Group structures of elementary supersingular abelian varieties over finite fields

Hui Zhu MSRI, 1000 Centennial Drive, Berkeley, CA 94720, USA

                     zhu@msri.org

Version Nov 29, 1998

Abstract Let A be a supersingular abelian variety over a finite field k which is isogenous to a power of a simple abelian variety over k. Write the characteristic polynomial of the Frobenius endomorphism of A relative to k as f = g^{e} for a monic irreducible polynomial g and a positive integer e, we show that the group of k-rational points A(k) on A is isomorphic to (Z/g(1)Z)^{e} unless A’s simple component is of dimension 1 or 2, in which case we prove that A(k) is isomorphic to (Z/g(1)Z)^{a} ×(Z/ \frac{g(1)}{2} Z×Z/2Z)^{b} for some non-negative integers a, b with a + b = e. In particular, if the characteristic of k is 2 or A is simple of dimension greater than 2, then A(k) ≅ (Z/g(1)Z)^{e}.

AMS classification: 14K22, 06B15.

1 Introduction

We list some notation and terminology for this paper as follows: k is a finite field of characteristic p with q elements. Let k̄ be an algebraic closure of k. Let A be an abelian variety of dimension d defined over k. Let π be the Frobenius endomorphism of A relative to k and f its characteristic polynomial.

An abelian variety over k is elementary if it is k-isogenous to a power of a simple abelian variety over k. This definition is different from that of [15] (see [16, page 54]). An abelian variety A is elementary if and only if f = g^{e} for some monic irreducible polynomial g over Q and some positive integer e. An arbitrary abelian variety is k-isogenous to a product of elementary abelian varieties, and f = ∏_{i=1}^{t} g_{i}^{e_{i}} for distinct monic irreducible polynomials g_{i} over Q and positive integers e_{i}. An abelian variety A over k is supersingular if each complex root of f can be written in the form ζ\sqrt{q}, the product of some root of unity ζ and the positive square root \sqrt{q}.

Theorem 1.1. Let A be an elementary supersingular abelian variety over k and f = g^{e} as above. Then A(k) is isomorphic as an abelian group to (Z/g(1)Z)^{e} except in the following cases:

(1) p ≡ 3 mod 4, q is not a square, and A is k-isogenous to a power of a supersingular elliptic curve with g = X^{2} + q,

(2) p ≡ 1 mod 4, q is not a square, and A is k-isogenous to a power of a two dimensional abelian variety with g = X^{2} − q.

In these two exceptional cases, there are non-negative integers a, b with a+b = e such that

A(k) ≅ (Z/g(1)Z)^{a} × (Z/\frac{g(1)}{2}Z × Z/2Z)^{b} .

This result is particularly striking when p = 2 or A is simple with d > 2 for then A(k) ≅_{Z} (Z/g(1)Z)^{e}. In the latter case A(k) will be either cyclic or a product of two cyclic groups, since e = 1 or 2. (See Proposition 3.3).

We call an elementary supersingular abelian variety A exceptional if it belongs to either of the two isogeny classes stated in Theorem 1.1 (1) and (2). We will show (see Proposition 3.9) that if A is exceptional, then for every pair of non-negative integers a′, b′ with a′ + b′ = e, there exists an abelian variety A′ which is k-isogenous to A and

A′(k) ≅ (Z/g(1)Z)^{a′} × (Z/\frac{g(1)}{2}Z × Z/2Z)^{b′} .

In this paper End_{k}(A) denotes the ring of k-endomorphisms of A. Write End_{k}^{0}(A) = End_{k}(A)⊗_{Z}Q. Let Q[π] be the Q-subalgebra of End_{k}^{0}(A) generated by π, let O be its maximal order, and Z[π] its Z-subalgebra generated by π. The group A(k̄) is naturally an End_{k}(A)-module. Our results describe A(k̄) as a module over any subring of End_{k}(A) ∩ Q[π] that contains π. For any prime number l we write R_{(l)} for the localization of a commutative ring R at l, this notation should not be confused with that for the l-adic completion R_{l}.

Theorem 1.2. Let A be an elementary supersingular abelian variety over k of dimension d. Let R be a ring with Z[π] ⊆ R ⊆ End_{k}(A) ∩ Q[π]. Then there is a surjective R-module homomorphism

φ : A(k̄) → (R_{(p)}/R)^{e}

such that the cardinality of the kernel of φ divides 2^{d}. Furthermore, φ is an isomorphism when p = 2.

Suppose A is a simple supersingular abelian variety over k and R the endomorphism ring End_{k}(A) ∩ Q[π]: if d ≠ 2, then A(k̄) ≅_{R} (R_{(p)}/R)^{e}; if d = 2, then A(k̄) ≅_{R} (R_{(p)}/R)^{a} × (O_{(p)}/O)^{b} for some non-negative integers a, b with a + b = e. (See Proposition 3.8.)

The group structure of the k-rational points and k̄-rational points on an elliptic curve were studied by [3] (see also [13, Chapter V] and [6]). The group structure of the k-rational points on a supersingular elliptic curve can be found in [12, Chapter 4]. Descriptions of that nature for higher dimensional abelian varieties remain open. (Recently, independent of our work, the group structure of dimensional two supersingular abelian varieties was studied in [17].)

We develop the following idea for studying a supersingular abelian variety A over k: we show that the ring Z[ζ\sqrt{q}] is locally almost a discrete valuation ring at every prime; in fact it is a Bass order over some suitable subring (see section 2). Next we describe the Tate modules of A over Z[π]. Finally the group structure of A(k) follows by viewing A(k) as the kernel of the isogeny π − 1 on A(k̄) (see section 3). The group and module structures on an arbitrary supersingular abelian varieties over finite fields are studied in [18].

This paper is based on a portion of the author’s Berkeley Ph.D thesis. The author is deeply grateful to her advisor Professor Hendrik Lenstra for inspiration and guidance. The author also wish to thank Bjorn Poonen and Phil Ryan for their comments on an earlier version of this paper. The author was supported as a MSRI postdoctoral fellow while preparing this paper.

2 Torsion-free modules over Bass orders

2.1 Orders

We begin this section by introducing some notions from commutative algebra, then we prove several auxiliary results from algebraic number theory. The material largely follows [2, Introduction and Chapter 3]. Here we assume all rings are commutative and modules are finitely generated. Let K be a local or global field of zero characteristic and O_{K} its ring of integers.

Suppose L is a finite dimensional separable K-algebra. An O_{K}-algebra Λ is called an O_{K}-order (in L) if it is a finitely generated projective O_{K}-module (and Λ ⊗_{O_{K}} K = L). A Z-order is simply called an order. Let Λ be an O_{K}-order in L. We denote the unique maximal O_{K}-order in L by O_{L}. If M is a Λ-module which is projective over O_{K}, then M is called a Λ-lattice.

For any prime ℘ of O_{K}, we denote by (O_{K})_{℘}, Λ_{℘}, M_{℘} their ℘-adic completions, respectively. If K = Q and ℘ = l for some prime number l, then we write (O_{K})_{l}, Λ_{l}, M_{l} for their l-adic completions.

