##
**Newton polygons and formal groups: Conjectures by Manin and Grothendieck.**
*(English)*
Zbl 0991.14016

The paper contains several important results and techniques concerning \(p\)-divisible (Barsotti-Tate) groups and their deformations. The paper begins with a survey of the properties, classification and deformation theory of \(p\)-divisible groups. There is a short exposition of the theory of displays, a theory that allows one to study deformations of \(p\)-divisible groups via explicit description of certain \(\sigma\)-linear operators. If \(G\) is a principally quasi-polarized \(p\)-divisible group (i.e., \(G \cong G^t\) by an anti-symmetric isomorphism) over a perfect characteristic \(p\) field \(k\), of dimension \(g\) and height \(2g\), a suitable choice of basis for its Dieudonné module, which is a module of rank \(2g\) over \(W(k)\), provides one with a matrix \(\left(\begin{smallmatrix} A & B \\ C & D\end{smallmatrix}\right)\) such that with respect to this basis the Frobenius map is given by \(\left(\begin{smallmatrix} A & pB \\ C & pD\end{smallmatrix}\right)\). This matrix is called a display, since it displays the structure of the Dieudonné module. The universal deformation space for \(G\), as a principally quasi-polarized \(p\)-divisible group, is the spectrum of \(k[[t_{ij}]]/(t_{ij} - t_{ji})\) \((i, j = 1, \dots, g\)), which is of dimension \(g(g+1)/2\). The display corresponding to the universal deformation is \(\left(\begin{smallmatrix} A+ TC & B + TD \\ C & D\end{smallmatrix}\right)\), where \(T\) is the symmetric \(g\times g\) matrix of the Teichmüller lifts of the variables \(t_{ij}\).

One can associate to a \(p\)-divisible group a Newton polygon that encodes its isogeny class. The Newton polygon of a principally quasi-polarized \(p\)-divisible group of dimension \(g\) is a lower convex polygon, starting at \((0, 0)\), ending at \((2g, g)\), has integral break points, and is symmetric in the sense that a slope \(\lambda\) appears as a slope of a segment of the polygon with the same multiplicity the slope \(1 - \lambda\) appears. The Manin conjecture mentioned in the title of the paper, is the conjecture that every such polygon appears as the Newton polygon associated to a principally polarized abelian variety. The Manin conjecture was proven long ago using Honda-Serre-Tate theory [see J. Tate, Sémin. Bourbaki 1968/69, No. 352, 95-110 (1971; Zbl 0212.25702)]. The set of symmetric Newton polygons for a fixed \(g\) forms a partially ordered set with respect to the notion of “lying above”. A theorem of Grothendieck and Katz [cf. N. M. Katz, Astérisque 63, 113-164 (1979; Zbl 0426.14007)] states that Newton polygons go up under specialization. Moreover, the set of closed points of the moduli space of principally polarized abelian varieties of dimension \(g\) in characteristic \(p\) where the Newton polygon of the \(p\)-divisible group of the associated abelian variety is equal to a given one, is a locally closed set. Manin’s conjecture asserts that it is non-empty. Grothendieck’s conjecture says these sets form a stratification in a natural way.

Let \(G\) be a \(p\)-divisible group over \(k\) and \(\eta\) a Newton polygon lying below that of \(G\). Grothendieck’s conjecture asserts that one can find a deformation of \(G\) over an integral local ring such that the generic fibre is a \(p\)-divisible group with Newton polygon \(\eta\). Using the Serre-Tate theorem one reduces a similar conjecture for principally polarized abelian varieties to the case of \(p\)-divisible groups.

Since the Newton polygon is determined by the operator \(F\) on the Dieudonné module, one is led to attack the question by using the universal deformation via displays. The difficulty is that one has to study the limit behavior of the iterations of \(F\) that are given by extremely complicated matrices. The author’s ingenious idea is to consider displays of a very special sort. These exist in the case of \(a\)-number \(1\), \(a(G)=1\). In this case, the matrices obtained are much less complicated and the author is able to show by calculation the existence of deformations with prescribed Newton polygons.

The assumption of \(a\)-number \(1\) gives that the Dieudonné module is a cyclic module over the ring \(W(k)[F, V]\), and seems to be essential for the methods of this paper. The author uses that to show the existence of a basis for the Dieudonné module in which the display has the special shape alluded to above. For this special shape the author finds a cyclic submodule of finite index, whose generator is annihilated by the “characteristic polynomial” of the matrix describing \(F\). In this case, one can resort to well known techniques [cf. M. Demazure, “Lectures on \(p\)-divisible curves”, Lect. Notes Math. 302 (1972; Zbl 0247.14010)] to identify the Newton polygon.

