##
**Monodromy and irreducibility of leaves.**
*(English)*
Zbl 1228.14021

In splendid style, the authors prove three striking and conclusive theorems about the Newton slope stratification of the moduli space of abelian \(g\)-folds over \({\mathbb F}_p\). Denote by \({\mathcal A}_{g,d,n}\) the moduli space of such varieties with a polarisation of degree \(d^2\) and a symplectic level-\(n\) structure, assuming \(n\) is coprime to \(pd\). The Newton slope strata are denoted by \({\mathcal W}^0_\xi({\mathcal A}_{g,n,d})\), indexed by symmetric Newton polygons \(\xi\) of height \(2g\). One of these strata is the supersingular stratum and must be excluded from what follows.

Theorem A: All the other Newton strata are geometrically irreducible if \(d=1\), i.e., in the principally polarised case.

Theorem B: The central leaf \({\mathcal C}(x)\) through any non-supersingular geometric point \(x\) over an algebraically closed field containing \({\mathbb F}_p\) is irreducible.

Theorem C: The \(p\)-adic monodromy for the leaf \({\mathcal C}(x)\) is maximal.

The central leaf is the stratum on which the \(p\)-divisible group associated to the polarised abelian variety remains constant, as well as the type of the polarisation (that is, the elementary divisors of the pairing it induced \(H_1(A,{\mathbb Z}_\ell)\) by the polarisation). The \(p\)-adic monodromy is the image of the action of the Galois group of \({\mathcal C}(x)\) on the \(p\)-divisible group at a generic point of the leaf: the statement that it is maximal means that it is the full automorphism group.

Of these results, Theorem B at least was already more or less known: there is a sketch of a proof in [C.-L. Chai, “Hecke orbits on Siegel modular varieties,” Geometric methods in algebra and number theory. Basel: Birkhäuser. Progress in Mathematics 235, 71–107 (2005; Zbl 1143.14021)]. A sketch of the proof of Theorem A was given by the second author as long ago as 2003, but was some way from being complete at that time. The proof of Theorem B here is new, though, and logically simpler although not at all easy. Theorem B is here deduced from Theorem A by using hypersymmetric points, corresponding to abelian varieties whose endomorphism rings are as large as possible for their \(p\)-divisible groups. These occur, by Theorem A, on every component of \({\mathcal C}(x)\), and they, or enough of them, are permuted by suitable Hecke correspondences. That, together with further results of the first author, proves the irreducibility in Theorem B. Hypersymmetric points also make an appearance in the proof of Theorem C, because the maximality can be interpreted as saying that certain finite covers of \({\mathcal C}(x)\) are irreducible: then the proof proceeds by showing that the Hecke correspondences act transitively on the fibres above the hypersymmetric points, and using the same result as before, that transitivity of the Hecke action on the set of irreducible components implies irreducibility.

After the introduction, of which the paragraphs above are a précis, there is a very helpful further introductory section in which all the definitions and general theory needed are collected, with suitable pointers to the literature. Theorems A, B and C are proved successively in the remaining sections, each clearly divided up into steps. There are a large number of technical details to be attended to, but the paper is, for all its difficulty, easy to read because it is at all times clear what task is being undertaken, and why.

Theorem A: All the other Newton strata are geometrically irreducible if \(d=1\), i.e., in the principally polarised case.

Theorem B: The central leaf \({\mathcal C}(x)\) through any non-supersingular geometric point \(x\) over an algebraically closed field containing \({\mathbb F}_p\) is irreducible.

Theorem C: The \(p\)-adic monodromy for the leaf \({\mathcal C}(x)\) is maximal.

The central leaf is the stratum on which the \(p\)-divisible group associated to the polarised abelian variety remains constant, as well as the type of the polarisation (that is, the elementary divisors of the pairing it induced \(H_1(A,{\mathbb Z}_\ell)\) by the polarisation). The \(p\)-adic monodromy is the image of the action of the Galois group of \({\mathcal C}(x)\) on the \(p\)-divisible group at a generic point of the leaf: the statement that it is maximal means that it is the full automorphism group.

Of these results, Theorem B at least was already more or less known: there is a sketch of a proof in [C.-L. Chai, “Hecke orbits on Siegel modular varieties,” Geometric methods in algebra and number theory. Basel: Birkhäuser. Progress in Mathematics 235, 71–107 (2005; Zbl 1143.14021)]. A sketch of the proof of Theorem A was given by the second author as long ago as 2003, but was some way from being complete at that time. The proof of Theorem B here is new, though, and logically simpler although not at all easy. Theorem B is here deduced from Theorem A by using hypersymmetric points, corresponding to abelian varieties whose endomorphism rings are as large as possible for their \(p\)-divisible groups. These occur, by Theorem A, on every component of \({\mathcal C}(x)\), and they, or enough of them, are permuted by suitable Hecke correspondences. That, together with further results of the first author, proves the irreducibility in Theorem B. Hypersymmetric points also make an appearance in the proof of Theorem C, because the maximality can be interpreted as saying that certain finite covers of \({\mathcal C}(x)\) are irreducible: then the proof proceeds by showing that the Hecke correspondences act transitively on the fibres above the hypersymmetric points, and using the same result as before, that transitivity of the Hecke action on the set of irreducible components implies irreducibility.

After the introduction, of which the paragraphs above are a précis, there is a very helpful further introductory section in which all the definitions and general theory needed are collected, with suitable pointers to the literature. Theorems A, B and C are proved successively in the remaining sections, each clearly divided up into steps. There are a large number of technical details to be attended to, but the paper is, for all its difficulty, easy to read because it is at all times clear what task is being undertaken, and why.

Reviewer: G. K. Sankaran (Bath)

### MSC:

14G35 | Modular and Shimura varieties |

11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |

14K15 | Arithmetic ground fields for abelian varieties |

14L05 | Formal groups, \(p\)-divisible groups |

### Citations:

Zbl 1143.14021
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\textit{C.-L. Chai} and \textit{F. Oort}, Ann. Math. (2) 173, No. 3, 1359--1396 (2011; Zbl 1228.14021)

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### References:

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