A Λ-module M is called torsion-free if αm ≠ 0 for any non-zerodivisor α ∈ Λ − {0} and m ∈ M − {0}. In particular, when Λ is a domain then this is equivalent to the standard definition. A O_{K}-module is projective if and only if it is torsion-free [4, II.4 (4.1)]. So M is a torsion-free Λ-module if and only if it is a Λ-lattice. If M is a torsion-free Λ-module, then it is torsion-free over O_{K}, hence there is a natural embedding M ↪ M ⊗_{O_{K}} K, where M ⊗_{O_{K}} K has a natural L-module structure. If M ⊗_{O_{K}} K is free of rank e over L for some integer e, then M is said of rank e. We shall note here that e is used to denote an arbitrary positive integer in this section.

Suppose M′ and M′′ are any torsion-free O_{K}-modules with M′ ⊗_{O_{K}} K = M′′ ⊗_{O_{K}} K, then the ideal index [M′′ : M′] is defined to be the fractional O_{K}-ideal generated by all determinants of the K-linear maps M′′ ⊗_{O_{K}} K →^{η} M′ ⊗_{O_{K}} K such that η(M′′) ⊆ M′. It is an integral O_{K}-ideal, that is, [M′′ : M′] ⊆ O_{K}, if and only if M′ ⊆ M′′. Let Tr_{L/K} denote the trace map from L to K. Denote the trace dual (also known as the complementary module in [7]) of Λ by Λ^{†} = {a ∈ L | Tr_{L/K}(aΛ) ⊆ O_{K}}. It is easy to see that Λ^{†} is a fractional ideal of Λ, and Λ ⊆ Λ^{†}. Define the discriminant (ideal) of Λ over O_{K} by Δ_{Λ/O_{K}} = [Λ^{†} : Λ]. Note that Λ ⊆ O_{L} ⊆ O_{L}^{†} ⊆ Λ^{†} and [Λ^{†} : O_{L}^{†}] = [O_{L} : Λ]. The discriminant of L over K is defined by Δ_{L/K} = Δ_{O_{L}/O_{K}} = [O_{L}^{†} : O_{L}] and so [O_{L} : Λ]^{2} Δ_{L/K} = Δ_{Λ/O_{K}}. Therefore, the relationship between the ideal index of Λ in O_{L} and the discriminant of Λ is [O_{L} : Λ]^{2} | Δ_{Λ/O_{K}} .

Suppose L is a finite field extension of K. Let α be an integral element in L and h ∈ O_{K}[X] be its (monic) minimal polynomial. Let Λ = O_{K}[α]. Then Δ_{Λ/O_{K}} = O_{K} Δ(h). Let ℘ be any non-zero prime ideal of O_{K}. Then Λ_{℘} is semisimple and Λ_{℘} ≅ ∏_{Q|℘} Λ_{℘} where the product ranges over all prime ideals Q of Λ lying over ℘. There is a bijective correspondence between these Q’s of Λ and the set of (monic) irreducible factors h̄_{0} of h̄ = (h mod ℘) ∈ (O_{K}/℘)[X]. (See [7, Chapter I, Proposition 25, page 27].) If Q corresponds to h̄_{0} in this bijection, then Q = (℘, h_{0}(α)) in Λ. We use the following notation throughout this paper: for any prime ideal v of O_{L} lying over a prime ℘ of O_{K}, let γ(v/℘), κ(v/℘) and ϱ(v/℘) denote the ramification index, residue field degree and decomposition degree, respectively. In particular, when Λ = O_{L} then κ(Q/℘) = dim_{O_{K}/℘} Λ/Q = deg(h̄_{0}) and γ(Q/℘) equals the multiplicity of h̄_{0} as a factor of h̄. We have the following lemma whose proof is due to Hendrik Lenstra.

Lemma 2.1. Let the notation be as above. Then Q is not invertible if and only if h̄_{0}^{2} | h̄ and all coefficients of the remainder of h upon division by h_{0} are in ℘^{2}. The (O_{K})_{℘}-order Λ_{℘} is not the maximal order if and only if there is a monic irreducible factor h̄_{0} of h̄ such that h̄_{0}^{2} | h̄, and all coefficients of the remainder of h upon division by h_{0} are in ℘^{2}.

Proof. Write J := (℘, h_{0}(X)) in O_{K}[X], it is a prime ideal. Then the natural surjective map O_{K}[X] → Λ induces a surjective map θ : J/J^{2} → Q/Q^{2} with Ker(θ) generated by h. Write h in base h_{0} and obtain h = r_{2}h_{0}^{2} + r_{1}h_{0} + r_{0} for some r_{2}, r_{1}, r_{0} ∈ O_{K}[X] with deg(r_{1}), deg(r_{0}) < deg(h̄_{0}). Then h ∈ J if and only if r_{0} ∈ ℘[X], while h ∈ J^{2} if and only if r_{1} ∈ ℘[X] and r_{0} ∈ ℘^{2}[X]. So dim_{Λ/Q} J/J^{2} = 1 + dim_{O_{K}/℘} ℘/℘^{2} = 2 and hence dim_{Λ/Q} Q/Q^{2} = dim_{Λ/Q} J/J^{2} − dim_{Λ/Q} Ker(θ) = 2 − dim_{Λ/Q} Ker(θ), where dim_{Λ/Q} Ker(θ) is 0 or 1. Therefore, dim_{Λ/Q} Q/Q^{2} ≠ 1 if and only if h ∈ J^{2}. We conclude that Q is not invertible if and only if h ∈ J^{2}, which is equivalent to h̄_{0}^{2} | h̄ and all coefficients of the remainder of h upon division by h_{0} are in ℘^{2}. Thus the semilocal ring Λ_{℘} is maximal if and only if Λ_{Q} is maximal for each prime ideal Q over ℘, which is equivalent to Q is invertible, and so follows our assertion.

Corollary 2.2. Let the notation be as in Lemma 2.1. If h_{0} = X − β with β ∈ O_{K}, then Q is not invertible if and only if h(β) ≡ 0 mod ℘^{2} and h′(β) ≡ 0 mod ℘, where h′ denotes the derivative of h.

Proof. The condition h̄_{0}^{2} | h̄ is equivalent to that h(β) ≡ 0 mod ℘ and h′(β) ≡ 0 mod ℘. The condition that all coefficients of the remainder of h upon division by h_{0} are in ℘^{2} is equivalent to h(β) ≡ 0 mod ℘^{2}.

The following is an easy exercise in algebraic number theory and so we leave its proof to the reader. In this paper (\frac{·}{p}) denotes the Jacobi symbol.

Lemma 2.3. Let p be a prime number in Z and m a positive integer. Then

(1) Δ_{Q(\sqrt{p})/Q} = p if p ≡ 1 mod 4 while Δ_{Q(\sqrt{p})/Q} = 4p if p ≢ 1 mod 4;

(2) \sqrt{p} ∈ Q(ζ_{m}) if and only if Δ_{Q(\sqrt{p})/Q} | m;

(3) let p ≠ 2, if p | m then Q(\sqrt{(\frac{−1}{p})p}) ∈ Q(ζ_{m}).

