Characteristic polynomials of abelian varieties over 𝔽_{p}

Chantal David, CICMA and Concordia University

February 22, 1999

1 Introduction

Let A be an abelian variety of dimension g over the finite field 𝔽_{p}, and let ϕ be the Frobenius endomorphism of A over 𝔽_{p}. Let P_{A}(x) be the characteristic polynomial of ϕ acting on the Tate module T_{ℓ}(A) for any ℓ ≠ p. Then, P_{A}(x) is a monic polynomial of degree 2g with coefficients in ℤ. It was shown by Tate [4] that P_{A}(x) characterizes A up to 𝔽_{p}-isogeny, i.e. that two abelian varieties A and B over 𝔽_{p} are 𝔽_{p}-isogenous if and only if P_{A}(x) = P_{B}(x).

We write

P_{A}(x) = ∏_{i=1,...,2g}(x − π_{i}).

By the Weil’s Riemann hypothesis, |π_{i}| = \sqrt{p} for i = 1, …, 2g. We then call a Weil number over 𝔽_{p} an algebraic integer π such that |σ(π)| = \sqrt{p} for all conjugates of π. Also, the roots of P_{A}(x) come in pairs (π, p/π), and P_{A}(x) writes as

P_{A}(x) = x^{2g} + a_{1}x^{2g−1} + … + a_{g}x^{g} + pa_{g−1}x^{g+1} + … + p^{g−1}a_{1}x + p^{g}.  (1)

We also say that the polynomial P_{A}(x) is the isogeny class of A.

By the triangle inequality,

|a_{n}| ≤ \binom{2g}{n} p^{n/2} for n = 1, …, g.  (2)

If A is an elliptic curve, i.e. g = 1, then a_{1} = a_{p}(E) and the triangle inequality is the Hasse inequality

|a_{p}(E)| ≤ 2\sqrt{p}.

Moreover, the triangle inequality is effective in this case, as it was shown by Deuring [1] that for any integer |a| ≤ 2\sqrt{p}, there is an elliptic curve E/𝔽_{p} such that P_{E}(x) = x^{2} − ax + p. This is not true for g > 1. For example, let P (x) = x^{4} + 3px^{2} + p^{2}. This is a polynomial of the form (1) satisfying (2), but the roots of P (x) are not Weil numbers over 𝔽_{p}, and then P (x) is not the isogeny class of any abelian variety over 𝔽_{p}.

We show in this paper that there exists contants such that the triangle inequality is effective for all dimensions g. This result also gives a lower bound for the number of isogeny classes of abelian varieties of dimension g over 𝔽_{p}.

Theorem 1.1 Fix any integer g ≥ 1. For any integers a_{1}, …, a_{g} and any prime p, let P (x) be the polynomial

P (x) = x^{2g} + a_{1}x^{2g−1} + … + a_{g}x^{g} + pa_{g−1}x^{g+1} + … + p^{g−1}a_{1}x + p^{g}.

Then, there exists positive constants C_{g}(n) (not depending on p) such

|a_{n}| ≤ C_{g}(n) p^{n/2} for n = 1, …, g

implies that P (x) is the isogeny class of an abelian variety over 𝔽_{p}.

Corollary 1.2 Let g be a positive integer. For any prime p, let I_{g}(p) be the number of 𝔽_{p}-isogeny classes of abelian varieties of dimension g over 𝔽_{p}. Then, there exists positive constants A_{g} and B_{g} depending only on g such that

A_{g} p^{g(g+1)/4} ≤ I_{g}(p) ≤ B_{g} p^{g(g+1)/4}.

Proof: The lower bound is from Theorem 1.1, and the upper bound from the triangle inequality. QED.

2 Honda’s Theorem

Theorem 2.1 (Honda [2]) Let q = p^{a} for some positive integer a, and let π be a Weil number over 𝔽_{q} with minimal polynomial m(x). Then, there exists an elementary abelian variety A over 𝔽_{q} such that P_{A}(x) = m(x)^{e}, where e is the least common denominator of the invariants i_{v} for each place v above p in E = End_{𝔽_{q}}(A) ⊗ ℚ and

i_{v} = \{ \begin{matrix}\frac{1}{2} & if v is real; \\ \frac{f_{v}ord_{v}(π)}{a} & otherwise \end{matrix}

where f_{v} is the residual degree at v.

This version of Honda’s Theorem is from [5, Theorem 1]. Then, over 𝔽_{p}, the correspondence between Weil numbers and abelian varieties can be stated as a correspondance between polynomials and abelian varieties. More precisely, we have the following corollary.

Corollary 2.2 Let P (x) ∈ ℤ[x] a polynomial of degree 2g of the form

P (x) = x^{2g} + a_{1}x^{2g−1} + … + a_{g}x^{g} + pa_{g−1}x^{g+1} + … + p^{g−1}a_{1}x + p^{g}

such that the roots of P (x) are Weil numbers over 𝔽_{p}. Then, P (x) = P_{A}(x) for some abelian variety of dimension g over 𝔽_{p}.

