–230. · Zbl 0641.10026 · doi:10.1215/S0012-7094-87-05413-5 |
| [11] | Goro Shimura,Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Japan, vol. 11, Princeton University Press, 1971. · Zbl 0221.10029 |
| [12] | –, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields,Nagoya Math. J. 43 (1971), 199–208. · Zbl 0225.14015 |
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| [15] | Jean-Pierre Serre andJohn Tate, Good reduction of abelian varieties,Ann. of Math. (2)88 (1968), 492–517. · Zbl 0172.46101 · doi:10.2307/1970722 |
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