2.2 Bass orders

A reference for concepts in this subsection is [2, Chapter 4]. We call an O_{K}- order Λ a Gorenstein order if every exact sequence of Λ-modules 0 → Λ → M → N → 0, in which M and N are Λ-lattices is split over Λ. If Λ has the additional property that every O_{K}-order in L containing Λ is also a Gorenstein order, then we call Λ a Bass order. Note that being a Bass order is a local property, in other words, Λ is a Bass O_{K}-order if and only if Λ_{℘} is a Bass (O_{K})_{℘}-order for every prime ℘ in O_{K}.

Proposition 2.4. The following are equivalent:

(1) Λ is a Bass O_{K}-order;

(2) O_{L}/Λ is a cyclic Λ-module;

(3) every ideal of Λ can be generated by two elements;

(4) for every maximal ideal Q of Λ we have dim_{Λ_{Q}/QΛ_{Q}}(O_{L})_{Q}/Q(O_{L})_{Q} ≤ 2;

(5) the multiplicity of Λ at each maximal ideal Q is ≤ 2.

Proof. The first three parts are equivalent according to [8, Theorem 2.1]. The last two parts are equivalent to (1) by [5, Theorem 2.1].

Remark 2.5. Here are some examples of Bass orders of interest.

(i) If L is a quadratic field extension over K, then (O_{L})_{℘}/Λ_{℘} is cyclic over Λ_{℘} for every prime ℘ of O_{K} and thus Λ is a Bass order over O_{K}.

(ii) All maximal orders in number fields are Bass orders.

We are interested in describing torsion-free modules M over Λ_{℘} of rank e. Recall that Λ_{℘} is a semilocal ring whose maximal ideals are those prime ideals Q lying over ℘, so there is a corresponding decomposition of M as M ≅ ∏_{Q|℘} M_{Q}.

If Λ_{℘} = (O_{L})_{℘}, then M_{Q} is torsion-free over the principal ideal domain (O_{L})_{Q} of rank e, so M_{Q} ≅ Λ_{Q}^{e} for all Q. Thus M ≅ Λ_{℘}^{e}. If Λ_{℘} ≠ (O_{L})_{℘}, then it is generally hard to classify such modules M in terms of orders in L_{℘} (see [2, Chapter 3]). However, torsion-free modules over Bass orders can be described as follows.

Theorem 2.6 (Bass). Let K be a local field, O_{K} its ring of integers, and Λ a Bass O_{K}-order in a finite field extension L over K. Then every indecomposable torsion-free Λ-module is a projective Λ′-module for some O_{K}-order Λ′ in L containing Λ.

Proof. Follows from [2, Theorem (37.13)] and the definition of Bass orders.

2.3 Supersingular q-numbers

In this subsection there are the definition of E and the proof of Lemma 2.8, the latter of which is crucial for the technical part of this paper.

For the rest of the paper l is a prime number different from p. Let ζ_{m} = e^{\frac{2π\sqrt{−1}}{m}} for any positive integer m. The primitive m-th roots of unity are the ζ_{m}^{ν} with positive integers ν that are coprime to m. We write Φ_{m}(·) for the m-th cyclotomic polynomial, i.e., the minimal polynomial of ζ_{m}.

An algebraic number α ∈ C is called a supersingular q-number if it is of the form ζ\sqrt{q}. Let π = ζ_{m}^{ν}\sqrt{q}. Let K = Q(π^{2}) and let O, O_{K} be the ring of integers of Q(π), K, respectively. Obviously K = Q(ζ_{m/(2,m)}) and [Q(π) : K] = 1 or 2. (In this paper (m_{1}, m_{2}) := gcd(m_{1}, m_{2}) for any integer m_{1}, m_{2}.)

Lemma 2.7. Suppose q is a non-square. Then Q(π) = K if and only if

(1) Δ_{Q(\sqrt{p})/Q} | m, and (2) Δ_{Q(\sqrt{p})/Q} ∤ m/(2, m) if 4 | m.

In this case 2 | γ(v/p) for any prime v of O lying over p.

Proof. We note Q(π) = Q (ζ_{m/(2,m)}, \sqrt{pζ_{m/(2,m)}}) = K (\sqrt{pζ_{m/(2,m)}}) . Thus Q(π) = K if and only if \sqrt{pζ_{m/(2,m)}} ∈ K, and if and only if

K (\sqrt{p}) = K (\sqrt{ζ_{m/(2,m)}}) ,

that is, Q (ζ_{m/(2,m)}, \sqrt{p}) = Q(ζ_{m}). This is equivalent to

(1a) \sqrt{p} ∈ Q(ζ_{m}), and (2a) \sqrt{p} ∉ Q(ζ_{m/(2,m)}) if 4 | m,

which is equivalent to (1) and (2) respectively by Lemma 2.3 (2).

To show the second assertion it is enough to prove it for just one prime v over p since all primes lying over p are conjugate as Q(π) is the cyclotomic field Q(ζ_{m/(2,m)}). We claim (2, p)p | \frac{m}{(2,m)}. In fact, if p = 2 then (1) implies 8 | m by Lemma 2.3 (1) so our claim follows; if p ≠ 2 then (1) implies p | m. But since p ≠ 2, we have (2, p)p | \frac{m}{(2,m)}. By Lemma 2.3 (3), we thus see that Q(ζ_{m/(2,m)})

contains a quadratic field Q(\sqrt{(\frac{−1}{p})p}) over Q where p is totally ramified. Hence 2 | γ(v/p).

For any prime number l, let m_{l} and m_{(l)} denote, respectively, the l-part and the non-l-part of a positive integer m. Let E be the set of supersingular q-numbers ζ_{m}^{ν}\sqrt{q} which satisfy the following conditions: p ≠ 2, q is not a square, p ∤ m, and

(1) 4 ∤ m when p ≡ 1 mod 4; and (2) 4||m when p ≡ 3 mod 4.

We remark here that α ∈ Λ_{℘} is a unit if and only if α is coprime to ℘.

Lemma 2.8. Let the notation be as above. If (l, π) ∉ {2} × E then Z[π]_{l} = O_{l}. If (l, π) ∈ {2} × E then Z[π]_{2} ⊊ O_{2}; let ℘ be any prime in O_{K} lying over 2, then

(1) Q(π) is a quadratic extension over K where ℘ is split if p ≡ ±1 mod 8, and ℘ is inert if p ≡ ±3 mod 8.

(2) Z[π]_{℘} is a local ring and a Bass (O_{K})_{℘}-order such that O_{℘} is the only (O_{K})_{℘}-order in Q(π)_{℘} that properly contains Z[π]_{℘}. Moreover, O_{℘}/Z[π]_{℘} ≅_{(O_{K})_{℘}} (O_{K})_{℘}/℘.