Proof: Write P (x) = (x^{2}−p)^{2n_{0}}P_{1}(x)^{n_{1}} … P_{r}(x)^{n_{r}} where n_{0} ∈ ℕ and for i = 1, …, r, P_{i}(x) is an irreducible polynomial with no real root, and n_{i} a positive integer. By Honda’s Theorem, there are elementary abelian varieties A_{1}, …, A_{r} over 𝔽_{p} such that P_{A_{i}}(x) = P_{i}(x)^{e_{i}} for i = 1, …, r. As a = 1 and P_{i}(x) has no real roots, e_{i} = 1 for i = 1, …, r. Similary, there is an abelian variety A_{0} over 𝔽_{p} such that P_{A_{0}}(x) = (x^{2} − p)^{2}. Then, P (x) = P_{A}(x) for

A = A_{0}^{n_{0}} × A_{1}^{n_{1}} × … × A_{r}^{n_{r}}.

QED.

It then suffices to prove the following proposition to obtain Theorem 1.1.

Proposition 2.3 Fix any integer g ≥ 1, and let a_{1}, …, a_{g} be any integers. For any prime p, let P (x) be the polynomial

P (x) = x^{2g} + a_{1}x^{2g−1} + … + a_{g}x^{g} + pa_{g−1}x^{g+1} + … + p^{g−1}a_{1}x + p^{g}.

There exists constants C_{g}(n), n = 1, …, g, (not depending on p) such that

|a_{n}| ≤ C_{g}(n) p^{n/2} for n = 1, …, g  (3)

implies that the roots of P (x) are Weil numbers for 𝔽_{p}.

3 Proof of Proposition 2.3

We number the roots π_{1}, …, π_{2g} of P (x) such that π_{g+i} = p/π_{i} for i = 1, …, g. Let Q(x) be the monic polynomial

Q(x) = ∏_{i=1,...,g} (x − (π_{i} + p/π_{i})) = ∏_{i=1,...,g} (x − β_{i}) .

Lemma 3.1 Let P (x) and Q(x) be the polynomials defined above. Then, |π_{i}| = \sqrt{p} for i = 1, …, 2g if and only if Q(x) is totally real, and |β_{i}| ≤ 2\sqrt{p} for i = 1, …, g.

Proof: The proof is a standard observation, and can be found for example in [6]. If |π_{i}| = \sqrt{p} for i = 1, …, 2g, then clearly |β_{i}| ≤ 2\sqrt{p} for i = 1, …, g. Also, as π_{i} = p/π_{i} if and only if π_{i} = ±\sqrt{p}, it follows that β_{i} = π_{i} + p/π_{i} is either the real part of the complex number π_{i} or ±2\sqrt{p}. Conversly, assume that all β_{i} are real and that |β_{i}| ≤ 2\sqrt{p}. Then, π_{i} and p/π_{i} are the roots of

x^{2} − β_{i}x + p,

which has either complex conjugate roots, i.e. |π| = |p/π| = \sqrt{p}, or a double real root, i.e. π = p/π = ±\sqrt{p}. QED.

It then suffices to prove that (3) implies that Q(x) is totally real with roots β_{i} such that |β_{i}| ≤ 2\sqrt{p} for i = 1, …, g. We write

Q(x) = ∏_{i=1,...,g}(x − β_{i}) = x^{g} + b_{1}x^{g−1} + … + b_{g−1}x + b_{g}.

We first need to express the coefficients b_{1}, …, b_{n} of Q(x) in term of a_{1}, …, a_{n}. We also define a_{0} = b_{0} = 1.

Lemma 3.2 For n = 1, …, g,

a_{n} = b_{n} + ∑_{i=1}^{[n/2]} \binom{g − (n − 2i)}{i} p^{i} b_{n−2i}

Proof: Let

A_{n} = (−1)^{n}a_{n} = ∑_{1≤i_{1}<...<i_{n}≤2g} π_{i_{1}} … π_{i_{n}},

and

B_{n} = (−1)^{n}b_{n} = ∑_{1≤i_{1}<...<i_{n}≤g} (π_{i_{1}} + p/π_{i_{1}}) … (π_{i_{n}} + p/π_{i_{n}}) = ∑_{1≤i_{1}<...<i_{n}≤2g}^{(0)} π_{i_{1}} … π_{i_{n}}

where ∑_{1≤i_{1}<...<i_{n}≤2g}^{(r)} means that exactly r pairs (i, g + i) occurs among i_{1}, …, i_{n}. Then,

A_{n} = B_{n} + ∑_{1≤i_{1}<...<i_{n}≤2g}^{(1)} π_{i_{1}} … π_{i_{n}} + ∑_{1≤i_{1}<...<i_{n}≤2g}^{(2)} π_{i_{1}} … π_{i_{n}} + …

= B_{n} + ∑_{i=1}^{[n/2]} \binom{g − (n − 2i)}{i} p^{i} B_{n−2i}.

