##
**Abelian varieties isogenous to a Jacobian.**
*(English)*
Zbl 1263.14032

It was until recently an open question (raised by Katz according to Oort and by Oort according to Katz) whether there exists an abelian variety \(A\) of dimension \(g\) over \({\overline{\mathbb Q}}\) that is not isogenous to the Jacobian of a stable curve. The expectation is that there should be such \(A\) (for \(g\geq 4\), obviously), and this has recently been proved by J. Tsimerman [Ann. Math. (2) 176, No. 1, 637–650 (2012; Zbl 1250.14032)], who makes use of the results of this paper. Here one can find, among other things, a proof of the same thing conditional on GRH (and thus superceded by Tsimerman’s work).

The paper under review addresses a rather more general question, posed by Poonen. Let \(X\) be a closed subset of \({\mathcal A}_g\) (the moduli space of principally polarised abelian \(g\)-folds) defined over some algebraically closed field \(k\). For \(x=(A,\lambda)\in{\mathcal A}_g(k)\), we define the isogeny orbit \({\mathcal I}(x)\) and the Hecke orbit \({\mathcal H}(x)\), and we ask whether either of these may, for some \(x\), have empty intersection with \(X\). More precisely we consider the statements

\(\mathrm{I}(k,g)\) [respectively \(\mathrm{SI}(k,g)\)]: for every closed proper subset \(X\subset {\mathcal A}_g\) over \(k\), there exists \(x=(A,\lambda)\in {\mathcal A}_g\) such that \({\mathcal H}(x)\cap X=\emptyset\) [respectively \({\mathcal I}(x)\cap X=\emptyset\)].

The question we started with is whether \(\mathrm{sI}(k,g)\) holds if we take \(X\) to be the closure \({\mathcal T}_g\) of the Torelli locus. The Hecke orbit of \(x\) includes only those \(y=(B,\mu)\) for which there is a quasi-isogeny \(A\to B\) preserving the principal polarisations: the isogeny orbit allows any isogeny, irrespective of what it does to the polarisations.

A major result of this paper is that \(\mathrm{I}{\overline{\mathbb Q}},g)\) follows from the André-Oort conjecture. In view of the proof of the André-Oort conjecture conditional on GRH, announced by Ullmo and Yafaev, this gives Tsimerman’s result subject to GRH as a special case (but, let it be emphasised, Tsimerman’s result, though limited to \(X={\mathcal T}_g\), is unconditional).

The value of the André-Oort conjecture in this context is that it reduces the problem from one about arbitrary subvarieties and points in \({\mathcal A}_g\) to “arithmetic” ones: we may assume that \(X\) is a union of Shimura varieties and that \(x\) is required to be a CM point. In fact one may further assume that \(x\) is a Weyl CM point, or Weyl special point: that is, that the endomorphism algebra is is a CM field \(L\) of degree \(2g\) over \({\mathbb Q}\) such that the Galois group of its normal closure is maximal, i.e.\(({\mathbb Z}/2{\mathbb Z})^g\rtimes S_g\).

Such points are common: “most” CM points are of this kind. However, the Shimura varieties they lie on are more restricted: an irreducible Shimura variety containing a Weyl CM point is a Hilbert modular variety attached to the maximal totally real subfield of \(L\). So to get the statement \(\mathrm{I}({\overline{\mathbb Q}},g)\) it is sufficient to take a Weyl CM point where the endomorphism algebra is a field \(L\) that does not give a component of \(X\), neither by being attached to a zero-dimensional component nor by containing a totally real field attached to a Hilbert modular variety component. There are plenty of such points.

A similar argument shows that for any \(g\geq 4\), there are only finitely many Weyl CM Jaobians of dimension \(g\), which seems, perhaps surprisingly, to be currently out of reach by any method not depending on GRH.

There is a notion of Weyl special point of a Shimura variety, generalising the case of the Siegel modular variety. A main result of this paper is the equivalent statement to \(\mathrm{I}({\overline{\mathbb Q}},k\)) (or rather, of its reformulation via the André-Oort conjecture) in that context: if \(Y\) is a union of Shimura varieties properly contained in a Shimura variety \(S\), then there exists a Weyl special point \(y\) in \(S\) whose Hecke orbit is disjoint from \(Y\). Both the definition of Weyl special point and the properties that make it tractable (playing the role of the field \(L\) above) are more complicated, necessarily involving the reductive group in the Shimura data: full details are of course in the paper, but an outline is given in the very clear introduction.

The paper under review addresses a rather more general question, posed by Poonen. Let \(X\) be a closed subset of \({\mathcal A}_g\) (the moduli space of principally polarised abelian \(g\)-folds) defined over some algebraically closed field \(k\). For \(x=(A,\lambda)\in{\mathcal A}_g(k)\), we define the isogeny orbit \({\mathcal I}(x)\) and the Hecke orbit \({\mathcal H}(x)\), and we ask whether either of these may, for some \(x\), have empty intersection with \(X\). More precisely we consider the statements

\(\mathrm{I}(k,g)\) [respectively \(\mathrm{SI}(k,g)\)]: for every closed proper subset \(X\subset {\mathcal A}_g\) over \(k\), there exists \(x=(A,\lambda)\in {\mathcal A}_g\) such that \({\mathcal H}(x)\cap X=\emptyset\) [respectively \({\mathcal I}(x)\cap X=\emptyset\)].

The question we started with is whether \(\mathrm{sI}(k,g)\) holds if we take \(X\) to be the closure \({\mathcal T}_g\) of the Torelli locus. The Hecke orbit of \(x\) includes only those \(y=(B,\mu)\) for which there is a quasi-isogeny \(A\to B\) preserving the principal polarisations: the isogeny orbit allows any isogeny, irrespective of what it does to the polarisations.

A major result of this paper is that \(\mathrm{I}{\overline{\mathbb Q}},g)\) follows from the André-Oort conjecture. In view of the proof of the André-Oort conjecture conditional on GRH, announced by Ullmo and Yafaev, this gives Tsimerman’s result subject to GRH as a special case (but, let it be emphasised, Tsimerman’s result, though limited to \(X={\mathcal T}_g\), is unconditional).

The value of the André-Oort conjecture in this context is that it reduces the problem from one about arbitrary subvarieties and points in \({\mathcal A}_g\) to “arithmetic” ones: we may assume that \(X\) is a union of Shimura varieties and that \(x\) is required to be a CM point. In fact one may further assume that \(x\) is a Weyl CM point, or Weyl special point: that is, that the endomorphism algebra is is a CM field \(L\) of degree \(2g\) over \({\mathbb Q}\) such that the Galois group of its normal closure is maximal, i.e.\(({\mathbb Z}/2{\mathbb Z})^g\rtimes S_g\).

Such points are common: “most” CM points are of this kind. However, the Shimura varieties they lie on are more restricted: an irreducible Shimura variety containing a Weyl CM point is a Hilbert modular variety attached to the maximal totally real subfield of \(L\). So to get the statement \(\mathrm{I}({\overline{\mathbb Q}},g)\) it is sufficient to take a Weyl CM point where the endomorphism algebra is a field \(L\) that does not give a component of \(X\), neither by being attached to a zero-dimensional component nor by containing a totally real field attached to a Hilbert modular variety component. There are plenty of such points.

A similar argument shows that for any \(g\geq 4\), there are only finitely many Weyl CM Jaobians of dimension \(g\), which seems, perhaps surprisingly, to be currently out of reach by any method not depending on GRH.

There is a notion of Weyl special point of a Shimura variety, generalising the case of the Siegel modular variety. A main result of this paper is the equivalent statement to \(\mathrm{I}({\overline{\mathbb Q}},k\)) (or rather, of its reformulation via the André-Oort conjecture) in that context: if \(Y\) is a union of Shimura varieties properly contained in a Shimura variety \(S\), then there exists a Weyl special point \(y\) in \(S\) whose Hecke orbit is disjoint from \(Y\). Both the definition of Weyl special point and the properties that make it tractable (playing the role of the field \(L\) above) are more complicated, necessarily involving the reductive group in the Shimura data: full details are of course in the paper, but an outline is given in the very clear introduction.

Reviewer: G. K. Sankaran (Bath)

### MSC:

14G35 | Modular and Shimura varieties |

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

11G15 | Complex multiplication and moduli of abelian varieties |

11G18 | Arithmetic aspects of modular and Shimura varieties |

14D10 | Arithmetic ground fields (finite, local, global) and families or fibrations |

14H40 | Jacobians, Prym varieties |

14K02 | Isogeny |

14K22 | Complex multiplication and abelian varieties |

### Citations:

Zbl 1250.14032
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\textit{C.-L. Chai} and \textit{F. Oort}, Ann. Math. (2) 176, No. 1, 589--635 (2012; Zbl 1263.14032)

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