Proof. If q is a square, then π ∉ E and Z[π]_{l} = Z[ζ_{m}]_{l} = O_{l}. For the rest of the proof, we assume q is not a square. We consider the following two cases.

Case 1. l ≠ 2 or p | m.

We claim Z[π]_{l} = O_{l}. Since l ≠ p, we note that Z[π^{2}]_{l} = Z[ζ_{m/(2,m)}]_{l} = (O_{K})_{l}. Suppose l ≠ 2, obviously Z[π^{2}]_{l} = (O_{K})_{l} ⊆ Z[π]_{l} ⊆ O_{l}. If Q(π) = K then O_{l} = (O_{K})_{l} and so Z[π]_{l} = O_{l}. If Q(π) ≠ K then [O : Z[π]]_{l}^{2} | Δ_{Z[π]/(O_{K})}; but since Z[π] ≅ O_{K}[X]/(X^{2} − qζ_{m/(2,m)}), we have Δ_{Z[π]/O_{K}} = O_{K}Δ(X^{2} − qζ_{m/(2,m)}^{ν}) = 4qO_{K}. So (Δ_{Z[π]/O_{K}})_{l} = (O_{K})_{l} since 4q is coprime to l. Therefore [O : Z[π]]_{l} is the unit ideal and so Z[π]_{l} = O_{l}.

Now let l = 2 and p | m. By Lemma 2.3 (3), \sqrt{(\frac{−1}{p})p} ∈ Z[ζ_{m/(2,m)}]_{2} =

Z[π^{2}]_{2} ⊆ Z[π]_{2}. Moreover, the norm of \sqrt{(\frac{−1}{p})p} over Q is ±p which is co-

prime to 2 so \sqrt{(\frac{−1}{p})p} is a unit in Z[π]_{2}. Therefore, Z[π]_{2} = Z[π\sqrt{(\frac{−1}{p})p}]_{2} =

Z[ζ_{m}^{ν}\sqrt{(\frac{−1}{p})}]_{2}. This proves our claim.

Case 2. l = 2 and p ∤ m.

Write m = 2^{j}m_{(2)}. It is easy to verify that Q(π) = Q(ζ_{m_{(2)}}, α) where α = ζ_{2^{j}}^{μ} \sqrt{p} for some 2^{j}-th primitive root of unity ζ_{2^{j}}^{μ}. We note that Q(ζ_{m_{(2)}}) and Q(α) are linearly disjoint and that the minimal polynomial of α over Q(ζ_{m_{(2)}}) is h = X^{2^{j−1}} + p^{2^{j−2}} if j ≥ 2, and is h = X^{2} − p if j < 2.

Let ℘′ be any prime ideal in the ring of integers of Q(ζ_{m_{(2)}}) lying over 2. We show Z[π]_{℘′} = Z[ζ_{m_{(2)}}, α]_{℘′}. The inclusion Z[π]_{℘′} ⊆ Z[ζ_{m_{(2)}}, α]_{℘′} is trivial. Conversely, since π^{2^{j}} = ζ_{m_{(2)}}^{2^{j}}q^{2^{j−1}} and α = πζ_{m_{(2)}}^{−μ}, we have ζ_{m_{(2)}}, α ∈ Z[π]_{℘′}. Thus Z[ζ_{m_{(2)}}, α]_{℘′} ⊆ Z[π]_{℘′}. That is, Z[π]_{℘′} = Z[ζ_{m_{(2)}}, α]_{℘′}. Hence, Z[π]_{℘′} = (Z[ζ_{m_{(2)}}]_{℘′})[α].

If j ≥ 2, then h ≡ (X − 1)^{2^{j−1}} mod ℘′. Note that Z[ζ_{m_{(2)}}]_{℘′} is a discrete valuation ring, so we have by Corollary 2.2 that Z[π]_{℘′} is not maximal if and only if h(1) = 1 + q^{2^{j−2}} ≡ 0 mod ℘′^{2}, that is, j = 2 and p ≡ 3 mod 4. Similarly, if j < 2 then h ≡ (X − 1)^{2} mod ℘′ and so Z[π]_{℘′} is not maximal if and only if p ≡ 1 mod 4. Note Z[π]_{2} = ∏_{℘′|2} Z[π]_{℘′}. By Lemma 2.1 and the above argument, Z[π]_{2} is not maximal if and only if π ∈ E.

In the special case π ∈ E, we have K = Q(ζ_{m_{(2)}}) and Q(π) = K(\sqrt{(\frac{−1}{p})p}) is

quadratic over K. Moreover, Z[π]_{℘} = (O_{K})_{℘}[\sqrt{(\frac{−1}{p})p}] and ℘ is totally ramified in Z[π]_{℘}. This proves that Z[π]_{℘} is a local ring. The decomposition of ℘ in the quadratic extension Q(π) over K corresponds to that of 2 in Q(\sqrt{(\frac{−1}{p})p}) over Q, which is as in our assertion. Since Z[π]_{℘} is a quadratic order over the complete discrete valuation ring (O_{K})_{℘}, it is a Bass order by Remark 2.5 (1). As (O_{K})_{℘}-orders, Z[π]_{℘} ⊂ O_{℘} ≅ (O_{K})_{℘}^{2}. There is an injection Z[π]_{℘}/(O_{K})_{℘} ↪ O_{℘}/(O_{K})_{℘} ≅ (O_{K})_{℘}, under which Z[π]_{℘}/(O_{K})_{℘} ≅ ℘^{i}(O_{K})_{℘} for some positive integer i. In other words, Z[π]_{℘} = (O_{K})_{℘}+℘^{i}O_{℘} and so O_{℘}/Z[π]_{℘} ≅ (O_{K})_{℘}/℘^{i}. But Δ_{Z[π]_{℘}/(O_{K})_{℘}} = (O_{K})_{℘}Δ(X^{2}−(\frac{−1}{p})p) = 4(O_{K})_{℘} and hence [O_{℘} : Z[π]_{℘}]^{2} = [(O_{K})_{℘} : ℘^{i}]^{2} = 2^{2i}(O_{K})_{℘} | 4(O_{K})_{℘}. Thus i = 1, that is, O_{℘}/Z[π]_{℘} ≅ (O_{K})_{℘}/℘ as (O_{K})_{℘}-modules. Hence O_{℘} is the only (O_{K})_{℘}-order in Q(π)_{℘} that properly contains Z[π]_{℘}.

2.4 Torsion-free modules over Bass orders

Let the notation be as in section 2.3. For any ring R we use R^{∗} to denote its group of units. Henceforth in this section we assume that R is an order in Q(π) containing Z[π]. Let M be a torsion-free R_{l}-modules (as defined in section 2.1) of rank e, our goal here is to describe all such modules. We recall that all modules are assumed to be finitely generated.