This proves the result. QED.

Lemma 3.3 For n = 1, …, g,

b_{n} = ∑_{i=0}^{[n/2]} d(n, i) p^{i} a_{n−2i}

where d(n, i) are integers independant of p.

Proof: Use Lemma 3.2 and induction on n. QED.

To show Proposition 2.3, we show that the conditions (3) on the coefficients of P (x) imply that the associated polynomial Q(x) is “close enough” to a given polynomial Q_{0}(x) which is totally real. Consider the polynomial

P_{0}(x) = x^{2g} + p^{g},

i.e. a_{i} = 0 for i = 1, …, n. The roots of P_{0}(x) are

π_{k} = ζ_{4g}^{2k−1}\sqrt{p} for k = 1, …, 2g,

where ζ_{4g} = exp (\frac{2πi}{4g}). Then, by Honda’s theorem, P_{0}(x) is the isogeny class of some abelian variety A_{0} over 𝔽_{p}. (A_{0} is a supersingular abelian variety, i.e. it becomes isogenous to a product of supersingular elliptic curves over some extension of 𝔽_{p}).

Then

Q_{0}(x) = ∏_{k=1,...,g}(x − (π_{k} + p/π_{k}))

have roots

β_{k} = (π_{k} + p/π_{k}) = 2\sqrt{p} cos (\frac{2π(2k − 1)}{4g}) for k = 1, …, g.

For k = 1, …, g − 1, let α_{k} be non-zero real numbers such that β_{k+1} < α_{k} < β_{k}, and let α_{0} = 2\sqrt{p} and α_{g} = −2\sqrt{p}. For k = 0, …, g, we also define γ_{k} by α_{k} = γ_{k}\sqrt{p}. Then, Q_{0}(α_{k}) and Q_{0}(α_{k−1}) have opposite signs for k = 1, …, g, and if

|Q(α_{k}) − Q_{0}(α_{k})| < |Q_{0}(α_{k})| for k = 0, …, g,  (4)

then Q(x) also has g real roots of absolute value ≤ 2\sqrt{p}.

We then have to prove that (3) implies (4). Using Lemma 3.3, we get

|Q_{0}(α_{k})| = | ∑_{n even^{n=0}}^{g} d(n, n/2) p^{n/2} γ_{k}^{g−n}\sqrt{p}^{g−n}|

= p^{g/2} | ∑_{n even^{n=0}}^{g} d(n, n/2) γ_{k}^{g−n}| = p^{g/2}K(g, k),  (5)

where K(g, k) is a non-zero constant independant of p. Similarly, if

|a_{n}| ≤ C_{g}(n)p^{n/2} for n = 1, …, g,

then

|Q(α_{k}) − Q_{0}(α_{k})| = |∑_{n=0}^{g} (∑_{2i≠n^{i=0}}^{[n/2]} d(n, i) p^{i} a_{n−2i}) γ_{k}^{g−n}\sqrt{p}^{g−n}|

≤ p^{g/2} ∑_{n=0}^{g} (∑_{2i≠n^{i=0}}^{[n/2]} |d(n, i)| C_{g}(n − 2i)) |γ_{k}|^{g−n}.  (6)

Comparing (5) and (6), it suffices to choose constants C_{g}(n) such that

∑_{n=0}^{g} (∑_{2i≠n^{i=0}}^{[n/2]} |d(n, i)| C_{g}(n − 2i)) |γ_{k}|^{g−n} < K(g, k) for k = 0, …, g.

Let N(k, g) be the number of terms on the sum above. Then, it suffices to choose constants C_{g}(n) such that for each i, n, k

|d(n, i)| C_{g}(n − 2i) |γ_{k}|^{g−n} < \frac{K(g, k)}{N(k, g)}.

This completes the proof of Proposition 2.3. QED. Acknowledgements: The author would like to thank Eyal Goren and Hershy Kisilevsky for many helpful discussions, and Francesco Pappalardi for a careful reading of the manuscript.

References

[1] M. Deuring, Die Typen de Multiplikatorenringe elliptischer Functionenkörper, Abh. Math. Sem. Hamburg 14 (1941), 197-272.

[2] T. Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83-95.

[3] J. S. Milne and W. C. Waterhouse, Abelian varieties over finite fields, Proc. Sympos. Pure Math. 20 (1971), 53–64.

[4] J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134-144.

[5] J. Tate, Classes d’isogénie des variétés abéliennes sur un corps fini, Séminaire Bourbaki, Lecture Notes in Math. 179, 1971.

[6] W. C. Waterhouse, Abelian varieties over finite fields, Ann. scient. École Norm. Sup. 2 (1969), 521-560.

Concordia University, Department of Mathematics, 1455 de Maisonneuve Blvd. West, Montréal, Quebec H3G 1M8, CANADA E–mail: chantal@cicma.concordia.ca