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Contemporary Mathematics
A short guide to p-torsion of abelian varieties in
characteristic p
Rachel Pries
Abstract. There are many equivalent ways to describe the p-torsion of a principally polarized abelian variety in characteristic p. We briefly explain these methods and then illustrate them for abelian varieties A of arbitrary dimension 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 complete tables for abelian varieties of dimension at most four.
- Introduction One attribute of every complex abelian variety of dimension g is that its p- torsion points form a group of order p 2 g^. In contrast, the p-torsion points on a g-dimensional abelian variety defined over an algebraically closed field k of charac- teristic p form a group of order at most p g^. Exceptional research has emerged in response to this phenomenon, from early work on Picard schemes to recent results on stratifications of moduli spaces of abelian varieties. The p-torsion of a principally polarized abelian variety defined over k can be described in terms of a group scheme or a Dieudonn´e module. It can be classified using its final type or Young type. It can be identified with an element in the Weyl group 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 give a thorough description of the p-torsion of a principally polarized abelian variety A of arbitrary dimension 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 complete tables for the p-torsion types that occur for g ≤ 4, including the sixteen types of p-torsion that occur for abelian varieties of dimension four.
1991 Mathematics Subject Classification. 11G15, 14K10. Key words and phrases. abelian variety, group scheme, p-torsion. The author would like to thank the NSF for its partial support from grant DMS-04- and J. Achter for his comments on previous versions of this paper.
⃝^ c 0000 (copyright holder) 1
2 RACHEL PRIES
- Methods to classify the p-torsion Let k be an algebraically closed field of characteristic p. Let Ag := Ag ⊗ k be the moduli space of principally polarized abelian varieties of dimension g defined over 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 flat morphism of degree p 2 g^. It factors as [p] = Ver ◦ Fr. Here Fr : A → A(p)^ is the relative Frobenius morphism coming from the p-power map on the structure sheaf; it is purely inseparable of degree p g^. The Verschiebung morphism Ver : A(p)^ → A is the 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 different ways of describing A[p].
2.1. Group schemes. The p-torsion A[p] is a finite commutative group scheme annihilated by p with rank p 2 g^ , again having morphisms Fr and Ver. Then A[p] is called a quasi-polarized BT 1 k-group scheme (short for quasi-polarized truncated Barsotti-Tate group of level 1). The quasi-polarization implies that A[p] is sym- metric. These group schemes were classified independently by Kraft (unpublished) [Kra] and by Oort [Oor01]. A complete description of this topic can be found in [Oor01] or [Moo01].
Example 2.1. Consider the constant group scheme Z/p = Spec(⊕ (^) γ∈Z/p k) with co-multiplication m ∗^ (γ) =
δ∈Z/p γδ^ ⊗^ δ^
− (^1) and co-inverse inv ∗ (^) (γ) = γ − (^1).
Also consider μp which is the kernel of Frobenius on G (^) m. 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 ordinary elliptic curve then E[p] ≃ Z/p ⊕ μp. We denote this group scheme by L.
Example 2.2. Let αp be the kernel of Frobenius on G (^) a. As a k-scheme, αp ≃ Spec(k[x]/xp^ ) with co-multiplication m ∗^ (x) = x ⊗ 1 + 1 ⊗ x and co-inverse inv ∗^ (x) = −x. The isomorphism type of the p-torsion of any two supersingular elliptic curves is the same. If E is a supersingular elliptic curve, we denote the isomorphism type of its p-torsion by I 1 , 1. By [Gor02, Ex. A.3.14], I 1 , 1 fits into a non-split exact sequence of the form 0 → αp → I 1 , 1 → αp → 0. The image of the embedded αp is unique and is the kernel of both Frobenius and Verschiebung.
Example 2.3. Let A be a supersingular non-superspecial abelian surface. In other words, A is isogenous, but not isomorphic, to the direct sum of two su- persingular elliptic curves. Let I 2 , 1 denote the isomorphism class of the group scheme A[p]. By [Gor02, Ex. A.3.15], there is a filtration H 1 ⊂ H 2 ⊂ I 2 , 1 where H 1 ≃ αp , H 2 /H 1 ≃ αp ⊕ αp , and I 2 , 1 /H 2 ≃ αp. Also H 2 contains both the kernel G 1 of Frobenius and the kernel G 2 of Verschiebung. There is an exact sequence 0 → H 1 → G 1 ⊕ G 2 → H 2 → 0.
The p-rank and a-number. Two invariants of (the p-torsion of) an abelian variety are the p-rank and a-number. The p-rank of A is f = dimF (^) p Hom(μp , A[p]). Then p f^ is the cardinality of Ap. The a-number of A is a = dim (^) k Hom(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 group scheme I 2 , 1 in Example 2.3 has p-rank 0 since it is an iterated extension of copies of αp and has a-number 1 since ker(V 2 ) = G 1 ⊕ G 2 has rank p 3.
4 RACHEL PRIES
The Young type of G was introduced by Van der Geer [vdG99] as a means of describing the Ekedahl-Oort strata in terms of degeneration loci for maps between flag varieties. The codimension in Ag of the stratum whose points have Young type μ is
∑a j=1 μj^. 2.5. Elements of the Weyl group. One can associate to μ an element ω of the Weyl group Wg of the sympletic group Sp 2 g [vdG99]. Here Wg is identified with the subgroup of all ω ∈ S (^2) g such that ω(i)+ω(2g+1−i) = 2g+1 for 1 ≤ i ≤ g. This subgroup is generated by the following involutions: s (^) i = (i, i + 1)(2g − i, 2 g + 1 − i) for 1 ≤ i < g; and s (^) g = (g, g + 1). Given a Young type μ, one defines ω as follows. For 1 ≤ i ≤ g, let ω(i) = c (respectively ω(i) = g + c) if i is the cth number such that μ (^) i = μi+1 (respectively μi ̸= μi+1 ). For 1 ≤ i ≤ g, let ω(2g + 1 − i) = 2g + 1 − ω(i). This yields an element of Wg. One can express ω as a word in the involutions s 1 ,... , s (^) g of S (^2) g , although this expression is not unique. For example, in the ordinary case where μ = ∅, then ω is given by ⟨ 1 ,... , 2 g⟩ ω → ⟨g + 1,... , 2 g, 1 ,... , g⟩. In the superspecial case where μ = {g,... , 1 }, then ω = id. Further examples with g ≤ 4 are in Section 4. We briefly explain the importance of the Weyl group characterization. There is a second filtration of D(G) which is stable under the action of F and V −^1 , which we denote by N 1 ′ ⊂ N 2 ′ ⊂ · · · ⊂ N 2 ′g. Then ω measures the interaction between these two filtrations. For example, when G is ordinary (f = g) then N (^) g ∩ N (^) g′ = 0. Informally speaking, this means that the intersection of Im(V ) and (a twist under σ of) Im(F ) is trivial. When G is superspecial (a = g), then dim(N (^) i ∩ N (^) g′ ) = i for 1 ≤ i ≤ g. Informally speaking, this implies that N (^) i is contained in (a twist under σ of) Im(F ). In general, dim(N (^) i ∩ N (^) g′ ) ≥ i − ν (^) i. The a-number is dim(V D(G) ∩ F D(G)) = dim(N (^) g ∩ N (^) g′ ). One can calculate the cycle classes of the closures of the Ekedahl-Oort strata in the tautological ring of Ag. Let λi for 1 ≤ i ≤ g be the Chern classes of the Hodge bundle of Ag. These classes generate the tautological subring of CH (^) Q∗ (Ag ) and satisfy (1 + λ 1 + · · · + λg )(1 − λ 1 + · · · + (−1)g^ λg ) = 1 [vdG99, Thm. 1.1].