Lemma 2.9. Let ℘ be any prime ideal in O_{K} lying over 2. Let N be an indecomposable torsion-free Z[π]_{℘}-module. Suppose (l, π) ∈ {2} × E. If ℘ is inert in Q(π), then N ≅ Z[π]_{℘} or O_{℘}. If ℘ is split, i.e., ℘ = ℘_{1}℘_{2} for some prime ideals ℘_{1}, ℘_{2} in Q(π), then N ≅ Z[π]_{℘}, O_{℘_{1}}, or O_{℘_{2}}.

Proof. By Lemma 2.8, the Z[π]_{℘} is a local ring and an (O_{K})_{℘}-order, so we invoke Theorem 2.6. If N is projective over the local ring Z[π]_{℘} then N ≅ Z[π]_{℘}. Otherwise, N is projective over O_{℘}. Suppose ℘ is inert in Q(π), that is, O_{℘} is a discrete valuation ring then N ≅_{O_{℘}} O_{℘}. If ℘ splits into ℘_{1} and ℘_{2} in Q(π), that is, if O_{℘} ≅ O_{℘_{1}} × O_{℘_{2}}, then N ≅_{O_{℘}} O_{℘_{1}} or O_{℘_{2}}. Therefore

N ≅_{Z[π]_{℘}} Z[π]_{℘}, O_{℘_{1}}, or O_{℘_{2}}.

This finishes the proof.

Proposition 2.10. There is the following isomorphism of R_{l}-modules:

M ≅_{R_{l}} \{ \begin{matrix}R_{l}^{e} & if (l, π) ∉ \{2\} × E, \\ ∏_{℘|l}(R_{℘}^{a_{℘}} × O_{℘}^{b_{℘}}) & if (l, π) ∈ \{2\} × E \end{matrix}

where ℘ ranges over all prime ideals in O_{K} lying over 2, and a_{℘}, b_{℘} are nonnegative integers such that a_{℘} + b_{℘} = e.

Proof. Suppose (l, π) ∉ {2} × E. By Lemma 2.8, the Z_{l}-order R_{l} is maximal and our assertion follows from the argument preceding Theorem 2.6.

Suppose (l, π) ∈ {2} × E. Since M_{℘} is a torsion-free R_{℘}-module of rank e, by the Krull-Schmidt-Azumaya theorem [2, Theorem (30.6)], M_{℘} can be expressed as a finite direct sum of indecomposables with the summands unique up to isomorphism and order of occurrence. If ℘ is inert in Q(π), then by Lemma 2.9 there are non-negative integers a_{℘}, b_{℘} with a_{℘} + b_{℘} = e such that M_{℘} ≅ R_{℘}^{a_{℘}} ×O_{℘}^{b_{℘}}. Now suppose ℘ is split in Q(π). Then M_{℘} ≅ R_{℘}^{a_{℘}} ×O_{℘_{1}}^{b_{℘}} ×O_{℘_{2}}^{c_{℘}} for some non-negative integers a_{℘}, b_{℘}, c_{℘}; by comparing ranks in Q(π)_{℘}^{e} ≅_{Q(π)_{℘}} Q(π)_{℘}^{a_{℘}} × Q(π)_{℘_{1}}^{b_{℘}} × Q(π)_{℘_{2}}^{c_{℘}}, we are forced to have b_{℘} = c_{℘}. Thus, M_{℘} ≅ R_{℘}^{a_{℘}} × (O_{℘_{1}} × O_{℘_{2}})^{b_{℘}} ≅ R_{℘}^{a_{℘}} × O_{℘}^{b_{℘}} for a_{℘}, b_{℘} with a_{℘} + b_{℘} = e. Therefore

M ≅ ∏_{℘|2} M_{℘} ≅_{R_{2}} ∏_{℘|2}(R_{℘}^{a_{℘}} × O_{℘}^{b_{℘}}).

This finishes our proof.

The following corollary is prepared for the next section.

Corollary 2.11. If M is a torsion-free R_{l}-module of rank e then we have M/(π − 1)M ≅_{R_{l}} (R_{l}/(π − 1))^{e} unless l = 2, q is not a square, and π = ±\sqrt{(\frac{−1}{p})q}, in which case there are non-negative integers a, b with a + b = e such that

M/(π − 1)M ≅_{R_{2}} (R_{2}/(π − 1))^{a} × (O_{2}/(π − 1))^{b}.

Proof. First of all we show that m ∉ 2^{Z} if and only if R_{2}/(π − 1) = 0, that is, π − 1 ∈ R_{2}^{∗}. Indeed, m ∉ 2^{Z} implies ζ_{m}^{ν} − 1 ∈ Z[π]_{2}^{∗} ⊆ R_{2}^{∗}. Write π − 1 = (ζ_{m}^{ν} − 1)\sqrt{q} + (\sqrt{q} − 1). If p = 2 then (ζ_{m}^{ν} − 1)\sqrt{q} lies in a prime over 2 while \sqrt{q} − 1 ∈ R_{2}^{∗} so their sum lies in R_{2}^{∗}; if p ≠ 2, then R_{2}^{∗}\sqrt{q} = R_{2}^{∗} and \sqrt{q} − 1 lies in a prime over 2 thus their sum also lies in R_{2}^{∗}. This proves our claim. Consequently, if m ∈ 2^{Z} then M/(π − 1)M ≅ (R_{2}/(π − 1))^{e} since they are both trivial. By Proposition 2.10, we have M/(π − 1)M ≅_{R_{l}} (R_{l}/(π − 1))^{e} unless l = 2, π ∈ E and m ∈ 2^{Z}. By the definition of E, we have π = ζ_{m}^{ν}\sqrt{q} ∈ E if and only if q is not a square and m = 1 or 2 if p ≡ 1 mod 4; while m = 4 if p ≡ 3 mod 4. That is, we have l = 2, q is not a square and π = ±\sqrt{(\frac{−1}{p})q}.

3 Elementary supersingular abelian varieties

3.1 Preliminaries

This section provides some auxiliary results on abelian varieties over finite fields. We shall quote from [9] and [10] without comment.

Recall that l is any prime different from p. If G is an abelian group we denote by G[l^{∞}] the subgroup of all elements in G whose order is a l-power. For every k-isogeny r : A → A, we denote by A[r] the kernel of the induced map on A(k̄) as abelian groups. The l-adic Tate module T_{l}(A) = \underleftarrow{lim}_{n}A[l^{n}] is free of rank 2d over Z_{l}. Since the Frobenius endomorphism π acts faithfully on it, T_{l}(A) is torsion-free Z[π]_{l}-module. V_{l}(A) = T_{l}(A)⊗_{Z_{l}} Q_{l} is a Q[π]_{l}-module. We also know that Q[π] is a semisimple Q-algebra. If the characteristic polynomial of the Frobenius is f = ∏_{i=1}^{t} g_{i}^{e_{i}} as in section 1, then

Q[π]_{l} ≅ ∏_{i=1}^{t} Q[π]_{l}/(g_{i}(π)), V_{l}(A) ≅ ∏_{i=1}^{t}(Q[π]_{l}/(g_{i}(π)))^{e_{i}}.  (1)

In particular, if A is elementary so Q[π] is a field, then V_{l}(A) ≅ Q(π)_{l}^{e}. Thus T_{l}(A) is a torsion-free module of rank e over any Z_{l}-order of Q(π)_{l} containing Z[π]_{l}.

