Supersingular abelian varieties over finite fields

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

                  Email: zhu@msri.org

Abstract

Let A be a supersingular abelian variety defined over a finite field k. We give an approximate description of the structure of the group A(k) of k-rational points of A in terms of the characteristic polynomial f of the Frobenius endomorphism of A relative to k. Write f = ∏ g_{i}^{e_{i}} for distinct monic irreducible polynomials g_{i} and positive integers e_{i}, we show that there is a group homomorphism φ : A(k) ⟶ ∏(Z/g_{i}(1)Z)^{e_{i}} that is “almost” an isomorphism in the sense that the size of the kernel and the cokernel of φ are bounded by an explicit function of dim A.

AMS classification: 14K22, 06B15.

1 Introduction

Let A be an abelian variety of dimension d defined over a finite field k of characteristic p with q elements. Let f be the characteristic polynomial of the Frobenius endomorphism of A relative to k. An abelian variety A over k is supersingular if each complex root of f can be written as ζ\sqrt{q}, the product of some root of unity ζ and the positive square root \sqrt{q}. This definition is equivalent to the standard ones as in [6] or [5]. The group structure of rational points on an elliptic curve over a finite field has been well studied (see [9, Chapter V]). We have studied the question for elementary supersingular abelian varieties in [10]. In this paper and [10], an elementary abelian variety is a power of a simple abelian variety. Here we study arbitrary supersingular abelian varieties.

For a finite abelian group G we write #G for its order. Let log(·) be the natural logarithm. Write f = ∏_{i=1}^{t} g_{i}^{e_{i}} for monic irreducible polynomials g_{i} with integer coefficients and positive integers e_{i}.

Theorem 1.1. Let A be an supersingular abelian variety over k of dimension d ≥ 2. Write f = ∏_{i=1}^{t} g_{i}^{e_{i}} as above. Then there exists a group homomorphism

φ : A(k) ⟶ ∏_{i=1}^{t}(Z/g_{i}(1)Z)^{e_{i}}

such that #Ker(φ) = #Coker(φ)

≤ \{ \begin{matrix}(2 log(2d − 2))^{2d−2} & if d > 4.35 × 10^{7} \\ (2 log(100d − 100))^{2d−2} & if 2 ≤ d ≤ 4.35 × 10^{7}. \end{matrix}

Further, for every prime l ≠ p, the l-part of #Ker(φ) divides

1  if l > d
l^{[^{\frac{2d−2}{l−1}}]} if 2 < l ≤ d or q is a square
2^{3d−3} if l = 2 and q is a non-square;

Let Z[π] be the Z-algebra generated by the Frobenius π in the endomorphism ring of A. Let k̄ be an algebraic closure of k.

Theorem 1.2. Let A be an supersingular abelian variety over k of dimension d ≥ 2. Write f = ∏_{i=1}^{t} g_{i}^{e_{i}} as above. Let R_{i} = Z[π]/(g_{i}(π)) and R_{i}_{(p)} its localization at p. There is a surjective Z[π]-module homomorphism

φ : A(k̄) ⟶ ∏_{i=1}^{t}(R_{i}_{(p)}/R_{i})^{e_{i}},

where #Ker(φ) < \{ \begin{matrix}(2 log(2d − 2))^{2d−2} & if d > 4.35 × 10^{7} \\ (2 log(100d − 100))^{2d−2} & if 2 ≤ d ≤ 4.35 × 10^{7}. \end{matrix}

Our theorems essentially demonstrate the following observation: The group structure of a supersingular abelian variety over a finite field is determined by the characteristic polynomial of its Frobenius endomorphism with an “error term” depending only on dim A, not on the size of the base field.

The organization of this paper is as the follows: Sections 2 and 3 are technical. Section 2 contains a lemma (see Lemma 2.1) from analytic number theory which will be used for section 3. In section 3 we will determine all possible irreducible factors of the characteristic polynomial f and compute their mutual resultants so as to give a useful approximation (see Lemma 3.3). In section 4, we consider finitely generated torsion-free modules over a fibre product of rings by applying Goursat’s lemma. Finally, by considering the l-adic Tate module of A as a torsion-free module over Z[π], we apply section 4 to our problem and prove the two theorems.

This paper is based on a portion of the author’s Berkeley Ph.D thesis. It is my pleasure to thank Professor Hendrik W. Lenstra, Jr. for his guidance and numerous suggestions. The author is supported as a MSRI postdoctoral fellow while preparing this paper and would like to thank the institute for its wonderful working environment.

2 A variation of Mertens theorem

Here we prove a lemma from analytic number theory that will be used in Lemma 3.3 in the next section. An immediate consequence is Corollary 2.2 which was initially conjectured by Hendrik Lenstra (see [7, §1] for its application). Mertens theorem implies that when n is large enough we have ∏_{l≤n} l^{\frac{1}{l}} < n, where l ranges over all primes ≤ n (see [2, Theorem 425] or [8, (2.5)]). Let ϕ(·) denote the Euler phi-function. In this section we will prove that when n is large enough we have ∏_{l|n} l^{\frac{1}{l−1}} < log ϕ(n). The subscript l | n denotes that l ranges over all distinct primes dividing n.

Lemma 2.1. Let n_{0} := p_{11} ∏_{i=1}^{8} p_{i} ≈ 3.007 × 10^{8}. Then

∏_{l|n} l^{\frac{1}{l−1}} < \{ \begin{matrix}log ϕ(n) & if n > n_{0}, \\ log(50ϕ(n)) & if 2 ≤ n ≤ n_{0}. \end{matrix}

Proof. Let C be Euler’s constant, and p_{i} the i-th prime number. Given an integer n ≥ 2 we find the positive integer t such that ∏_{i=1}^{t} p_{i} ≤ n < ∏_{i=1}^{t+1} p_{i}. Since n has at most t distinct prime factors and \frac{log l}{l−1} is a decreasing function,

∑_{l|n} \frac{log l}{l − 1} ≤ ∑_{i=1}^{t} \frac{log p_{i}}{p_{i} − 1}.  (1)

By [8, (2.8) and (3.23)], we have for t ≥ 12 that

∑_{i=1}^{t} \frac{log p_{i}}{p_{i} − 1} = ∑_{i=1}^{t} \frac{log p_{i}}{p_{i}} + ∑_{m=2}^{∞} ∑_{i=1}^{t} \frac{log p_{i}}{p_{i}^{m}} < log p_{t} + \frac{1}{log p_{t}} − C.  (2)

Suppose n ≥ ∏_{i=1}^{13} p_{i}. The two auxiliary functions F (n) := F_{0}(n)+ \frac{1}{F_{0}(n)} −C and F_{0}(n) := log log n − log(1 − \frac{1}{log log n−0.7093}) are increasing with respect to n. By Bertrand’s Postulate (see [2, 22.3]) and [8, (3.32)], we obtain p_{t} > \frac{p_{t+1}}{2} > \frac{log n}{2.0325}; thus by [8, (3.16)], we have log p_{t} < log log n−log(1− \frac{1}{log p_{t}}) < F_{0}(n). Combining (1) and (2) yields

∑_{l|n} \frac{log l}{l − 1} < F (n).  (3)

Define H(n) := exp(C) log log n + \frac{2.5}{log log n}. Since \frac{n}{H(n)} is an increasing function for n ≥ 30, and by [8, (3.41)], we get

\frac{∏_{i=1}^{t} p_{i}}{H(∏_{i=1}^{t} p_{i})} < \frac{n}{H(n)} < ϕ(n) for all n ≠ ∏_{i=1}^{9} p_{i}.  (4)

Suppose n > ∏_{i=1}^{25} p_{i}. It is not hard to show that n > (H(n))^{21}, and so

n < ϕ(n)H(n) < ϕ(n)(n/H(n))^{\frac{1}{20}} < ϕ(n)^{1.05}.  (5)

Now F (n)−log log n is decreasing, so F (n) < log log n−0.0529. Then (3) and (5) yield ∑_{l|n} \frac{log l}{l−1} < log log n − 0.0529 < log(1.05 log ϕ(n)) − 0.0529 < log log ϕ(n).

Suppose ∏_{i=1}^{10} p_{i} ≤ n < ∏_{i=1}^{25} p_{i}, by explicit calculation for each 10 ≤ t ≤ 24 and by (4), we have

∏_{l|n} l^{\frac{1}{l−1}} ≤ ∏_{i=1}^{t} pi^{\frac{1}{p_{i}−1}} < log \frac{∏_{i=1}^{t} p_{i}}{H(∏_{i=1}^{t} p_{i})} < log ϕ(n).

Suppose n_{0} < n < ∏_{i=1}^{10} p_{i}. This implies that n has at most 9 distinct prime factors. Since l^{\frac{1}{l−1}} > 1 and l^{\frac{1}{l−1}} is decreasing in l, the product ∏_{l|n} l^{\frac{1}{l−1}} attains its maximum when n is the product of the smallest possible 9 primes among p_{1}, ⋯, p_{10} such that n > n_{0}. This forces the first inequality below, while the rest follows from explicit computation and (3):

∏_{l|n} l^{\frac{1}{l−1}} ≤ ∏_{{1≤i≤10 \atop i≠8}} pi^{\frac{1}{p_{i}−1}} < log n_{0} − 1.85 < log \frac{n_{0}}{H(∏_{i=1}^{10} p_{i})} < log \frac{n_{0}}{n/ϕ(n)},

which is less than log ϕ(n) and the first part of the lemma is proved.

Suppose 30 ≤ n ≤ ∏_{i=1}^{10} p_{i} and n ≠ ∏_{i=1}^{9} p_{i}. By (1), (5) and explicit computation on each 3 ≤ t ≤ 9, we obtain

∏_{l|n} l^{\frac{1}{l−1}} ≤ ∏_{i=1}^{t} pi^{\frac{1}{p_{i}−1}} < log \frac{50 ∏_{i=1}^{t} p_{i}}{H(∏_{i=1}^{t} p_{i})} < log(50ϕ(n)).

It is easy to verify for n = ∏_{i=1}^{9} p_{i} and 2 ≤ n < 30.

By a similar but easier calculation, we can show ∏_{l|n} l^{\frac{1}{l−1}} < log n for all n so that p_{8} ∏_{i=1}^{6} p_{i} < n ≤ n_{0} and thus for all n > n_{0} by the above lemma. This gives the following corollary.

Corollary 2.2. For all n > p_{8} ∏_{i=1}^{6} p_{i} = 570570, we have ∏_{l|n} l^{\frac{1}{l−1}} < log n.

Remark 2.3. The minimal bounds for n in Lemma 2.1 and Corollary 2.2 are both sharp. It is not hard to verify the following: if n = n_{0} = p_{11} ∏_{i=1}^{8} p_{i}, then ∏_{l|n} l^{\frac{1}{l−1}} > log ϕ(n); if n = p_{8} ∏_{i=1}^{6} p_{i}, then ∏_{l|n} l^{\frac{1}{l−1}} > log n.

3 Supersingular polynomials

In this section, we will quote algebraic number theory from [1] or [3] without comment. Recall that q is a power of the prime p. An algebraic number in C is called a supersingular q-number if it is of the form ζ\sqrt{q}, the product of some root of unity ζ and the positive square root of q. Obviously it is an algebraic integer. Here we determine all minimal polynomials of supersingular q-numbers, calculate their mutual resultants, and prove Lemma 3.3. This lemma is a core technical point for our proof of Theorems 1.1 and 1.2 in section 5.

Let (\frac{a}{b}) be the Jacobi symbol for an integer a and odd integer b, further, define (\frac{a}{2}) = 0 if 2 | a and (\frac{a}{2}) = (−1)^{(a^{2}−1)/8} if 2 ∤ a. Denote by ζ_{m} the primitive m-th root of unity, exp(\frac{2π\sqrt{−1}}{m}). The Galois group Gal(Q(ζ_{m})/Q) consists of the σ_{i} defined by σ_{i}(ζ_{m}) = ζ_{m}^{i} with i coprime to m and 1 ≤ i ≤ m. We claim that \sqrt{p} ∈ Q(ζ_{m}) implies σ_{i}(\sqrt{p}) = (\frac{p}{i})\sqrt{p}. Since they are both multiplicative, it suffices to show σ_{l}(\sqrt{p}) = (\frac{p}{l})\sqrt{p} for each prime l dividing i. If l is odd then σ_{l}(\sqrt{p}) ≡ (\sqrt{p})^{l} = p^{\frac{l−1}{2}} \sqrt{p} mod l and thus σ_{l}(\sqrt{p}) = (\frac{p}{l})\sqrt{p}. Suppose l = 2. Since m is odd, our hypothesis implies Q(\sqrt{p}) ⊆ Q(ζ_{p}) ⊆ Q(ζ_{m}). Denote by σ̄_{2} the image of σ_{2} in Gal(Q(ζ_{p})/Q), then σ_{2}(\sqrt{p}) = σ̄_{2}(\sqrt{p}) = \sqrt{p} or −\sqrt{p}. It equals the former if and only if σ̄_{2} ∈ Gal(Q(ζ_{p})/Q(\sqrt{p})), that is, if and only if 2 is a square in (Z/pZ)^{∗}. Thus σ_{2}(\sqrt{p}) = (\frac{p}{2})\sqrt{p}.

Let Φ_{m} be the m-th cyclotomic polynomial. We write (m_{1}, m_{2}) for the greatest common divisor for integers or polynomials m_{1} and m_{2}. Let C(π) be the conjugacy class of π in C.

