##
**Pinceaux de variétés abéliennes. (Pencils of abelian varieties).**
*(French)*
Zbl 0595.14032

Astérisque, 129. Publié avec le concours du Centre National de la Recherche Scientifique. Paris: Société Mathématique de France. 266 p. FF 155.00; $ 18.00 (1985).

Denote by \({\mathcal A}={\mathcal A}_{g,d,n}\) the (fine) moduli space of abelian varieties of dimension g, polarized of degree \(d^ 2\) with \(level\quad n\) structure (n\(\geq 3)\). It is known that \({\mathcal A}\) is a quasiprojective scheme (of finite type) over Spec(\({\mathbb{Z}}[1/n])\). With the universal family of abelian varieties \(f:\quad {\mathcal X}\to {\mathcal A}\) we have the invertible sheaf \({\bar \omega}=f_*\Omega^ g_{{\mathcal X}/{\mathcal A}}\). - One of the main theorems of this article is: The sheaf \({\bar \omega}\) is ample on \({\mathcal A}.\)

For the proof the author constructs a totally symmetric (notion defined by the author; to say roughly, symmetric with cubic structure) invertible sheaf \({\mathcal L}\) on \({\mathcal X}\) whose polarization is 4 times the canonical one and shows the following two statements: \((A)\quad \det (f_*{\mathcal L})^{\vee}\) is ample with respect to f; \((B)\quad (\det f_*{\mathcal L})^{\otimes 2}\otimes {\bar \omega}^{\otimes 4^ gd}\) is a torsion in Pic(\({\mathcal A}).\)

In the classical case (over \({\mathbb{C}})\) such procedure (and the above theorem) corresponds to main results of the theory of theta functions: for every ample line bundle the sections are given by theta functions; the theta zero-values generate essentially the sheaf of automorphic forms (\({\bar \omega}\)) which is shown to be ample.

The statement (B) is a result of a more general one: Let \(f:\quad {\mathcal A}\to S\) be a family of g-dimensional abelian schemes over a scheme S and \({\mathcal L}^ a \)symmetric invertible sheaf on \({\mathcal A}\) which is ample with respect to S of degree \(d^ 2\). We consider \({\bar \omega}{}_{{\mathcal A}/S}=f_*\Omega^ g_{{\mathcal A}/S}\). Then \((\bigwedge^ df_*L)^{\otimes 2}\otimes {\bar \omega}^{\otimes d}_{{\mathcal A}/S}\) is a torsion in Pic(S). - The author proves this claim when d is invertible on S or S is normal and excellent. He then tries to extend the results in the case when the abelian varieties have only stable reductions. Let us consider such a family \(f:\quad {\mathcal A}\to S\) with ample symmetric sheaf \({\mathcal L}_{\eta}\) over the generic point. For that purpose the author first generalizes the notion of cubic structure in such case and proves that a cubic structure on the generic fibre extends uniquely to a cubic \({\mathcal L}\) over S (after a finite base change). We assume here that, by denoting K(\({\mathcal L})\) the kernel of the associated polarization \(\phi_{{\mathcal L}}:\quad {\mathcal A}\to {\mathcal A}^ t,\) K(\({\mathcal L})\) is finite over S. The author proves the following under a slightly stronger assumption that K(\({\mathcal L}^{\otimes 2})\) is finite over S which is normal, noetherian and excellent: \((\bigwedge^ df_*{\mathcal L})^{\otimes 2}\otimes {\bar \omega}^{\otimes d}_{{\mathcal A}/S}\) is torsion in Pic(S).- From this one can show: Theorem. Under the assumption above, for some \(n\leq 1\), \({\bar \omega}{}^{\otimes d}_{{\mathcal A}/S}\) is generated by sections.

As an application the author reproves a theorem of Zarkhin on the finiteness of abelian varieties with given isogeny class over a function field over a finite field, which was according to the author one of the motivations of his study.

For the proof the author constructs a totally symmetric (notion defined by the author; to say roughly, symmetric with cubic structure) invertible sheaf \({\mathcal L}\) on \({\mathcal X}\) whose polarization is 4 times the canonical one and shows the following two statements: \((A)\quad \det (f_*{\mathcal L})^{\vee}\) is ample with respect to f; \((B)\quad (\det f_*{\mathcal L})^{\otimes 2}\otimes {\bar \omega}^{\otimes 4^ gd}\) is a torsion in Pic(\({\mathcal A}).\)

In the classical case (over \({\mathbb{C}})\) such procedure (and the above theorem) corresponds to main results of the theory of theta functions: for every ample line bundle the sections are given by theta functions; the theta zero-values generate essentially the sheaf of automorphic forms (\({\bar \omega}\)) which is shown to be ample.

The statement (B) is a result of a more general one: Let \(f:\quad {\mathcal A}\to S\) be a family of g-dimensional abelian schemes over a scheme S and \({\mathcal L}^ a \)symmetric invertible sheaf on \({\mathcal A}\) which is ample with respect to S of degree \(d^ 2\). We consider \({\bar \omega}{}_{{\mathcal A}/S}=f_*\Omega^ g_{{\mathcal A}/S}\). Then \((\bigwedge^ df_*L)^{\otimes 2}\otimes {\bar \omega}^{\otimes d}_{{\mathcal A}/S}\) is a torsion in Pic(S). - The author proves this claim when d is invertible on S or S is normal and excellent. He then tries to extend the results in the case when the abelian varieties have only stable reductions. Let us consider such a family \(f:\quad {\mathcal A}\to S\) with ample symmetric sheaf \({\mathcal L}_{\eta}\) over the generic point. For that purpose the author first generalizes the notion of cubic structure in such case and proves that a cubic structure on the generic fibre extends uniquely to a cubic \({\mathcal L}\) over S (after a finite base change). We assume here that, by denoting K(\({\mathcal L})\) the kernel of the associated polarization \(\phi_{{\mathcal L}}:\quad {\mathcal A}\to {\mathcal A}^ t,\) K(\({\mathcal L})\) is finite over S. The author proves the following under a slightly stronger assumption that K(\({\mathcal L}^{\otimes 2})\) is finite over S which is normal, noetherian and excellent: \((\bigwedge^ df_*{\mathcal L})^{\otimes 2}\otimes {\bar \omega}^{\otimes d}_{{\mathcal A}/S}\) is torsion in Pic(S).- From this one can show: Theorem. Under the assumption above, for some \(n\leq 1\), \({\bar \omega}{}^{\otimes d}_{{\mathcal A}/S}\) is generated by sections.

As an application the author reproves a theorem of Zarkhin on the finiteness of abelian varieties with given isogeny class over a function field over a finite field, which was according to the author one of the motivations of his study.

Reviewer: Y.Namikawa

### MSC:

14K10 | Algebraic moduli of abelian varieties, classification |

14K05 | Algebraic theory of abelian varieties |

14K15 | Arithmetic ground fields for abelian varieties |