T_{l} defines a (covariant) functor from the category of abelian varieties A′ over k with a k-isogeny r : A → A′ to the category of Z[π]_{l}-modules T_{l}(A′) with an injective Z[π]_{l}-module homomorphism T_{l}(A) → T_{l}(A′). For each positive integer n we have l^{−n}T_{l}(A)/T_{l}(A) ≅ A[l^{n}] and so, taking direct limits, V_{l}(A)/T_{l}(A) ≅ A[l^{∞}]. Mapping the exact sequence 0 → T_{l}(A) → V_{l}(A) → A[l^{∞}] → 0 to that of A′ by r induces an injective Z[π]_{l}-module homomorphism r : T_{l}(A) → T_{l}(A′) with cokernel T_{l}(A′)/rT_{l}(A) and an isomorphism r : V_{l}(A) → V_{l}(A′). Let r^{−1}T_{l}(A′) be the pullback of T_{l}(A′) ⊂ V_{l}(A′) under this isomorphism, there is an isomorphism T_{l}(A′)/rT_{l}(A) ≅ r^{−1}T_{l}(A′)/T_{l}(A). By applying the Snake Lemma to the above resulting diagram, we have Ker(r)[l^{∞}] ≅ r^{−1}T_{l}(A′)/T_{l}(A). In fact, every Z[π]_{l}-lattice (as Z_{l}-order) of V_{l}(A) containing T_{l}(A) arises in this way:

Proposition 3.1. Let θ : V_{l}(A)/T_{l}(A) →^{∼} A[l^{∞}] be the isomorphism as above. For every Z[π]_{l}-lattice M containing T_{l}(A), there is an abelian variety A′ with a k-isogeny r : A → A′ such that M = r^{−1}T_{l}(A′) in V_{l}(A) and θ(M/T_{l}(A)) = Ker(r)(k̄).

Proof. Write G = θ(M/T_{l}(A)), it is a finite subgroup of A[l^{∞}]. By our assumption, there is an étale subgroup scheme G of A over k with G(k̄) = G. Take A′ = A/G and the k-isogeny r : A → A′ with Ker(r) = G. The argument preceding the proposition indicates θ(r^{−1}T_{l}(A′)/T_{l}(A)) = G[l^{∞}] = G.

Define T_{p}(A) = \underleftarrow{lim}_{n}A[p^{n}] in an analogous manner. It is free Z_{p}-module of rank between 0 and d (inclusive).

Proposition 3.2. For any k-isogeny r : A → A, there is an isomorphism of abelian groups:

A[r] ≅ ∏_{l} T_{l}(A)/rT_{l}(A)

where l ranges over all prime numbers.

Proof. The finite abelian group A[r] has the decomposition A[r] ≅ ∏_{l}(A[r])[l^{∞}], where each component is isomorphic to T_{l}(A)/rT_{l}(A).

Note that A(k) = A[π − 1].

3.2 Elementary supersingular abelian varieties

An abelian variety over k is supersingular if and only if the eigenvalues of its Frobenius π are supersingular q-numbers. This is equivalent to that the Newton polygon of A is a straight line of slope 1/2, which is equivalent to that A is k̄- isogenous to a power of a supersingular elliptic curve (as a special case of [11, Theorem 4.2]). A supersingular abelian variety A over k has A[p] = 0. We remark in brief that the converse is not true when d > 2. In fact, an abelian variety has A[p] = 0 if and only if its Newton polygon has no 0-slope segment, which does not imply being a straight line of slope 1/2 when d > 2.

For the rest of this section we assume that A is an elementary supersingular abelian variety over k whose Frobenius relative to k is f = g^{e} for some monic irreducible polynomial g over Q and a positive integer e.

Since Q[π] = Q(π) is a field, we fix an embedding of Q(π) in C and identifying π with its image, we write π = ζ_{m}^{ν}\sqrt{q} for some primitive m-th root of unity ζ_{m}^{ν}. We resume the notation from section 2.3: K = Q(π^{2}) = Q(ζ_{m/(2,m)}), its ring of integers O_{K} = Z[ζ_{m/(2,m)}], and O the ring of integers of Q(π).

Let Q be the set of all supersingular q-numbers ζ_{m}^{ν}\sqrt{q} for some primitive root of unity ζ_{m}^{ν} such that either of the following two conditions is satisfied: (1) m = 1 or 2; (2) q is a square, (2, p)p ∤ m and p is of odd order in the group (Z/m_{(p)}Z)^{∗}.

Proposition 3.3. Suppose A is simple supersingular over k.

(1) When π ∈ Q then e = 2 and End_{k}^{0}(A) is a quaternion algebra over Q(π);

(2) When π ∉ Q then e = 1 and End_{k}^{0}(A) is commutative and equal to Q(π).

Proof. Let v be any place of Q(π) (including both finite and infinite primes). Let e_{v} denote the denominator of the Hasse invariant, inv_{v}(End_{k}^{0}(A)), of End_{k}^{0}(A) at v. By [14, Théorème 1] we have

inv_{v}(End_{k}^{0}(A)) = \frac{ord_{v}(π)[Q(π)_{v} : Q_{p}]}{ord_{v}(q)} = \frac{[Q(π)_{v} : Q_{p}]}{2} = \frac{γ(v/p)κ(v/p)}{2} mod 1,

for all primes v lying over p, so e_{v} = 1 or 2. Now e_{v} = 1 for all complex v and also for all finite primes v not lying over p, while e_{v} = 2 for all real v. We have e = lcm_{v}(e_{v}) = 2 if either (1)’ v is real or (2)’ γ(v/℘)κ(v/℘) is odd; and e = 1 otherwise. It is obvious that (1)’ is equivalent to (1). We show below that if v is not real prime then (2)’ is equivalent to (2):

Suppose q is not a square: we claim that e_{v} = 1 for all finite primes v over p. Now [Q(π) : K] = 1 or 2. The former implies 2 | γ(v/p). Consider the latter case, if \sqrt{p} ∈ Q(π), then 2 | γ(v/p) and so e_{v} = 1; otherwise, we would have quadratic extensions Q(ζ_{m}, \sqrt{p}) ⊃ Q(π) ⊃ K. But if p was unramified in Q(π)/K, then it would be unramified in Q(ζ_{m}, \sqrt{p})/Q(ζ_{m}), which is absurd; so we must conclude that p is totally ramified in Q(π)/K and hence 2 | γ(v/p) and so e_{v} = 1.

Suppose q is a square: so that Q(π) = Q(ζ_{m}). Then for any finite prime v over p, we have that κ(v/p) equals the order of p in (Z/m_{(p)}Z)^{∗}; Let ϕ(·) denote the Euler phi-function here, then γ(v/p) = ϕ(m_{p}), which is odd if and only if (2, p)p ∤ m. This finishes our proof.

