Contemporary MathematicsA short guide to p-torsion of abelian varieties incharacteristic pRachel PriesAbstract. There are many equival ent ways to describe the p-torsion of aprincipally polarized abelian variety in characteristic p. We briefly explainthese methods and then illustrate them for abelian varieties A of arbitrarydimension g in several important cases, including when A has p-rank f and a-number 1 and when A has p-rank f and a-number g − f. We provide completetables for abelian varieties of dimension at most four.1. Introductio nOne attribute of e very complex abelian variety of dimension g is that its p-torsion points form a group of order p2g. In contrast, the p-torsion points on ag-dimensional a belian va riety defined over an algebraically closed field k of charac-teristic p for m a group of or der at mos t pg. Exceptio nal research has emerged inresponse to this phenomenon, from early work on Picard schemes to r e c e nt resultson stratifications of moduli spaces of abelian varieties.The p-torsion of a principally polariz e d abelian variety defined over k can bedescribed in terms of a group scheme or a Dieudonn´e module. It can b e classifiedusing its final type or Young type. It can be identified with an element in the Weylgroup of the sympletic group or with a cycle class in the tautological ring of Ag.In this paper, we briefly summarize the main types of classification. We givea thorough description of the p-torsion of a principally polarized abelian variety Aof arbitrary dimension g in several important cases, including when A has p -rankf and a-number 1, and when A has p-rank f and a-number g − f. We providecomplete ta bles for the p-torsion types that occur for g ≤ 4, including the sixteentypes of p-tor sion that occur for abelian varieties of dimension four.1991 Mathematics Subjec t Classification. 11G15, 14K10.Key words and phrases. abelian var iety, group scheme, p-torsion.The author would like to thank the NSF for its partial support from grant DMS-04-00461and J. Achter for his comments on previous versions of this paper.c0000 (copyright holder)12 RACHEL PRIES2. Methods to classify the p-torsionLet k be an algebraically closed field of characteristic p. Let Ag:= Ag⊗ k bethe moduli space of principally polarized abelian varieties of dimension g definedover k. Let A ∈ Ag(k) be an abelian variety of dimension g defined over k.Consider the multiplication-by-p morphism [p] : A → A which is a proper flatmorphism of degr ee p2g. It factors as [p] = Ver ◦ Fr. Here Fr : A → A(p)is therelative Frobenius morphism coming from the p-power map on the structure sheaf;it is pure ly inseparable of degree pg. The Vers chiebung morphism Ver : A(p)→ A isthe dual of Fr. If A is principally polarized, then im(Fr ) = ker(Ver) and im(Ver) =ker(Fr).The kernel of [p] is A[p], the p-torsion of A. We summarize several differentways of describing A[p].2.1. Group schemes. The p-torsion A[p] is a finite commutative group schemeannihilated by p with rank p2g, aga in having morphisms Fr and Ver. Then A[p] iscalled a quasi-polarized BT1k-group scheme (short for quasi-polarized truncatedBarsotti-Tate group of level 1). The quasi-pola rization implies that A[p] is s ym-metric. These group schemes were c lassified independently by Kraft (unpublished)[Kra] and by Oor t [Oor01]. A complete description o f this topic can be found in[Oor01] or [Moo01].Example 2.1. Consider the constant group scheme Z/p = Spec(⊕γ∈Z/pk)with co-multiplication m∗(γ) =Pδ∈Z/pγδ ⊗ δ−1and co-inverse inv∗(γ) = γ−1.Also consider µpwhich is the kernel of Frobe nius on Gm. As a k-scheme, µp≃Spec(k[x]/(xp− 1)) with co- multiplication m∗(x) = x ⊗ x and co-inverse inv∗(x) =x−1. If E is an ordinar y elliptic curve then E[p] ≃ Z/p ⊕ µp. We denote this groupscheme by L.Example 2.2. Let αpbe the kernel of Frobenius on Ga. As a k-scheme,αp≃ Spec(k[x]/xp) with co-multiplication m∗(x) = x ⊗ 1 + 1 ⊗ x and co-inverseinv∗(x) = −x. The isomorphism type of the p-torsion of any two supersingularelliptic curves is the same. If E is a supersingular elliptic curve, we denote theisomorphism type of its p-torsion by I1,1. By [Gor02, Ex. A.3.14], I1,1fits into anon-split e xact sequence of the form 0 → αp→ I1,1→ αp→ 0. The image of theembedded αpis unique and is the kernel of b oth Frobenius and Verschiebung.Example 2.3. Let A be a supersingular non-superspecial abelian surface. Inother words, A is isogenous, but not isomorphic, to the direct sum of two su-persingular elliptic curves. Let I2,1denote the isomorphism class of the groupscheme A[p]. By [Gor02, Ex. A.3.15], there is a filtration H1⊂ H2⊂ I2,1whereH1≃ αp, H2/H1≃ αp⊕ αp, and I2,1/H2≃ αp. Also H2contains both the kernelG1of Frobenius and the kernel G2of Verschiebung. There is an exact sequence0 → H1→ G1⊕ G2→ H2→ 0.The p-rank and a-number. Two invariants of (the p-torsion of) an abelianvariety are the p-ra nk and a-number. The p-rank of A is f = dimFpHom(µp, A[p]).Then pfis the cardinality of A[p](k). The a -number of A is a = dimkHom(αp, A[p]).It is well-known that 0 ≤ f ≤ g and 0 ≤ a + f ≤ g.In Example 2.1, f = 1 and a = 0. In Example 2.2, f = 0 and a = 1. The groupscheme I2,1in Example 2.3 has p-rank 0 since it is an iterated extension of copiesof αpand has a-number 1 since ker(V2) = G1⊕ G2has rank p3.A SHORT GUIDE TO p-TORSION OF ABELIAN VARIETIES IN CHARACTERISTIC p 3An abelian variety A of dimension g is ordinary if A[p] has p-rank f = g. IfA is ordinary then A[p] ≃ Lg. At the other ex treme, A is superspecial if A[p] hasa-number a = g. In this case, A ≃ Egfor a supersingular elliptic curve E andA[p] ≃ Ig1,1[LO98, 1.6].2.2. Covariant Dieudonn´e modules. One can describe the p-torsion A[p]using the theory of covariant Dieudonn´e modules. This is the dual of the con-travariant theory found in [Dem86]; s e e also [Gor02, A.5]. Briefly, let σ denotethe Fro benius automorphism of k. Consider the non-commutative ring E = k[F, V ]generated by semi-linear operator s F and V with the relations F V = V F = 0and F λ = λσF and λV = V λσfor all λ ∈ k. Let E(A, B) denote the left idealEA + EB of E generated by A and B. A deep result is tha t the Dieudonn´e functorD gives an equivalence of catego ries between BT1group schemes G (with rank p2g)and finite left E-modules D(G) (having dimension 2g as a k-vector space). If G isquasi-polarize d, then