Proposition 3.1. Let g be the minimal polynomial of a given supersingular q-number π.

(1) If q is a square, then C(π) = C(ζ_{m}\sqrt{q}) for some m, and

g = Ψ_{m}(X) := (\sqrt{q})^{ϕ(m)}Φ_{m}(\frac{X}{\sqrt{q}});

(2) If q is a non-square, then C(π) = C(ζ_{m}^{ν}\sqrt{q}) for some primitive m-th root of unity ζ_{m}^{ν} with m ≢ 2 mod 4;

(i) if Q(π) ≠ Q(π^{2}), then

g = G_{m}(X) := q^{\frac{ϕ(m)}{(2,m)}} Φ_{m/(2,m)}(\frac{X^{2}}{q})
= ∏_{{(i,m/(2,m))=1 \atop 1≤i≤m/(2,m)}} (X − ζ_{m/(2,m)}^{i}\sqrt{q})(X + ζ_{m/(2,m)}^{i}\sqrt{q});  (6)

(ii) if Q(π) = Q(π^{2}), then

g = E_{m,±1}(X) := ∏_{{(i,m/(2,m))=1 \atop 1≤i≤m/(2,m)}}(X ∓ (\frac{q}{i})ζ_{m}^{i}\sqrt{q}).  (7)

Proof. Part (1) is straightforward. We shall show part (2). Write π = ζ_{m}^{ν}\sqrt{q} for some primitive m-th root of unity ζ_{m}^{ν}. If 2||m, then π = −ζ_{m/2}^{ν(m+2)/4}\sqrt{q}; but since m/2 is odd, π is conjugate to ζ_{m/2}^{μ}\sqrt{q} for some primitive (m/2)-th root of unity ζ_{m/2}^{μ}. Thus we may assume m ≢ 2 mod 4 for the rest of the proof.

Now [Q(π) : Q(π^{2})] = 1 or 2. It can be shown that Q(π) = Q(π^{2}) if and only if Δ_{Q(\sqrt{p})/Q} | m and 2Δ_{Q(\sqrt{p}/Q)} ∤ m (see [10, Lemma 2.6]). Suppose [Q(π) : Q(π^{2})] = 2. It is not hard to see that π is a root of G_{m}, and its minimal polynomial is G_{m} since G_{m} has degree 2ϕ(m)/(2, m) and

[Q(π) : Q] = \frac{[Q(π) : Q(ζ_{m/(2,m)})][Q(ζ_{m}) : Q]}{[Q(ζ_{m}) : Q(ζ_{m/(2,m)})]} = 2ϕ(m)/(2, m).

Suppose [Q(π) : Q(π^{2})] = 1. Then \sqrt{p} ∈ Q(ζ_{m/(2,m)}) and so by the argument preceding this proposition, we have σ_{i}(π) = σ_{i}(ζ_{m}^{ν}\sqrt{q}) = (\frac{q}{i})ζ_{m}^{iν} \sqrt{q} for all σ_{i} ∈ Gal(Q(ζ_{m/(2,m)})/Q). The degree of π is ϕ(m/(2, m)), so its minimal polynomial is E_{m,( \frac{q}{ν})} = ∏(X − (\frac{q}{i})ζ_{m}^{νi}\sqrt{q}) where the product ranges over i with (i, m/(2, m)) = 1 and 1 ≤ i ≤ m/(2, m).

Remark 3.2. Let E denote the set of supersingular q-numbers ζ_{m}^{ν}\sqrt{q} for some primitive m-th root of unity ζ_{m}^{ν} where p ∤ m, p ≠ 2, q is not a square, such that (I) 4 ∤ m when p ≡ 1 mod 4 while (II) 4||m when p ≡ 3 mod 4.

Let Q be the set of supersingular q-numbers ζ_{m}^{ν}\sqrt{q} for some primitive m-th root of unity ζ_{m}^{ν} where p ∤ m and such that either (I) m = 1, 2 or (II) q is a square, (2, p)p ∤ m and ord(p mod m_{(p)}) is odd.

We note that π ∈ Q (respectively, E) if and only if C(π) ⊂ Q (respectively, E). In other words, these definitions are independent of the choice of π from its conjugacy class.

For i = 1, 2, ⋯, t, let C_{i} be conjugacy classes of supersingular q-numbers with minimal polynomials g_{i}. By Proposition 3.1, C_{i} = C(ζ_{m_{i}}^{ν_{i}} \sqrt{q}) where m_{i} ≢ 2 mod 4 when q is a non-square. We order the C_{i}’s so that m_{1} ≤ ⋯ ≤ m_{t}. For i = 1, 2, ⋯, t, let e_{i} be positive integers such that (I) e_{i} ≤ e_{i+1} when m_{i} = m_{i+1}; and (II) e_{i} is even when π_{i} ∈ Q. Under these conditions, the numbers defined by d := ∑_{i≥1} e_{i} deg(g_{i})/2 and d_{E} := ∑_{i≥1,π_{i}∈E} e_{i} deg(g_{i})/2 are positive integers (see [10, Proposition 3.3]). These two technically defined numbers will be used in section 5.

Let R(·, ·) denote the resultant of two polynomials. For any prime number l, we write n_{l} and n_{(l)} for the l-part and the non-l-part of a positive integer n, respectively. For any real number r we denote the largest integer ≤ r by [r].

Lemma 3.3. Let the notation be as above, and let d ≥ 2. Then

(2^{d_{E}} ∏_{i=2}^{t} ∏_{j=1}^{i−1} |R(g_{i}, g_{j})|^{e_{i}})_{(p)} < \{ \begin{matrix}(2 log(2d − 2))^{2d−2} & if d > 4.35 × 10^{7}; \\ (2 log(100d − 100))^{2d−2} & if d ≤ 4.35 × 10^{7}. \end{matrix}

In particular, for any prime l different from p, then

(2^{d_{E}} ∏_{i=2}^{t} ∏_{j=1}^{i−1} |R(g_{i}, g_{j})|^{e_{i}})_{l} divides \{ \begin{matrix}1 & if l > d \\ l^{[ \frac{2d−2}{l−1} ]} & if 2 < l ≤ d or q is a square \\ 2^{3d−3} & if l = 2 and q is a non-square. \end{matrix}

Remark 3.4. The strategy of our proof is to first compute the resultants of cyclotomic polynomials (in Lemma 3.5), then reduce our problem to the cyclotomic case (see Lemma 3.6). Finally we apply Lemma 2.1 to approximate our desired bounds.

Lemma 3.5. For any positive integers m > n, we have

R(Φ_{m}, Φ_{n}) = \{ \begin{matrix}(−1)^{ϕ(n)ϕ(m)}l^{ϕ(n)} & if m/n is a power of a prime l, \\ 1 & otherwise. \end{matrix}

Proof. Let l be a prime number. Write m = m_{(l)}l^{α} and n = n_{(l)}l^{β}, then

Φ_{m}(X) = \frac{Φ_{m_{(l)}} (X^{l^{α}})}{Φ_{m_{(l)}} (X^{l^{α−1}} )} ≡ \frac{Φ_{m_{(l)}} (X)^{l^{α}}}{Φ_{m_{(l)}} (X)^{l^{α−1}}} = Φ_{m_{(l)}}(X)^{\frac{ϕ(m)}{ϕ(m_{(l)})}} mod l.

Hence, l | R(Φ_{m}, Φ_{n}) if and only if m_{(l)} = n_{(l)}, that is, m/n ∈ l^{Z}. Thus |R(Φ_{m}, Φ_{n})| = 1 if m/n is not a prime power. Now assume m_{(l)} = n_{(l)}. Then

R(Φ_{n}(X), \frac{X^{m}−1}{X^{m/l}−1}) = R(Φ_{n}(X), Φ_{l}(X^{m/l})) = ∏_{(i,n)=1}Φ_{l}(ζ_{n}^{im/l}) = l^{ϕ(n)}.

According to the factorization \frac{X^{m}−1}{X^{m/l}−1} = Φ_{m}(X) ∏_{s}Φ_{m/s}(X) where s ranges over divisors of m_{(l)} that are not equal to 1, we get

l^{ϕ(n)} = |R(\frac{X^{m}−1}{X^{m/l}−1}, Φ_{n}(X))| = |R(Φ_{m}, Φ_{n})| ∏_{s} |R(Φ_{m/s}, Φ_{n})|.

The last product is trivial since n/(m/s) is not a prime power. This proves our assertion up to a sign. It remains to show that R(Φ_{m}, Φ_{n}) is positive if and only if m ≠ 2. Indeed, if m ≥ 3 then complex conjugation is contained in Gal(Q(ζ_{m})/Q) and so R(Φ_{m}, Φ_{n}) = N_{Q(ζ_{m})/Q}(Φ_{n}(ζ_{m})) is positive. If m = 2 then it is trivial to see that R(Φ_{2}, Φ_{1}) = −2. This finishes our proof.

Recall E from Remark 3.2. Let Ē be the complement of E in the set of all supersingular q-numbers with non-square q.

Lemma 3.6. Let the notation be as in Lemma 3.3,

(∏_{i>j} |R(g_{i}, g_{j})|^{e_{i}})_{(p)} divides 2_{(p)}^{[^{\frac{∑_{i≥2,π_{i}∈Ē} e_{i} deg(g_{i})}{2}} ]} ∏_{i≥2,l|2m_{i}} l_{(p)}^{[ \frac{e_{i} deg(g_{i})}{l−1} ]}.

Proof. (a) Denote by F the set of π_{i}’s with m_{i} = m_{i+1}. Let l be a prime different from p. For any fixed π_{i}, by Lemma 3.5, the product ∏ |R(Φ_{m_{i}}, Φ_{m_{j}})|_{l} over all j with 1 ≤ j ≤ i − 1, π_{j} ∉ F attains its maximum when each m_{j} = \frac{m_{i}}{l^{i−j}}. In this case, we have ∑_{j=1}^{i−1}ϕ(m_{j}) ≤ ϕ(\frac{m_{i}}{l}) + ϕ(\frac{m_{i}}{l^{2}} ) + ⋯ + ϕ(\frac{m_{i}}{l^{i−1}} ) ≤ \frac{ϕ(m_{i})}{l−1}. Hence

∏_{1≤j≤i−1,π_{j} ∉F} |R(Φ_{m_{i}}, Φ_{m_{j}})|_{l} divides l^{\frac{ϕ(m_{i})}{l−1}} for prime l | m_{i}.  (8)

(b) Assume q is a square. Since F is empty, by Proposition 3.1 (1) and (8),

∏_{i>j} |R(g_{i}, g_{j})|_{(p)}^{e_{i}} = ∏_{i>j} |R(Ψ_{m_{i}}, Ψ_{m_{j}})|_{(p)}^{e_{i}} = ∏_{i>j} |R(Φ_{m_{i}}, Φ_{m_{j}})|_{(p)}^{e_{i}},

which divides ∏_{i≥2,l|m_{i}} l_{(p)}^{[ \frac{e_{i} deg(g_{i})}{l−1} ]}.

(c) Assume q is a non-square. If q is a non-square then π_{i} ∈ F if and only if the pairs π_{i}, π_{i+1} have minimal polynomials g_{i} = E_{m_{i},±1} and g_{i+1} = E_{m_{i},∓1}. Since ∏_{i>j} |R(g_{i}, g_{j})|_{(p)}^{e_{i}} divides

∏_{π_{i−1}∈F} |R(E_{m_{i},1}, E_{m_{i},−1})|_{(p)}^{e_{i}} ∏_{i>j,π_{j}∉F} |R(g_{i}, G_{m_{j}})|_{(p)}^{e_{i}},

it suffices to show the following two divisibilities hold:

∏_{π_{i−1}∈F} |R(E_{m_{i},1}, E_{m_{i},−1})|_{(p)}^{e_{i}} divides 2_{(p)}^{[^{\frac{∑_{i≥2,π_{i}∈Ē} e_{i} deg(g_{i})}{2}} ]}  (9)

∏_{i>j,π_{j}∉F} |R(g_{i}, G_{m_{j}})|_{(p)}^{e_{i}} divides ∏_{i≥2,l|2m_{i}} l_{(p)}^{[ \frac{e_{i} deg(g_{i})}{l−1} ]}.  (10)

We firstly prove (9): Let σ and δ range over the embeddings of Q(π_{i}) in C, by (7) we have

∏_{π_{i−1}∈F} |R(E_{m_{i},1}, E_{m_{i},−1})|_{(p)}^{e_{i}} = ∏_{{π_{i−1}∈F \atop σ,δ}} |π_{i}^{σ} − (−π_{i})^{δ}|_{(p)}^{e_{i}}.

Splitting the product into two parts according to σ = δ and σ ≠ δ, they are

= ∏_{π_{i−1}∈F,σ} |2π_{i}^{σ}|_{(p)}^{e_{i}} · ∏_{π_{i−1}∈F,σ≠δ} |\frac{π_{i}^{2σ} − π_{i}^{2δ}}{π_{i}^{σ} − π_{i}^{δ}} |_{(p)}^{e_{i}}

= 2_{(p)}^{∑_{π_{i−1}∈F} e_{i} deg(g_{i})} · ∏_{π_{i−1}∈F} |\frac{Δ_{Z[π_{i}^{2}]/Z}}{Δ_{Z[π_{i}]/Z}} |_{(p)}^{e_{i}} ,

where Δ(·) denotes the discriminant. The last product is trivial since the inclusion chain Z[π_{i}^{2}] = Z[qζ_{m_{i}/(2,m_{i})}] ⊆ Z[π_{i}] ⊆ Z[ζ_{m_{i}/(2,m_{i})}] has p-power index. Note that π_{i−1} ∈ F implies π_{i−1}, π_{i} ∈ Ē; but e_{i−1} ≤ e_{i} by our hypothesis, so we have ∑_{π_{i−1}∈F} e_{i} deg(g_{i}) ≤ ∑_{i≥2,π∈Ē} e_{i} deg(g_{i})/2. Then (9) follows.

Secondly we prove (10): Let n_{(2,p)} denote the non-2 and non-p part of the integer n. Now we claim that for any i > j we have

|R(g_{i}, G_{m_{j}})|_{(p)} divides 2^{deg(g_{i})}|R(Φ_{m_{i}}, Φ_{m_{j}})|(2,p)^{\frac{deg(g_{i})}{ϕ(m_{i})}} .  (11)

By Proposition 3.1 (2), it suffices to consider the following two cases: Case 1. Suppose g_{i} = G_{m_{i}}. By the definition in (6), we have

|R(G_{m_{i}}, G_{m_{j}})|_{(p)} = |R(Φ_{m_{i}}(X^{2/(2,m_{i})}), Φ_{m_{j}}(X^{2/(2,m_{j})}))|_{(p)}.

Note Φ_{m_{i}}(X^{2}) = Φ_{m_{i}}(X)Φ_{2m_{i}}(X) when 2 ∤ m_{i}. Further calculations via Lemma 3.5 yield that

|R(G_{m_{i}}, G_{m_{j}})|_{(p)} divides 2^{deg(G_{m_{i}})}|R(Φ_{m_{i}}, Φ_{m_{j}})|(2,p)^{\frac{2}{(2,m_{i})}} .

Case 2. Suppose g_{i} = E_{m_{i},±1}. From (6) and (7), G_{m_{i}} = E_{m_{i},1}E_{m_{i},−1} and |R(E_{m_{i},1}, G_{m_{j}})|_{(p)} = |R(E_{m_{i},−1}, G_{m_{j}})|_{(p)}, so we have

|R(E_{m_{i},±1}, G_{m_{j}})|_{(p)} = |R(G_{m_{i}}, G_{m_{j}})|_{(p)}^{1/2}.

Our claim then follows from Case 1.

By (11) and (8), the product ∏_{i>j,π_{j}∉F} |R(g_{i}, G_{m_{j}})|_{(p)}^{e_{i}} divides

2^{∑_{i≥2} e_{i} deg(g_{i})} ∏_{i≥2,l|m_{i}} l_{(2,p)}^{[ \frac{e_{i} deg(g_{i})}{l−1} ]} = ∏_{i≥2,l|2m_{i}} l_{(p)}^{[ \frac{e_{i} deg(g_{i})}{l−1} ]}.

This finishes our proof.

Proof of Lemma 3.3. If t = 1, then 2^{d_{E}} ∏_{i>j} |R(g_{i}, g_{j})|^{e_{i}} = 2^{d_{E}} divides 2^{d} since d_{E} ≤ d. In this case it is straightforward to verify our assertion. For the rest of the proof we assume t ≥ 2. We shall prove the local bound first.

(I) Let l > d and l ≠ p, we claim ∏_{i>j} |R(g_{i}, g_{j})|_{l}^{e_{i}} = 1. Suppose the contrary. As l > d ≥ 2, we know that l is an odd prime. By Lemma 3.6, we have that l | m_{i} for some i. Suppose m_{i} = l or 2l. We separate two cases: If q is a square, then e_{i} is even since π_{i} ∈ Q, thus d ≥ deg(g_{i})+1 = ϕ(m_{i})+1 = l which is a contradiction; If q is a non-square, then Q(π_{i}) ≠ Q(π_{i}^{2}) and so g_{i} = G_{m_{i}} by Proposition 3.1; thence d ≥ deg(g_{i})/2 + 1 = ϕ(m_{i}) + 1 = l, which contradicts our assumption that l > d. Suppose m_{i} ≥ 3l, then l ≤ ϕ(m_{i})/2+1 ≤ deg(g_{i})/2+1 ≤ d which is absurd. Secondly, if q is a square or 2 < l ≠ p, then by Lemma 3.6 the lexponent of 2^{d_{E}} ∏_{i>j} |R(g_{i}, g_{j})|^{e_{i}} ≤ [^{\frac{∑_{i≥2} e_{i} deg(g_{i})}{l−1}}] ≤ [\frac{2d−2}{l−1}] since e_{1} deg(g_{1}) ≥ 2. Similarly, by Lemma 3.6, the 2-exponent of 2^{d_{E}} ∏_{i>j} |R(g_{i}, g_{j})|_{2}^{e_{i}} is equal to [∑_{π_{i}∈E} e_{i} deg(g_{i})] + [∑_{i≥2,π_{i}∈Ē} e_{i} deg(g_{i})] + ∑_{i≥2} e_{i} deg(g_{i}) ≤ 3d − 3.

(II) Now we prove the global bound. Let m′_{i} be the non-2-part of m_{i}. Let the notation be as in Lemma 3.6, (2^{d_{E}} ∏_{i>j} |R(g_{i}, g_{j})|^{e_{i}})_{(p)} divides

2^{[^{\frac{∑_{i} e_{i} deg(g_{i})}{2}} ]} ∏_{i≥2,l|2m′_{i}} l^{[ \frac{e_{i} deg(g_{i})}{l−1} ]} < 2^{d} ∏_{i≥2}( ∏_{l|2m′_{i}} l^{\frac{1}{l−1}} )^{e_{i} deg(g_{i})};

Note ϕ(2m′_{i}) = ϕ(m′_{i}) ≤ 2d − 2, so by Lemma 2.1 we have

∏_{l|2m′_{i}} l^{\frac{1}{l−1}} < log(50ϕ(2m′_{i})) < log(100d − 100)
< 2^{d}(log(100d − 100))^{∑^{i≥2} e_{i} deg(g_{i})} ≤ (2 log(100d − 100))^{2d−2}.

Now assume that d > 4.35 × 10^{7}. If 2m′_{i} > n_{0} then Lemma 2.1 implies that ∏_{l|2m′_{i}} l^{\frac{1}{l−1}} < log(2d − 2); otherwise, by inequality (1) in the proof of the same

lemma and explicit computation, ∏_{l|2m′_{i}} l^{\frac{1}{l−1}} ≤ ∏_{i=1}^{9} pi^{\frac{1}{p_{i}−1}} ≤ log(2d−2). Therefore,

2^{d} ∏_{i≥2}( ∏_{l|2m′_{i}} l^{\frac{1}{l−1}} )^{e_{i} deg(g_{i})} < 2^{d}(log(2d − 2))^{2d−2} ≤ (2 log(2d − 2))^{2d−2}.

