Moduli of abelian varieties Ching-Li Chai and Frans Oort Version 6, Nov 2017, edited by CLC changes: Conjecture 8.3 ! Conjecture 8B Next: further revision by FO Have revised and imposed some uniformity in the bibliography Add more questions? Introduction We discuss three questions listed in the 1995 collection [107] and (partial) answers to them. We will also describe ideas connected with other problems on related topics. Throughout this article p denotes a fixed prime number. 0.1. [107, Conjecture 8B] (irreducibility of Newton Polygon strata). Let β be a non- supersingular symmetric Newton polygon. Denote by Wβ;1 the Newton polygon stratum in the moduli space Ag;1 ⊗Fp of g-dimensional principally polarized abelian varieties in characteristic p. We made the following conjectured in 1995. Every non-supersingular NP stratum Wβ;1 in Ag;1 ⊗ Fp is geometrically irreducible. We knew at that time that (for large g) the supersingular locus Wσ in Ag;1 ⊗ Fp is geo- metrically reducible; see 6.7 for references. And we had good reasons to believe that every non-supersingular Newton polygon strata in Ag;1⊗Fp should be geometrically irreducible. This conjecture was proved in [14, Thm. 3.1]. See Section 6 for a sketch of a proof, its ingredients and other details. 0.2. [107, x 12] (CM liftings). Tate proved that every abelian variety A defined over a finite 0 κ field is of CM type, that is its endomorphism algebra Endκ(A) := Endκ(A) ⊗Z Q contains a commutative semisimple algebra over Q whose dimension over Q is twice the dimension of A ; see [145], Th. 2 on page 140. Honda and Tate proved that every abelian variety over Fp is isogenous to an abelian variety which admits a CM lifting to characteristic zero; see [44, Main Theorem, p. 83] and [146, Thm. 2, p. 102]. More precisely: For any abelian variety A over a finite field κ = Fq, there exists a finite extension field K ⊃ κ, an abelian variety B0 over K, a K-isogeny between A ⊗κ K and B0, and a CM lift of B0 to characteristic zero. Two follow-up questions arise naturally. Is a field extension K/κ necessary (for the existence of a CM lift)? Is (a modification by) an isogeny necessary (for the existence of a CM lift)? 1 The answer to the second question above is \yes, an isogeny is necessary in general": we knew in 1992 [104] that there exist abelian varieties defined over Fp which do not admit CM lifting to characteristic zero. Later, more precise questions were formulated, in 0.2.1 and 0.2.2 below. 0.2.1. Suppose that A is an abelian variety over a finite field κ. Does there exists a κ-isogeny A ! B0 such that B0 admits a CM lift to characteristic zero? In other words, is a field extension necessary in the Honda-Tate theorem? 0.2.2. Suppose that A is an abelian variety over a finite field κ. Does there exist a κ-isogeny A ! B0 and a CM lifting of B0 to a normal mixed characteristics local domain with residue field κ? Complete answers to 0.2.1, 0.2.2 and some further CM lifting questions are given in [10]. Below, in Section 7 we will describe the questions, methods and results. 0.3. [107, x 13] (generalized canonical coordinates). Around a moduli point x0 = [X0; λ0] of an ordinary abelian variety over a perfect field k ⊃ Fp, the formal completion =x0 ∼ ^g(g+1)=2 Ag;1 = (Gm) of the moduli space Ag;1 over W (k) is a formal torus, where the origin is chosen to be the canonical lift of (X0; λ0); see [68], Chapter V; as proved by Drinfeld [23], see [56], also see [48]. In [107, 13A] we asked for \canonical coordinates" around any moduli point in Ag ⊗Fp, with a note that this question \should be made much more precise before it can be taken seriously." We will show results and limitations: (a) We do not know how to produce \canonical coordinates" in mixed characteristics (0; p) for the formal completion of Ag;1 at a non-ordinary point, nor do we know a how to produce good or useful \canonical coordinates" for the formal completion of Ag;1 ⊗ Fp at a non-ordinary point. (b) We (think we) can construct the right concept around an Fp point of a central leaf in Ag ⊗ Fp, in the sense that the formal completion at y0 of this central leaf has a natural structure, being built up from p-divisible formal groups by a family of fibrations. Note that the central leaf through the moduli point of an ordinary abelian variety is the dense open subset of the moduli space consisting of all