Remark 3.4. Suppose A is simple supersingular over k. If π ∈ E, then π ∈ Q if and only if d = 2. This follows from the above proposition and the definitions of E and Q.

Remark 3.5. Let A be a simple supersingular abelian variety with odd dimension d > 2, then e = 1 and End_{k}^{0}(A) must be commutative. Indeed, recall [14, Théorème 1] that 2d = e[Q(π) : Q] and so it suffices to show 2 | [Q(π) : K][Q(ζ_{m/(2,m)}) : Q]. Either [Q(π) : K] = 1 or 2, in the former case [Q(ζ_{m/(2,m)}) : Q] = ϕ(m/(2, m)) > 1 and so is even.

3.3 Module structures

Let R be a subring in Q(π) with Z[π] ⊆ R ⊆ End_{k}(A) ∩ Q(π). For any finite group G, we write #G for its order.

Lemma 3.6. Let M′ ⊆ M′′ be modules over any ring R. Let r ∈ R be such that rR is of finite index in R and r acts faithfully on M′, M′′.

(1) If M′ contains a free R-module of rank s as a submodule of finite index,

then

#M′/rM′ = (#(R/rR))^{s}.

(2) If M′, M′′ contain a free R-module of rank s as a submodule of finite index in M′, M′′, respectively, then there are homomorphisms ρ′ : M′/rM′ → M′′/rM′′ and ρ′′ : M′′/rM′′ → M′/rM′ with

#Ker(ρ′) = #Coker(ρ′) = #Ker(ρ′′) = #Coker(ρ′′) | #M′′/M′.

Proof. (1) Since r acts faithfully on M′ and R^{s}, the injective map r : M′ → M′ induces an injective map r : R^{s} ↪ R^{s}. On the other hand, the given injection R^{s} ↪ M′ is of finite index, we thus have

#(M′/rM′) · #(M′/R^{s}) = #(R^{s}/rR^{s}) · #(M′/R^{s}).

Therefore, #M′/rM′ = #(R/rR)^{s}.

(2) Let r act on the short exact sequence of R-modules

0 → M′ → M′′ → M′′/M′ → 0,

and apply the Snake lemma. We then get the desired map ρ′ with #Coker(ρ′) dividing #M′′/M′. By part (1), we have #M′/rM′ = #M′′/rM′′ as they both equal #(R/rR)^{s}. Thus #Ker(ρ′) = #Coker(ρ′). Any finite R/rR-module N has an isomorphic dual Hom_{Z}(N, Q/Z), our assertion on ρ′′ follows by taking the dual of ρ′.

Proposition 3.7. Let r be an isogeny in R. Then there is an R-module homomorphism

φ_{r} : A[r] ⟶ ∏_{l≠p}(R_{l}/rR_{l})^{e}

which is an isomorphism except when π ∈ E in which case #Ker(φ_{r}) and #Coker(φ_{r}) are equal and divide 2_{(p)}^{d}.

Proof. By Proposition 3.2 and the fact A[p] = 0, we have A[r] ≅ ∏_{l≠p} T_{l}/rT_{l}. Recall that T_{l} is a torsion-free R_{l}-module of rank e, so we invoke Proposition 2.10. If π ∉ E or p = 2, then T_{l}/rT_{l} →^{∼} (R_{l}/rR_{l})^{e} for each l ≠ p, and we obtain the desired isomophism φ_{r}. Now suppose π ∈ E. Lemma 2.8 (2) implies #O_{℘}/R_{℘} | #O_{℘}/Z[π]_{℘} = #(O_{K})_{℘}/℘ = 2^{κ(℘/2)}. Clearly κ(℘/2)ϱ(℘/2) | [K : Q] and [K : Q] = [Q(π) : Q]/2 by Lemma 2.8 (1). For each l, we have a map T_{l}/rT_{l}→(R_{l}/rR_{l})^{e} which is an isomorphism if l ≠ 2. When l = 2, Lemma 3.6 indicates the size of its kernel and cokernel are equal and divide (#T_{2}/R_{2}^{e})_{(p)}. Taking product over all l ≠ p we obtain the desired map φ_{r} with #Ker(φ_{r}) = #Coker(φ_{r}) and divides

(#O_{2}/R_{2})_{(p)}^{e} | 2_{(p)}^{κ(℘/2)ϱ(℘/2)e} | 2_{(p)}^{e[K:Q]} | 2_{(p)}^{e[Q(π):Q]/2}

where the last number equals 2_{(p)}^{d}.

Proof of Theorem 1.2. Let S = Z − pZ. By Proposition 3.7, there is an R-module homomorphism φ_{n} : A[n] → ((\frac{1}{n}R)/R)^{e} for every n ∈ S. Let W_{n} be the set of such homomorphisms. If m | n, then by passing to the largest submodule annihilated by m we see that any R-module homomorphism φ_{n} maps the submodule A[m] of A[n] to ((\frac{1}{m})R/R)^{e}, so there is a restriction map W_{n} → W_{m}. Since the projective limit of a system of non-empty finite sets is non-empty, the projective limit of the sets W_{n} is non-empty. Therefore we can make a simultaneous choice of R-module homomorphisms φ_{n} that commute with the inclusions A[m] ⊆ A[n] and ((\frac{1}{m}R)/R)^{e} ⊆ ((\frac{1}{n}R)/R)^{e}. Taking the injective limit over n ∈ S, we get an R-module homomorphism \underrightarrow{lim}A[n] → \underrightarrow{lim}((\frac{1}{n}R)/R)^{e}, that is φ : A(k̄) →^{∼} (R_{(p)}/R)^{e}. Since A(k̄) and (R_{(p)}/R)^{e} are both divisible as abelian groups, the cokernel of φ is also divisible, but it is finite and hence trivial. So Coker(φ) ≅ \underrightarrow{lim}Coker(φ_{n}) is trivial and φ is surjective. In A(k̄) we have Ker(φ) ≅ \underrightarrow{lim}Ker(φ_{n}). Thus φ is an isomorphism except when π ∈ E, in which case #Ker(φ) divides 2_{(p)}^{d} since #Ker(φ_{n}) divides 2_{(p)}^{d} for each n.

Proposition 3.8. Let A be a simple supersingular abelian variety over k with f = g^{e}. Let R = End_{k}(A) ∩ Q(π). If p = 2 or d ≠ 2, then A(k̄) ≅_{R} (R_{(p)}/R)^{e}. If p ≠ 2 and d = 2, then there are non-negative integers a, b with a + b = e and

A(k̄) ≅_{R} (R_{(p)}/R)^{a} × (O_{(p)}/O)^{b}.