This finishes our proof.

Example 3.7. Those local upper bounds in Lemma 3.3 (1) are sharp. The second bound is achieved in the following example: Let q be a square. Let l be an odd prime different from p. Let π_{i} = ζ_{l^{i−1}} \sqrt{q} and e_{i} even positive integers for i = 1, ⋯, t. Then we have 2^{d_{E}} ∏_{i>j} |R(g_{i}, g_{j})|_{l}^{e_{i}} = l^{\frac{2d−2}{l−1}} .

Here is a non-trivial example in which the third bound is approached very closely: Consider π_{1} = ζ_{3}\sqrt{3}, π_{2} = ζ_{12}\sqrt{3}, π_{3} = ζ_{12}^{7}\sqrt{3} It can be checked that π_{i} ∈ Ē so 2^{d_{E}} ∏_{i>j} R(g_{i}, g_{j})^{e_{i}} = 2^{2e_{2}+4e_{3}}; while 2^{3d−3} = 2^{6e_{1}+3e_{2}+3e_{3}−3}. Note e_{2} ≥ e_{3} by our hypothesis.

4 Torsion-free modules and fibre products

All rings are commutative with 1. Let t ≥ 2. Let 𝔞_{i} be an ideal of a ring R_{i} for i = 1, ⋯, t. Inductively the fibre product R_{1} ×_{R_{2}/𝔞_{2}} R_{2} × ⋯ ×_{R_{t}/𝔞_{t}} R_{t} is

R′_{t−1} ×_{R_{t}/𝔞_{t}} R_{t} := {(r′_{t−1}, r_{t}) ∈ R′_{t−1} × R_{t} | γ_{t}(r′_{t−1} + 𝔞′_{t−1}) = r_{t} + 𝔞_{t}},

where R′_{t−1} = R_{1} ×_{R_{2}/𝔞_{2}} R_{2} × ⋯ ×_{R_{t−1}/𝔞_{t−1}} R_{t−1} has an ideal 𝔞′_{t−1} such that there is an isomorphism R′_{t−1}/𝔞′_{t−1} →^{γ_{t}} R_{t}/𝔞_{t}.

Given an R_{i}-module M_{i} with a submodule N_{i} ⊇ 𝔞_{i}M_{i} for i = 1, ⋯, t, define the fibre product of modules M_{1} ×_{M_{2}/N_{2}} M_{2} × ⋯ ×_{M_{t}/N_{t}} M_{t} analogously as

M′_{t−1} ×_{M_{t}/N_{t}} M_{t} := {(x′_{t−1}, x_{t}) ∈ M′_{t−1} × M_{t} | θ_{t}(x′_{t−1} + N′_{t−1}) = (x_{t} + N_{t})};

where M′_{t−1} = M_{1} ×_{M_{2}/N_{2}} M_{2} × ⋯ ×_{M_{t−1}/N_{t−1}} M_{t−1} has a submodule N′_{t−1} ⊇ 𝔞_{t−1}M_{t−1} such that there is a γ_{t}-linear isomorphism θ_{t} : M′_{t−1}/N′_{t−1} → M_{t}/N_{t}. (Note: γ_{t}-linear means θ_{t}r′_{t−1} = (γ_{t}r′_{t−1})θ_{t} for every r′_{t−1} ∈ R′_{t−1}.) Then M_{1} ×_{M_{2}/N_{2}} M_{2} × ⋯ ×_{M_{t}/N_{t}} M_{t} is a module over R_{1} ×_{R_{2}/𝔞_{2}} R_{2} × ⋯ ×_{R_{t}/𝔞_{t}} R_{t}.

We have the following Goursat’s Lemma for rings (also see [4, Exercise 5, page 75] for Goursat’s lemma for groups).

Lemma 4.1. Let R_{1}, …, R_{t} be rings. Suppose R is a subring of ∏_{i=1}^{t} R_{i} such that the projections R →^{ρ_{i}} R_{i} are surjective. Let R′_{i} be the image of the projection R → ∏_{i=1}^{j} R_{i}. Denote the projection maps from R′_{i} to R′_{i−1} and R_{i} by ρ′_{i−1} and ρ′′_{i}, respectively. We may identify 𝔞_{i} = Ker(ρ′_{i−1}) and 𝔞′_{i−1} = Ker(ρ′′_{i}) as ideals in R_{i} and R′_{i−1}, respectively. We obtain isomorphisms R′_{i−1}/𝔞′_{i−1} →^{γ_{i}} R_{i}/𝔞_{i} for i = 2, ⋯, t which define an isomorphism R ≅ R_{1} ×_{R_{2}/𝔞_{2}} R_{2} × ⋯ ×_{R_{t}/𝔞_{t}} R_{t}. As abelian groups, (R′_{i−1} × R_{i})/R′_{i} ≅ R_{i}/𝔞_{i} for i = 2, …, t.

Proof. From the inductive definition of fibre product, it suffices to prove for t = 2. It is clear that 𝔞_{1} = R ∩ (R_{1} × {0}), and by assumption, it can be identified as an ideal in R_{1}. Similarly we identify 𝔞_{2} with an ideal in R_{2}. Thus 𝔞_{1} × 𝔞_{2} is the largest ideal of R_{1} × R_{2} that is also an ideal in R. The natural map θ : R → R_{1}/𝔞_{1} × R_{2}/𝔞_{2} defines an isomorphism γ : R_{1}/𝔞_{1} → R_{2}/𝔞_{2} as the image of R. If two elements (r_{1}, r_{2}), (r_{1}, r_{3}) ∈ R_{1} × R_{2} lie in R, then (0, r_{2} −r_{3}) ∈ R. Hence r_{2} −r_{3} ∈ 𝔞_{2}, otherwise contradicting the fact that 𝔞_{1} ×𝔞_{2} is the largest ideal of R_{1} × R_{2} which is also an ideal in R. This shows that γ is well-defined. Using the same argument, we see that γ is injective and surjective. From our construction R_{1} ×_{R_{2}/𝔞_{2}} R_{2} is exactly the pullback of the map θ and hence identical with R.

We have an analogous Goursat’s lemma for modules.

Lemma 4.2. Let R be as in Lemma 4.1. Let M_{i} be an R_{i}-module and M be an R-submodule of ∏_{i=1}^{t} M_{i} such that the projections M →^{ϱ_{i}} M_{i} are surjective. Let M′_{i} denote the image of the projection M → ∏_{i=1}^{j} M_{i}. Denote the projection maps from M′_{i} to M′_{i−1} and M_{i} by ϱ′_{i−1} and ϱ′′_{i}, respectively. We may identify N_{i} = Ker(ϱ′_{i−1}) and N′_{i−1} = Ker(ϱ′′_{i}) as submodules of M_{i} and M_{i−1}, respec-

tively. We obtain ρ_{i}-linear isomorphisms M′_{i−1}/N′_{i−1} →^{θ_{i}} M_{i}/N_{i} which define an R-module isomorphism M ≅ M_{1} ×_{M_{2}/N_{2}} M_{2} × ⋯ ×_{M_{t}/N_{t}} M_{t}.

Remark 4.3. Any subring R of ∏_{i=1}^{t} R_{i} with surjective projections R → R_{i} is isomorphic to a fibre product of R_{1}, ⋯, R_{t} as defined in Lemma 4.1. For the rest of the paper, we define the fibre product R = R_{1} ×_{R_{2}/𝔞_{2}} R_{2} × ⋯ ×_{R_{t}/𝔞_{t}} R_{t} by the projections R →^{ρ_{i}} R_{i}. Similarly we define a fibre product of R_{i}-modules M_{i} by the projections M →^{ρ_{i}} M_{i}.

Assume all modules are finitely generated. Let l be a prime. Suppose K is a finite dimensional separable Q_{l}-algebra. Let R be an Z_{l}-order in K, that is, a Z_{l}-algebra that spans K over Q_{l}. An R-module M is torsion-free if αm ≠ 0 for all α ∈ R − {0} and m ∈ M − {0}. (If R is a domain then this is equivalent to the standard notation.) If M is a torsion-free R-module, then there is a natural injective map M → M ⊗_{R} K; if moreover M ⊗_{R} K ≅ K^{e} for some integer e then we say that M is of rank e. See [10, Lemma 3.6] for the proof of the following auxiliary lemma.

Lemma 4.4. Let R, K be as above. Let r ∈ R − {0}. Let M ⊆ M′ be torsion-free R-modules of rank e, then #M/rM = (#(R/rR))^{e}. There exist homomorphisms ρ : M/rM → M′/rM′ and ρ′ : M′/rM′ → M/rM with #Ker(ρ) = #Coker(ρ) and #Ker(ρ′) = #Coker(ρ′) dividing #(M′/M).

Proposition 4.5. For i = 1, ⋯, t, let R_{i} be a Z_{l}-order in a separable Q_{l}- algebra K_{i}. Let 𝔞_{i} be an ideal in R_{i} such that R = R_{1} ×_{R_{2}/𝔞_{2}} R_{2} × ⋯ ×_{R_{t}/𝔞_{t}} R_{t}. Let M be a torsion-free R-module and denote by M_{i} the image of the injection M ⊗_{R} R_{i} → M ⊗_{K} K_{i}. The projections M → M_{i} define an R-module isomorphism M ≅ M_{1} ×_{M_{2}/N_{2}} M_{2} × ⋯ ×_{M_{t}/N_{t}} M_{t} for some R_{i}-submodules N_{i} in M_{i}. Further, if M_{i} is of rank e_{i} then #((∏_{i=1}^{t} M_{i})/M) divides ∏_{i=2}^{t} #(R_{i}/𝔞_{i})^{e_{i}}.

Proof. By hypothesis, M ⊆ ∏_{i=1}^{t} M_{i}. We use induction on t to show that M is the desired fibre product and #((∏_{i=1}^{t} M_{i})/M) = ∏_{i=2}^{t} #(M_{i}/N_{i}). Suppose t = 2. By Lemma 4.2, M ≅ M_{1} ×_{M_{2}/N_{2}} M_{2} for some submodule N_{2}. Write 𝔞 = 𝔞_{1} ×𝔞_{2}. Since M_{1} ×_{M_{2}/N_{2}} M_{2} = M ⊇ 𝔞M = 𝔞(R_{1} ×R_{2})M = 𝔞(M_{1} ×M_{2}) = 𝔞_{1}M_{1} × 𝔞_{2}M_{2}, we get 𝔞_{1}M_{1} ⊆ N_{1}, 𝔞_{2}M_{2} ⊆ N_{2}, and #((M_{1} × M_{2})/M) = #(M_{2}/N_{2}). Denote by M′_{i} the image of the projection M → ∏_{j=1}^{i} M_{j}. Suppose there are R_{i}-submodules N_{i} in M_{i} such that, for i = 2, ⋯, t − 1, we have M′_{i} = M_{1} ×_{M_{2}/N_{2}} M_{2} × ⋯ ×_{M_{i}/N_{i}} M_{i}, and #(∏_{j=1}^{i−1} M_{j})/M′_{i−1} = ∏_{j=2}^{i−1}(#M_{j}/N_{j}).

Then M ≅ M′_{t−1} ×_{M_{t}/N_{t}} M_{t} and

#((∏_{i=1}^{t} M_{i})/M) = #((∏_{i=1}^{t−1} M_{i})/M′_{t−1}) · #((M′_{t−1} × M_{t})/M)
= (∏_{i=2}^{t−1} #(M_{i}/N_{i})) · #(M_{t}/N_{t})
= ∏_{i=2}^{t} #(M_{i}/N_{i}).

This finishes our induction. But #M_{i}/N_{i} | #(M_{i}/𝔞_{i}M_{i}) = #(R_{i}/𝔞_{i})^{e_{i}} by Lemma 4.4, so our assertion follows.

Below is an explicit example of a fibre product of rings.

Proposition 4.6. Let g_{1}, …, g_{t} ∈ Z[X] be arbitrary monic polynomials in one variable such that (g_{i}, g_{j}) = 1 in Q[X] for i ≠ j. Denote by π and π_{i} the images of X in the Z-algebras Z[X]/(∏_{i=1}^{t} g_{i}) and Z[X]/(g_{i}), respectively. Let R = Z[π]_{l}, R_{i} = Z[π_{i}]_{l} and 𝔞_{i} = (∏_{j=1}^{i−1} g_{j}(π_{i}))R_{i}. The natural projections

R →^{ρ_{i}} R_{i} define an isomorphism R ≅ R_{1} ×_{R_{2}/𝔞_{2}} R_{2} × ⋯ ×_{R_{t}/𝔞_{t}} R_{t} such that

#R_{i}/𝔞_{i} = ∏_{j=1}^{i−1} |R(g_{i}, g_{j})|_{l} for all i ≥ 2.

Proof. Sending π to (π_{1}, ⋯, π_{t}) defines a ring homomorphism R → ∏_{i=1}^{t} R_{i}. It is injective since (g_{i}, g_{j}) = 1 for all i ≠ j. For each i, this map induces surjective projections R → R_{i}. The asserted isomorphism follows from induction on t by invoking Lemma 4.1. Thus #R_{i}/𝔞_{i} = #R_{i}/(∏_{j=1}^{i−1} g_{j}(π_{i})) = ∏_{j=1}^{i−1} |N_{Q(π_{i})/Q}(g_{j}(π_{i}))|_{l} = ∏_{j=1}^{i−1} |R(g_{j}, g_{i})|_{l}.

5 Arbitrary supersingular abelian varieties

In this section, we shall prove Theorems 1.1 and 1.2. Let l be a prime ≠ p. Let T_{l} := T_{l}(A) be the l-adic Tate module of A and V_{l} := T_{l} ⊗_{Z_{l}} Q_{l}. There is a k-isogeny A →^{γ} ∏_{i=1}^{t} A_{i}, where A_{i} is an elementary abelian variety with characteristic polynomial g_{i}^{e_{i}}. Let Q[π] be the Q-subalgebra generated by π in the endomorphism algebra of A. Write R := Z[π] and R_{i} := Z[π]/g_{i}(π)Z[π]. Let R_{l} and R_{l,i} be the l-adic completions of R and R_{i}, respectively. The isogeny γ gives an isomorphism of Q[π]-modules, γ : V_{l} ⟶^{∼} ∏_{i=1}^{t} V_{l}(A_{i}). On T_{l}, the isogeny γ induces an injective map of R-modules T_{l} ⟶ ∏_{i=1}^{t} T_{l,i} where T_{l,i} = T_{l}(A_{i}), it can be identified as the image of T_{l} in the projection V_{l} → V_{l}(A_{i}). Clearly, T_{l} is a torsion-free R_{l}-module. Let π_{i} denote the image of π in Q[π]/(g_{i}(π)), then Q[π]/(g_{i}(π)) = Q(π_{i}) is actually a field, and we fix their embedding in C. Since V_{l}(A_{i}) ≅_{Q(π_{i})_{l}} Q(π_{i})_{l}^{e_{i}}, we note that T_{l,i} is a torsion-free R_{l,i}-module of rank e_{i} for each i.

Lemma 5.1. Let the notation be as above. Let r ∈ R. There is an R_{l}-module homomorphism

φ_{l} : T_{l}/rT_{l} ⟶^{α_{l}} ∏_{i=1}^{t}(T_{l,i}/rT_{l,i}) ⟶^{β_{l}} ∏_{i=1}^{t}(R_{l}/rR_{l})^{e_{i}}

with #Ker(φ_{l}) = #Coker(φ_{l}) dividing (2^{d_{E}} ∏_{i=2}^{t} ∏_{j=1}^{i−1} |R(g_{i}, g_{j})|^{e_{i}})_{l}.

Proof. By Propositions 4.5 and 4.6, R_{l} = R_{l,1} ×_{R_{l,2}/𝔞_{2}} R_{l,2} × ⋯ ×_{R_{l,t}/𝔞_{t}} R_{l,t} and T_{l} ≅ T_{l,1} ×_{T_{l,2}/N_{l,2}} T_{l,2} × ⋯ ×_{T_{l,t}/N_{l,t}} T_{l,t} for some R_{l,i}-submodules N_{l,i} in T_{l,i} such that

#((∏_{i=1}^{t} T_{l,i})/T_{l}) divides ∏_{i=2}^{t} ∏_{j=1}^{i−1} |R(g_{i}, g_{j})|_{l}^{e_{i}}.  (12)

Applying Lemma 4.4, there is a map α_{l} with #Ker(α_{l}) = #Coker(α_{l}) dividing ∏_{i=2}^{t} ∏_{j=1}^{i−1} |R(g_{i}, g_{j})|_{l}^{e_{i}}. On the other hand, by [10, Proposition 3.11], we have #(T_{l,i}/R_{l,i}^{e_{i}}) | 2^{e_{i} deg(g_{i})/2} if (l, π_{i}) ∈ {2}×E; and it equals 1 otherwise. Applying Lemma 4.4 again, we get a map β_{l} with #Ker(β_{l}) = #Coker(β_{l}) dividing

∏_{i=1}^{t} #(T_{l,i}/R_{l,i}^{e_{i}}) = ∏_{π∈E} 2_{l}^{e_{i} deg(g_{i})/2} = 2_{l}^{d_{E}} .  (13)

Hence the composition map φ_{l} = β_{l} · α_{l} has #Ker(φ_{l}) = #Coker(φ_{l}) dividing the product of the last numbers of (12) and (13).

Remark 5.2. If we order the e_{i} such that e_{1} ≥ e_{2} ≥ … ≥ e_{t}, and denote by R′_{l,i} the image of the projection R_{l} → R_{l,1} × ⋯ × R_{l,i}, then the divisibility in (12) is actually equality if T_{l,i} ≅_{R_{l,i}} R_{l,i}^{e_{i}} and

T_{l} ≅_{R_{l}} R_{l,1}^{e_{1}−e_{2}} × (R′_{l,2})^{e_{2}−e_{3}} × (R′_{l,3})^{e_{3}−e_{4}} × … × (R′_{l,t−1})^{e_{t−1}−e_{t}} × R_{l}^{e_{t}}.

Proof of Theorem 1.1: Suppose d = dim A ≥ 2. Noting that the l-Sylow subgroup of A(k) is isomorphic to T_{l}/(π − 1)T_{l} and that the p-Sylow subgroup is trivial, we define φ := ∏_{l≠p} φ_{l}, the φ_{l} as in Lemma 5.1. Our assertion follows from Lemma 3.3. Now suppose d = 1, then f has only one irreducible factor g. By [10, Theorem 1.1] there is a group homomorphism φ : A(k) → (Z/g(1)Z)^{e} with #Ker(φ) = #Coker(φ) dividing 2_{(p)}. Our assertion follows.

The proof of Theorem 1.2 is almost identical to that of [10, Theorem 1.2]. We provide a sketch of its proof. For any integer n coprime to p, we find an R-module homomorphism A[n] → ∏_{i=1}^{t}(R_{i}/nR_{i})^{e_{i}} with kernel and cokernel bounded as in the assertion. These bounds do not depend on n. After taking the suitable injective limit on both sides over n we get the desired homomorphism φ having the same bounds.

References

[1] H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, 1993.

[2] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, fifth ed., Oxford Science Publications, 1979.

[3] K. Ireland and M. Rosen, A classical introduction to modern number theory, second ed., Graduate Texts in Mathematics, vol. 84, Springer, 1992.

[4] S. Lang, Algebra, third ed., Addison-Wesley, 1993.

[5] K-Z Li and F. Oort, Moduli of supersingular abelian varieties, Lecture Notes in Math., vol. 1680, Springer-Verlag, 1998.

[6] F. Oort, Subvarieties of moduli spaces, Invent. Math. 24 (1974), 95–119.

[7] C. Powell, Bounds for multiplicative cosets over fields of prime order, Math. Comp. 66 (1997), 807–822.

[8] J.B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.

[9] J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, 1986.

[10] H. Zhu, Group structures of elementary supersingular abelian varieties over finite fields, MSRI Preprint #1998-065.