- Important examples 3.1. Abelian varieties with p-rank f. Given g and f such that 0 ≤ f ≤ g, let Vg,f denote the stratum of Ag whose points correspond to principally polarized abelian varieties A of dimension g with f (^) A ≤ f. The generic point of Vg,g = Ag corresponds to an abelian variety with p-rank g, a-number 0, and A[p] ∼= L g^. If f < g, every component of Vg,f has codimension g − f , and its generic point has a-number 1, [NO80, Thm. 4.1]. In this section, we describe the p-torsion group scheme that occurs for the generic point(s) of Vg,f.
Lemma 3.1. Let r ∈ N. There is a unique symmetric BT 1 group scheme of rank p 2 r^ with p-rank 0 and a-number 1, which we denote Ir, 1. The covariant Dieudonn´e module of Ir, 1 is E/E(F r^ + V r^ ).
Proof. Let Ir, 1 be a symmetric BT 1 group scheme of rank p 2 r^ with p-rank 0 and a-number 1. It is sufficient to show that the final type of Ir, 1 is uniquely determined. The p-rank 0 condition implies that V acts nilpotently on D(Ir, 1 ), so ν 1 = 0. The a-number 1 condition implies that r − 1 is the dimension of V 2 D(Ir, 1 ),
A SHORT GUIDE TO p-TORSION OF ABELIAN VARIETIES IN CHARACTERISTIC p 5
so νr = r − 1. The restrictions on ν (^) i imply that there is a unique final type possible for Ir, 1 , namely ν = [0, 1 ,... , r − 1]. Consider D = E/E(F r^ + V r^ ). Note that F r+1^ = 0 and V r+1^ = 0 on D. Then D is an E-module with dimension 2r as a k-vector space. It has basis {F,... , F r^ , 1 , V,... , V r−^1 }. Then V D has basis {V,... , V r−^1 , F r^ } and V 2 D has basis {V 2 ,... , V r−^1 , F r^ }. Thus D has a-number 1. Continuing, one sees that V is nilpotent on D and thus the p-rank of D is 0. Thus D must be the covariant Dieudonn´e module D(Ir, 1 ).!
Proposition 3.2. Let A ∈ Ag (k) be a principally polarized abelian variety of dimension g with p-rank f and a-number 1. Then A[p] ≃ L f^ ⊕Ig−f, 1. The covariant Dieudonn´e module of A[p] is
D ≃ (E/E(F, 1 − V ) ⊕ E/E(V, 1 − F ))f^ ⊕ E/E(F g−f^ − V g−f^ ).
The final type of A[p] is ν = [1,... , f, f,... , g − 1] (or [0,... , 0] if f = 0). The Young type is μ = {g − f } (or ∅ if f = g).
Proof. The group scheme A[p] must include f copies of L along with a group scheme of rank p 2(g−f^ )^ with p-rank 0 and a-number 1. By Lemma 3.1, the only possibility for the latter is Ig−f, 1. The statement about the Dieudonn´e module follows immediately. For the final type, note that ν (^) g = g − 1 since A[p] has a- number 1 and ν (^) f = f since A[p] has p-rank f. The numerical restrictions on ν (^) i then imply that ν = [1,... , f, f,... , g − 1]. The Young type follows by direct calculation.!
If f < g, one can show that the group scheme L f^ ⊕ Ig−f, 1 corresponds to the element ω of the Weyl group such that ω(f + 1) = 1 and ω : { 1 ,... , g} − {f + 1} 1 → {g + 1,... , 2 g − 1 } is increasing. The cycle class of the (reduced) stratum Vg,f in the tautological ring of Ag is given by (p − 1)(p 2 − 1)... (p g−f^ − 1)λg−f [vdG99, Thm. 2.4].