Proof. Let φ : A(k̄) → (R_{(p)}/R)^{e} be defined as in Theorem 1.2, which is an isomorphism unless π ∈ E. Suppose π ∈ E. Then #Ker(φ) = #Coker(φ) is a 2-power. Suppose d ≠ 2. Then e = 1 by Remark 3.4 and so T_{2} is a torsion-free R_{2}-module of rank 1. Recall that K is a cyclotomic field. For any prime ℘ in O_{K} lying over 2, write T_{℘} for the ℘-adic completion of T_{2} and T_{℘} : T_{℘} = {r ∈ R_{℘} | rT_{℘} ⊆ T_{℘}}. Then T_{℘} : T_{℘} = R_{℘}, so T_{℘} is a fractional ideal of R_{℘}. Recall from Lemma 2.8 that R_{℘} is a Bass (O_{K})_{℘}-order and thus T_{℘} ≅_{R_{℘}} R_{℘} by [1, § 2.6]. So T_{2} ≅_{R_{2}} R_{2} and this induces isomorphism T_{2}/2T_{2} →^{∼} R_{2}/2R_{2} by Lemma 3.6. Thus φ is an isomorphism. Suppose d = 2. Then π ∈ E implies π = ±\sqrt{(\frac{−1}{p})q} and e = 2 by Remark 3.4. In this case, ℘ = 2, so by

Proposition 2.10, we have A[n] ≅ ∏_{l≠p} T_{l}/nT_{l} ≅ (\frac{1}{n}R/R)^{a} × (\frac{1}{n}O/O)^{b} for all n ∈ S = Z − pZ. Take injective limit both sides over n ∈ S, we have

A(k̄) ≅ \underrightarrow{lim}((\frac{1}{n}R/R)^{a} × (\frac{1}{n}O/O)^{b}) ≅_{R} (R_{(p)}/R)^{a} × (O_{(p)}/O)^{b}.

This finishes our proof.

3.4 Group structures

In this subsection we shall apply the results of the previous subsection to our study of the group structure of A(k).

If A is exceptional, Q(π) = Q(\sqrt{(\frac{−1}{p})q}) = Q(\sqrt{(\frac{−1}{p})p}), so O = Z[^{\frac{1+\sqrt{( \frac{−1}{p})p}}{2}}]. By Lemma 2.8 (2) we notice O_{2}/Z[π]_{2} ≅ Z_{2}/2Z_{2} ≅ Z/2Z.

Proof of Theorem 1.1. Apply Corollary 2.11 to M = T_{l}(A) and R = Z[π]. Now

A(k) ≅_{Z[π]} (Z[π]/(π − 1))^{e} ≅_{Z} (Z/g(1)Z)^{e}

unless A is exceptional, in which case the argument preceding the proof implies that (π−1)/2 ∈ O_{2} while (π−1)/4 ∉ O_{2}. Since #O_{2}/(π−1) = #Z[π]_{2}/(π−1) = |g(1)|_{2}, we have

O_{2}/(π − 1) ≅_{Z_{2}} Z_{2}/2Z_{2} × Z_{2}/\frac{g(1)}{2}Z_{2}.

Hence there are non-negative integers a, b with a + b = e such that

A(k) ≅_{Z[π]} (Z[π]_{2}/(π − 1))^{a} × (O_{2}/(π − 1))^{b} × ∏_{l≠2}(Z[π]_{l}/(π − 1))^{e}
≅_{Z} ((Z_{2}/g(1)Z_{2})^{a} × (Z_{2}/\frac{g(1)}{2}Z_{2} × Z_{2}/2Z_{2})^{b}) × ∏_{l≠2}(Z_{l}/g(1)Z_{l})^{e}
≅_{Z} (Z/g(1)Z)^{a} × (Z/\frac{g(1)}{2}Z × Z/2Z)^{b}.

This proves our theorem.

Proposition 3.9. Let the notation be as in Theorem 1.1. If A is exceptional, then for every pair of non-negative integers a′, b′ with a′ + b′ = e there exists an abelian variety A′ isogenous over k to A such that

A′(k) ≅_{Z} (Z/g(1)Z)^{a′} × (Z/\frac{g(1)}{2}Z × Z/2Z)^{b′}.

Proof. Let A be exceptional. By Theorem 1.1, there are non-negative integers a, b with a + b = e such that T_{2} ≅_{Z[π]_{2}} Z[π]_{2}^{a} × O_{2}^{b} and

A(k) ≅_{Z} (Z/g(1)Z)^{a} × (Z/\frac{g(1)}{2}Z × Z/2Z)^{b}.

If b′ = b, then we are done. If b′ < b, let

M = Z[π]_{2}^{a} × O_{2}^{b′} × (\frac{1}{2}Z[π]_{2})^{b−b′};

if b′ > b, let

M = Z[π]_{2}^{a′} × O_{2}^{b′},

in either case M ≅_{Z[π]_{2}} Z[π]_{2}^{a′} × O_{2}^{b′}. By the argument preceding the proof of Theorem 1.1, we know that O_{2} ⊆ \frac{1}{2}Z[π]_{2} ⊂ Q(π)_{2}. By Proposition 3.1, there exits an abelian variety A′ over k with T_{2}(A′) = M and a k-isogeny A →^{r} A′ with Ker(r)(k̄) ≅ T_{2}(A′)/T_{2}(A) while T_{l}(A′) = T_{l}(A) for all l ≠ 2. Thus

A′(k) ≅  ∏_{l≠p} T_{l}(A′)/(π − 1)T_{l}(A′)
≅_{Z[π]} (Z_{2}[π]/(π − 1))^{a′} × (O_{2}/(π − 1))^{b′} × ∏_{l≠2}(Z[π]_{l}/(π − 1))^{e}
≅_{Z} (Z/g(1)Z)^{a′} × (Z/\frac{g(1)}{2}Z × Z/2Z)^{b′}.

This finishes the proof.

Corollary 3.10. Suppose A is a simple supersingular abelian variety over k of dimension d > 2 with f = g^{e}, then A(k) ≅_{Z} (Z/g(1)Z)^{e} with e = 1 or 2. If d = 1 or 2 then A(k) ≅_{Z} (Z/g(1)Z)^{e} or A(k) ≅_{Z} Z/\frac{q+1}{2}Z × Z/2Z.

Proof. If A is simple over k of dimension d > 2, then A is never exceptional, so A(k) ≅_{Z} (Z/g(1)Z)^{e}, where e = 1 or 2 as we have seen in Proposition 3.3.

If A is an elliptic curve, then A(k) ≅_{Z} (Z/g(1)Z)^{e} unless A is exceptional in which case A(k) ≅_{Z} (Z/g(1)Z)^{e} or A(k) ≅_{Z} Z/\frac{q+1}{2}Z × Z/2Z. Both cases may occur because of Proposition 3.9. (This result can be found in [12, Chapter 4].)

If A is of dimension 2, then A(k) ≅_{Z} (Z/g(1)Z)^{2} unless A is exceptional in which case A(k) ≅_{Z} (Z/g(1)Z)^{2} or A(k) ≅_{Z} Z/\frac{q−1}{2}Z × Z/2Z. Both cases may occur because of Proposition 3.9.

In particular, by Remark 3.5, if A is simple supersingular of odd dimension d > 2, then A(k) ≅ Z/g(1)Z is cyclic.

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