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**Galois properties of division fields of elliptic curves.**
*(English)*
Zbl 0809.14026

In 1972, J.-P. Serre [Invent. Math. 15, 259–331 (1972; Zbl 0235.14012)] proved that given a non CM elliptic curve \(E\) defined over a number field \(K\), the Galois group of the field of rationality of the \(l\)-torsion points of \(E\) \((l\) prime) is as big as possible for \(l>l_ 0 (K)\) large enough, i.e. \(\simeq \text{GL}_ 2 (\mathbb F_ l)\). However, except in the case \(K = \mathbb Q\), no effective upper bound for \(l_ 0(K)\) was known (although Serre did announce such an estimate in an oral talk in Paris, 1988).

The object of the paper under review is to provide such a bound. The authors rely on a series of previous papers (in fact a real program), they started with a crucial though rather technical estimate on periods of abelian varieties [Ann. Math. (2) 137, No. 2, 407–458 (1993; Zbl 0796.11023)]. This estimate, which controls the degree of the smallest abelian subvariety containing a prescribed period of a given abelian variety \(A\) in its tangent space in terms of the Faltings height of \(A\), enables them to get a bound on the degree of a minimal isogeny linking two abelian varieties \(A\) and \(B\) (say principally polarized) in terms of the Faltings height of, say \(A\) [Ann. Math. (2) 137, No. 3, 459–472 (1993; Zbl 0804.14019)]. The first step is then to use Zarkhin’s trick to get a more technical but partially unpolarized isogeny estimate. The two following sections of the paper make use of the list of possible subgroups for \(\text{GL}_ 2 (\mathbb F_ l)\) to systematically eliminate all subgroups not containing \(\text{SL}_ 2 (\mathbb F_ l)\). The isogeny estimates take care of Borels and unsplit Cartans. The proof can then be completed following Serre’s paper (loc. cit.) and one gets a bound of the form \(l_ 0<c \max \{d, h(j_ E)\}^ \gamma\), where \(c\) and \(\gamma\) are universal constants, \(d\) is the degree of \(K/ \mathbb Q\), and \(h(j_ E)\) is the Weil height of the \(j\)-invariant of the curve \(E\). The exponent \(\gamma\) is not given, and is certainly large. However, it is expected that a “customized” approach would give a reasonable exponent. On the other hand, the constant \(c\) is ineffective. This comes from the fact that the main period estimate itself involves some constants in the dimension of the abelian variety whose existence is given by compactness arguments. Until these are replaced, this theorem wouldn’t yield an algorithm usable on a computer.

The two last sections of the paper are devoted to analogs of the main result on products of (non CM) elliptic curves and to Kummer theory, i.e. Galois properties of division points of a finite set of linearly independent points of infinite order of a (non CM) elliptic curve, extending results of M. I. Bashmakov [Russ. Math. Surv. 27(1972), 25–70 (1973); translation from Usp. Mat. Nauk 27, No. 6(168), 25–66 (1972; Zbl 0256.14016)] and D. Bertrand [Glasg. Math. J. 22, 83–88 (1981; Zbl 0453.14019)].

The object of the paper under review is to provide such a bound. The authors rely on a series of previous papers (in fact a real program), they started with a crucial though rather technical estimate on periods of abelian varieties [Ann. Math. (2) 137, No. 2, 407–458 (1993; Zbl 0796.11023)]. This estimate, which controls the degree of the smallest abelian subvariety containing a prescribed period of a given abelian variety \(A\) in its tangent space in terms of the Faltings height of \(A\), enables them to get a bound on the degree of a minimal isogeny linking two abelian varieties \(A\) and \(B\) (say principally polarized) in terms of the Faltings height of, say \(A\) [Ann. Math. (2) 137, No. 3, 459–472 (1993; Zbl 0804.14019)]. The first step is then to use Zarkhin’s trick to get a more technical but partially unpolarized isogeny estimate. The two following sections of the paper make use of the list of possible subgroups for \(\text{GL}_ 2 (\mathbb F_ l)\) to systematically eliminate all subgroups not containing \(\text{SL}_ 2 (\mathbb F_ l)\). The isogeny estimates take care of Borels and unsplit Cartans. The proof can then be completed following Serre’s paper (loc. cit.) and one gets a bound of the form \(l_ 0<c \max \{d, h(j_ E)\}^ \gamma\), where \(c\) and \(\gamma\) are universal constants, \(d\) is the degree of \(K/ \mathbb Q\), and \(h(j_ E)\) is the Weil height of the \(j\)-invariant of the curve \(E\). The exponent \(\gamma\) is not given, and is certainly large. However, it is expected that a “customized” approach would give a reasonable exponent. On the other hand, the constant \(c\) is ineffective. This comes from the fact that the main period estimate itself involves some constants in the dimension of the abelian variety whose existence is given by compactness arguments. Until these are replaced, this theorem wouldn’t yield an algorithm usable on a computer.

The two last sections of the paper are devoted to analogs of the main result on products of (non CM) elliptic curves and to Kummer theory, i.e. Galois properties of division points of a finite set of linearly independent points of infinite order of a (non CM) elliptic curve, extending results of M. I. Bashmakov [Russ. Math. Surv. 27(1972), 25–70 (1973); translation from Usp. Mat. Nauk 27, No. 6(168), 25–66 (1972; Zbl 0256.14016)] and D. Bertrand [Glasg. Math. J. 22, 83–88 (1981; Zbl 0453.14019)].

Reviewer: S.David (Paris)

### MSC:

14H52 | Elliptic curves |

14K02 | Isogeny |

14G05 | Rational points |

11G05 | Elliptic curves over global fields |

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