The author obtains a ‘pure characteristic \(p\)’ proof of the Manin conjecture, and is able to establish the Grothendieck conjecture for \(a\)-number less or equal to one. In a later paper [F. Oort in: Moduli of abelian varieties, Proc. 3rd Texel Conf., Texel Island 1999, Prog. Math. 195, 417-440 (2001; Zbl 1086.14037)], the author uses this to establish the original Grothendieck conjecture for \(p\)-divisible groups and the general Grothendieck conjecture for principally polarized abelian varieties.

One can associate to a \(p\)-divisible group a Newton polygon that encodes its isogeny class. The Newton polygon of a principally quasi-polarized \(p\)-divisible group of dimension \(g\) is a lower convex polygon, starting at \((0, 0)\), ending at \((2g, g)\), has integral break points, and is symmetric in the sense that a slope \(\lambda\) appears as a slope of a segment of the polygon with the same multiplicity the slope \(1 - \lambda\) appears. The Manin conjecture mentioned in the title of the paper, is the conjecture that every such polygon appears as the Newton polygon associated to a principally polarized abelian variety. The Manin conjecture was proven long ago using Honda-Serre-Tate theory [see J. Tate, Sémin. Bourbaki 1968/69, No. 352, 95-110 (1971; Zbl 0212.25702)]. The set of symmetric Newton polygons for a fixed \(g\) forms a partially ordered set with respect to the notion of “lying above”. A theorem of Grothendieck and Katz [cf. N. M. Katz, Astérisque 63, 113-164 (1979; Zbl 0426.14007)] states that Newton polygons go up under specialization. Moreover, the set of closed points of the moduli space of principally polarized abelian varieties of dimension \(g\) in characteristic \(p\) where the Newton polygon of the \(p\)-divisible group of the associated abelian variety is equal to a given one, is a locally closed set. Manin’s conjecture asserts that it is non-empty. Grothendieck’s conjecture says these sets form a stratification in a natural way.

Let \(G\) be a \(p\)-divisible group over \(k\) and \(\eta\) a Newton polygon lying below that of \(G\). Grothendieck’s conjecture asserts that one can find a deformation of \(G\) over an integral local ring such that the generic fibre is a \(p\)-divisible group with Newton polygon \(\eta\). Using the Serre-Tate theorem one reduces a similar conjecture for principally polarized abelian varieties to the case of \(p\)-divisible groups.

Since the Newton polygon is determined by the operator \(F\) on the Dieudonné module, one is led to attack the question by using the universal deformation via displays. The difficulty is that one has to study the limit behavior of the iterations of \(F\) that are given by extremely complicated matrices. The author’s ingenious idea is to consider displays of a very special sort. These exist in the case of \(a\)-number \(1\), \(a(G)=1\). In this case, the matrices obtained are much less complicated and the author is able to show by calculation the existence of deformations with prescribed Newton polygons.

The assumption of \(a\)-number \(1\) gives that the Dieudonné module is a cyclic module over the ring \(W(k)[F, V]\), and seems to be essential for the methods of this paper. The author uses that to show the existence of a basis for the Dieudonné module in which the display has the special shape alluded to above. For this special shape the author finds a cyclic submodule of finite index, whose generator is annihilated by the “characteristic polynomial” of the matrix describing \(F\). In this case, one can resort to well known techniques [cf. M. Demazure, “Lectures on \(p\)-divisible curves”, Lect. Notes Math. 302 (1972; Zbl 0247.14010)] to identify the Newton polygon.

The author obtains a ‘pure characteristic \(p\)’ proof of the Manin conjecture, and is able to establish the Grothendieck conjecture for \(a\)-number less or equal to one. In a later paper [F. Oort in: Moduli of abelian varieties, Proc. 3rd Texel Conf., Texel Island 1999, Prog. Math. 195, 417-440 (2001; Zbl 1086.14037)], the author uses this to establish the original Grothendieck conjecture for \(p\)-divisible groups and the general Grothendieck conjecture for principally polarized abelian varieties.

Reviewer: Eyal Z.Goren (Montreal)

### MSC:

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

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |