##
**Large period matrices and a conjecture of Lang.**
*(English)*
Zbl 0827.11037

David, Sinnou (ed.), Séminaire de théorie des nombres, Paris, France, 1991-92. Boston, MA: Birkhäuser. Prog. Math. 116, 153-177 (1994).

The author studies abelian varieties with “large period matrices”. More precisely, if \(A\) is a principally polarized abelian variety defined over \(\mathbb{C}\), it is isomorphic to a torus \(\mathbb{C}^g/ \mathbb{Z}^g+ \tau \mathbb{Z}^g\), where \(\tau\) can be chosen in the standard fundamental domain of the Siegel space \(S_g\). Now, if \(A\) is in fact defined over a number field \(k\), it can be shown that the norm \(|\tau|\leq c(g) [k: \mathbb{Q}] h(A/ k)\), where \(h(A/ k)\) is the (stable) Faltings height of \(A/k\) [the author, Diophantine approximation and transcendence theory, Lect. Notes Math. 1290, 109-148 (1987; Zbl 0639.14025)].

The question is now to characterize those abelian varieties for which the reverse inequality holds. For elliptic curves, this question is easily solved using the \(q\)-expansion formula for the \(j\)-invariant. But for higher dimensional varieties, no such information was known (despite the interest of the question, which can be linked with “effective Mordell”), and the author shows that there are infintiely many (simple) abelian varieties of all dimensions for which the above inequality is true. The applications are numerous: a conjecture of S. Lang [Elliptic curves: Diophantine analysis (Springer 1978; Zbl 0388.10001)] states that if \(E/k\) is an elliptic curve defined over \(k\) and \(P\in E(k)\) is of infinite order, then the Néron-Tate height of \(P\) satisfies \(\widehat {h} (P)\geq c(k) h(E/ k)\). J. H. Silverman [Duke Math. J. 51, 395-403 (1984; Zbl 0579.14035)] has generalized this conjecture for abelian varieties, and the reviewer has shown [Bull. Soc. Math. Fr. 121, 509-544 (1993; Zbl 0803.11031)] that this conjecture holds up to a corrective factor of the form \(({{|\tau|} \over {h(A/ k)}})^{c(g)}\).

The results of this paper thus show that the conjecture of Lang-Silverman holds for infinitely many simple abelian varieties of any dimension. It also shows that in the theorem of the author and G. Wüstholz [Ann. Math., II. Ser. 137, 459-472 (1993; Zbl 0804.14019)], one cannot expect to have a bound for the degree of the smallest abelian subvariety of \(A\) containing a given period \(\omega\) in its tangent space independent of the height of \(A\) (see §10 of the latter).

The underlying idea of the proof of the results in this paper are the following: since the question is easily solved for elliptic curves, start with an elliptic curve with integral (large enough) \(j\)-invariant, take its \(g\)-th power and make small perturbations of the resulting point in the moduli space (given by the theta embedding). One then shows that sufficiently many of the resulting abelian varieties are defined over \(\overline {\mathbb{Q}}\), that their height (as well as the size of the period matrix) does not change too much, nor the degree of the field of definition. To ensure the simplicity of the resulting abelian variety, one uses the fact that the generic endomorphism ring is \(\mathbb{Z}\) and therefore sufficiently many of the abelian varieties constructed that way have \(\mathbb{Z}\) as their endomorphism ring and are thus simple.

In the next result, the author uses Jacobians of hyperelliptic curves to make a similar construction. This corrects the drawback of the previous construction (one has no control over the discriminant of the field of definition of the resulting varieties). The author then uses a result of S. Mori [Jap. J. Math., New Ser. 2, 109-130 (1976; Zbl 0339.14016)], which shows that the generic endomorphism ring of the Jacobian of hyperelliptic curves is still \(\mathbb{Z}\). The resulting construction then yields infinitely many Jacobians of hyperelliptic curves which are all defined over \(\mathbb{Q}\), and which satisfy the desired property. The perturbation computations on the resulting period matrices are on the other hand much more difficult and heavily rely on subtle computations on abelian integrals.

For the entire collection see [Zbl 0807.00013].

The question is now to characterize those abelian varieties for which the reverse inequality holds. For elliptic curves, this question is easily solved using the \(q\)-expansion formula for the \(j\)-invariant. But for higher dimensional varieties, no such information was known (despite the interest of the question, which can be linked with “effective Mordell”), and the author shows that there are infintiely many (simple) abelian varieties of all dimensions for which the above inequality is true. The applications are numerous: a conjecture of S. Lang [Elliptic curves: Diophantine analysis (Springer 1978; Zbl 0388.10001)] states that if \(E/k\) is an elliptic curve defined over \(k\) and \(P\in E(k)\) is of infinite order, then the Néron-Tate height of \(P\) satisfies \(\widehat {h} (P)\geq c(k) h(E/ k)\). J. H. Silverman [Duke Math. J. 51, 395-403 (1984; Zbl 0579.14035)] has generalized this conjecture for abelian varieties, and the reviewer has shown [Bull. Soc. Math. Fr. 121, 509-544 (1993; Zbl 0803.11031)] that this conjecture holds up to a corrective factor of the form \(({{|\tau|} \over {h(A/ k)}})^{c(g)}\).

The results of this paper thus show that the conjecture of Lang-Silverman holds for infinitely many simple abelian varieties of any dimension. It also shows that in the theorem of the author and G. Wüstholz [Ann. Math., II. Ser. 137, 459-472 (1993; Zbl 0804.14019)], one cannot expect to have a bound for the degree of the smallest abelian subvariety of \(A\) containing a given period \(\omega\) in its tangent space independent of the height of \(A\) (see §10 of the latter).

The underlying idea of the proof of the results in this paper are the following: since the question is easily solved for elliptic curves, start with an elliptic curve with integral (large enough) \(j\)-invariant, take its \(g\)-th power and make small perturbations of the resulting point in the moduli space (given by the theta embedding). One then shows that sufficiently many of the resulting abelian varieties are defined over \(\overline {\mathbb{Q}}\), that their height (as well as the size of the period matrix) does not change too much, nor the degree of the field of definition. To ensure the simplicity of the resulting abelian variety, one uses the fact that the generic endomorphism ring is \(\mathbb{Z}\) and therefore sufficiently many of the abelian varieties constructed that way have \(\mathbb{Z}\) as their endomorphism ring and are thus simple.

In the next result, the author uses Jacobians of hyperelliptic curves to make a similar construction. This corrects the drawback of the previous construction (one has no control over the discriminant of the field of definition of the resulting varieties). The author then uses a result of S. Mori [Jap. J. Math., New Ser. 2, 109-130 (1976; Zbl 0339.14016)], which shows that the generic endomorphism ring of the Jacobian of hyperelliptic curves is still \(\mathbb{Z}\). The resulting construction then yields infinitely many Jacobians of hyperelliptic curves which are all defined over \(\mathbb{Q}\), and which satisfy the desired property. The perturbation computations on the resulting period matrices are on the other hand much more difficult and heavily rely on subtle computations on abelian integrals.

For the entire collection see [Zbl 0807.00013].

Reviewer: S.David (Paris)

### MSC:

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

14K15 | Arithmetic ground fields for abelian varieties |

14K02 | Isogeny |