3.2. Abelian varieties with given a-number. Given g and f such that 0 ≤ f ≤ g, let T (^) g,a denote the stratum of Ag whose points correspond to principally polarized abelian varieties of dimension g with a (^) A ≥ a. Then T (^) g,a is irreducible unless a = g [vdG99, Thm. 2.11]. In this section, we describe the p-torsion that occurs for the generic point(s) of T (^) g,a. It is known that T (^) g,a has codimension a(a + 1)/2. The generic point(s) of T (^) g,a have a-number a and p-rank g − a.
Proposition 3.3. Let A ∈ Ag (k) be an abelian variety of dimension g with p- rank f and a-number g − f. Then A[p] ≃ L f^ ⊕ (I 1 , 1 )g−f^. The covariant Dieudonn´e module of A[p] is
D ≃ (E/E(F, 1 − V ) ⊕ E/E(V, 1 − F ))f^ ⊕ (E/E(F + V ))g−f^.
The final type is ν = [1,... , f,... , f ] (or [0,... , 0] if f = 0). The Young type is μ = {g − f, g − f − 1 ,... , 1 } (or ∅ if g = f ).
Proof. The group scheme A[p] must include f copies of L along with a group scheme of rank p 2(g−f^ )^ with p-rank 0 and a-number g − f. The only possibility for the latter is g − f copies of D(I 1 , 1 ). The statement about the Dieudonn´e module follows immediately. For the final type, note that ν (^) g = f since A[p] has a-number g − f and ν (^) f = f since A[p] has p-rank f. The numerical restrictions on ν (^) i then imply that ν = [1,... , f,... , f ]. The Young type follows by direct calculation.!
A SHORT GUIDE TO p-TORSION OF ABELIAN VARIETIES IN CHARACTERISTIC p 7
One can check that V 2 D has basis 0 ⊕ 0 ⊕ ⟨V 2 ⟩ and thus I 4 , 3 has a-number 3. Also I 4 , 3 has p-rank 0 since V acts nilpotently on D(I 4 , 3 ). By the process of elimination, I 4 , 3 has final type [0, 0 , 1 , 1] and Young type { 4 , 3 , 1 }.
- Complete tables for dimension at most four For convenience, we provide tables for dimension g ≤ 4. Some parts of these tables can be found in [EvdG]. The second column gives the codimension of the strata in Ag.
4.1. The case g = 1:
Name codim f a ν μ ω cycle class (reduced) L 0 1 0 [1] ∅ s 1 λ 0 I 1 , 1 1 0 1 [0] { 1 } 1 (p − 1)λ 1
4.2. The case g = 2:
Name codim f a ν μ ω cycle class (reduced) L 2 0 2 0 [1, 2] ∅ s 2 s 1 s 2 λ 0 L ⊕ I 1 , 1 1 1 1 [1, 1] { 1 } s 1 s 2 (p − 1)λ 1 I 2 , 1 2 0 1 [0, 1] { 2 } s 2 (p − 1)(p 2 − 1)λ 2 I (^12) , 1 3 0 2 [0, 0] { 2 , 1 } 1 (p − 1)(p 2 + 1)λ 1 λ 2
This is the smallest dimension for which the Newton polygon of A does not determine the group scheme A[p]. The Newton polygon 2G 1 , 1 (supersingular, with four slopes of 1/2) occurs for both (I 1 , 1 )^2 and I 2 , 1.