ordinary point. So this statement (b) can be regarded as a generalization of the Serre-Tate canonical coordinates. This is a weak generalization; its limitation is given in (a). { The central leaf passing through an almost ordinary abelian variety (i.e. the p-rank equals dim(A) − 1) is a dense open subset in the of the Newton polygon stratum in characteristic p attached to the given abelian variety. The \generalized canonical coordinates" we produce is only for equi-characteristic p, in the sense that it gives ^ ^ a natural structure of the formal completion C(x0) of the leaf C(x0) at the given ^ almost ordinary point x0. One can show that C(x0) does not admit a functorial lift to characteristic 0. Similarly the locus of all almost ordinaries in Ag does not admit a lift to a flat scheme over the ring of integers of a finite extension field of Qp which is invariant under all prime-to-p Hecke correspondences. 2 { For other moduli points in positive characteristic central leaves are smaller than the related Newton polygon stratum; we do construct a generalization (analogue) of canonical coordinates on (the formal completion of) central leaves in characteristic p. But we do not construct something like \canonical coordinates" on the whole Newton polygon stratum if the p-rank is smaller than dim(A) − 1. We will explain why and provide references. For much more information see [17]. 1 p-divisible groups In this section we briefly recall notations we are going to use. 1 1.1. For given integers d 2 Z>0 and h 2 Z≥d, a Newton polygon ζ of height h and dimension d 2 is a lower convex polygon on R starting at (0; 0) and ending at (h; d) and having break points in Z × Z; lower convex means that the whole polygon lies on or above every line spanned by a segment of the polygon. Equivalently: there exist rational numbers 1 ≥ s1 ≥ · · · ≥ sh ≥ 0, called the slopes of ζ such that • s1 + ··· + sh = d, sh + ··· + sb 2 Z for all b such that sb+1 < sb, and • The slopes of the polygon ζ in the open intervals (0; 1);:::; (h − 1; h) are sh; sh−1; : : : ; s1 respectively. Thus ζ is the graph of the convex piecewise linear function ζ on [0; h] whose derivative is equal to sh−i+1 on the open interval (i − 1; i) for i = 1; : : : ; h. Explicitly, X ζ(t) = (t + b − h)sb + si b+1≤i≤h for all integer b 2 [1; h] and all real numbers t 2 [h − b; h − b + 1]. The reason to list the slopes in the \reverse order" comes from geometry: the first non- trivial sub-p-divisible subgroup in the slope filtration of a p-divisible group X over a field of characteristic p with Newton polygon ζ has the largest (Frobenius) slope among the slopes of ζ; see [165], [130]. A Newton polygon with slope sequence s1; : : : ; sh as above will be denoted by NP(s1; : : : ; sh): Below are some further definitions and terminologies. 0 0 0 (i) Suppose that 1 ≥ s1 ≥ s2 ≥ · · · ≥ sa ≥ 0 are rational numbers and h1; : : : ; ha ≥ 1 are 0 positive integers, and the sequence 1 ≥ s1; : : : ; sh resulting from repeating si hi-times 0 0 for each i = 1; : : : ; r will be denoted by h1 ∗s1; : : : ; ha ∗sa. If the latter is the sequence of slopes of a Newton polygon of height h1 + ··· + ha, the corresponding Newton polygon will be denoted by 0 0 NP h1 ∗ s1; : : : ; ha ∗ sa : (ii) Suppose that (m1; n1);:::; (ma; na) are a pairs of relatively prime natural numbers such that m1 > m2 > ··· > ma , and µ ; : : : ; µ are positive integers. We introduce m1+n1 m2+n2 ma+na 1 a another notation NP µ1 ∗ (m1; n1) + ··· + µa ∗ (ma; na) 1Here we only consider Newton polygons whose slopes are between 0 and 1, suitable for p-divisible groups. Under this restriction the notion of dimension and height of a Newton polygon corresponds to the dimension and hight of p-divisible groups. 3 for the Newton polygon which according to the shorthand introduced in (i) is NPµ (m + n ) ∗ m1 ; : : : ; µ (m + n ) ∗ ma : 1 1 1 m1+n1 1 a a ma+na µ1 µa This is the Newton polygon attached to the p-divisible group Gm1;n1 × · · · × Gma;na ; see the paragraph after the statement of theorem 1.4. (iii) For any Newton polygon ζ with slopes 1 ≥ s1 ≥ · · · ≥ sh of height h, the p-rank of ζ, denoted by f(ζ), is f(ζ) := fi j si = 0; 1 ≤ i ≤ hg, the number of slopes of ζ which are equal to 0. Clearly f(ζ) ≤ ht(ζ) − dim(ζ). (iv) A Newton Polygon is called isoclinic if all slopes are equal; in this case sj = d=h for every j.