4.3. The case g = 3:
Name codim f a ν μ ω L 3 0 3 0 [1, 2 , 3] ∅ s 3 s 2 s 3 s 1 s 2 s (^3) L 2 ⊕ I 1 , 1 1 2 1 [1, 2 , 2] { 1 } s 2 s 3 s 1 s 2 s (^3) L ⊕ I 2 , 1 2 1 1 [1, 1 , 2] { 2 } s 3 s 1 s 2 s (^3) L ⊕ I (^12) , 1 3 1 2 [1, 1 , 1] { 2 , 1 } s 1 s 2 s (^3) I 3 , 1 3 0 1 [0, 1 , 2] { 3 } s 3 s 2 s (^3) I 3 , 2 4 0 2 [0, 1 , 1] { 3 , 1 } s 2 s (^3) I 1 , 1 ⊕ I 2 , 1 5 0 2 [0, 0 , 1] { 3 , 2 } s (^3) I 1 3 , 1 6 0 3 [0, 0 , 0] { 3 , 2 , 1 } 1
This is the smallest dimension for which the group scheme A[p] does not de- termine the Newton polygon of A. If A[p] ≃ I 3 , 1 , then the Newton polygon of A is usually G 1 , 2 + G 2 , 1 (three slopes of 1/3 and of 2/3) but by [Oor91, Thm. 5.12] it can also be 3G 1 , 1 (supersingular, with six slopes of 1/2). The cycle classes for this table can be found in [EvdG, 15.2].
8 RACHEL PRIES
4.4. The case g = 4: Name codim f a ν μ ω L 4 0 4 0 [1, 2 , 3 , 4] ∅ s 4 s 3 s 4 s 2 s 3 s 4 s 1 s 2 s 3 s (^4) L 3 ⊕ I 1 , 1 1 3 1 [1, 2 , 3 , 3] { 1 } s 3 s 4 s 2 s 3 s 4 s 1 s 2 s 3 s (^4) L 2 ⊕ I 2 , 1 2 2 1 [1, 2 , 2 , 3] { 2 } s 4 s 2 s 3 s 4 s 1 s 2 s 3 s (^4) L 2 ⊕ I (^21) , 1 3 2 2 [1, 2 , 2 , 2] { 2 , 1 } s 2 s 3 s 4 s 1 s 2 s 3 s (^4) L ⊕ I 3 , 1 3 1 1 [1, 1 , 2 , 3] { 3 } s 4 s 3 s 4 s 1 s 2 s 3 s (^4) L ⊕ I 3 , 2 4 1 2 [1, 1 , 2 , 2] { 3 , 1 } s 3 s 4 s 1 s 2 s 3 s (^4) I 4 , 1 4 0 1 [0, 1 , 2 , 3] { 4 } s 4 s 3 s 4 s 2 s 3 s (^4) L ⊕ I 1 , 1 ⊕ I 2 , 1 5 1 2 [1, 1 , 1 , 2] { 3 , 2 } s 4 s 1 s 2 s 3 s (^4) I 4 , 2 5 0 2 [0, 1 , 2 , 2] { 4 , 1 } s 3 s 4 s 2 s 3 s (^4) L ⊕ I (^13) , 1 6 1 3 [1, 1 , 1 , 1] { 3 , 2 , 1 } s 1 s 2 s 3 s (^4) I 1 , 1 ⊕ I 3 , 1 6 0 2 [0, 1 , 1 , 2] { 4 , 2 } s 4 s 2 s 3 s (^4) I 1 , 1 ⊕ I 3 , 2 7 0 3 [0, 1 , 1 , 1] { 4 , 2 , 1 } s 2 s 3 s (^4) I 2 , 1 ⊕ I 2 , 1 7 0 2 [0, 0 , 1 , 2] { 4 , 3 } s 4 s 3 s (^4) I 4 , 3 8 0 3 [0, 0 , 1 , 1] { 4 , 3 , 1 } s 3 s (^4) I (^12) , 1 ⊕ I 2 , 1 9 0 3 [0, 0 , 0 , 1] { 4 , 3 , 2 } s (^4) I (^14) , 1 10 0 4 [0, 0 , 0 , 0] { 4 , 3 , 2 , 1 } 1
The cycle classes for this table can be found in [EvdG, 15.3]. It is not straight-forward to determine which Ekedahl-Oort strata lie in the boundary of which others. When g = 4, the answer to this question is given by the natural partial ordering on the Young type, which matches the Bruhat-Chevalley order on the elements of the Weyl group.
References
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