RIGIDITY THEOREMS FOR ABELIAN VARIETIES

YU. G. ZARHIN AND A. SILVERBERG

A theme running through this paper is the fact that a root of unity congruent to 1 modulo n is 1, if n ≥ 3. This theme can be traced back to Minkowski [4], who proved that an integral matrix of finite multiplicative order which is congruent to the identity modulo n is the identity, if n ≥ 3. Serre applied this idea to abelian varieties, and showed ([3], 4.7.1) that if an automorphism of finite order of a semiabelian variety induces the identity on the scheme-theoretic kernel of multiplication by n, then it is the identity if n ≥ 3, and its square is the identity if n = 2.

Using Serre’s result, Raynaud augmented Grothendieck’s study of Neron models of abelian varieties, by proving a useful criterion of semistable reduction ([3], 4.7). Raynaud’s criterion says that if an abelian variety and all its n-torsion points are defined over a field with a discrete valuation of residue characteristic not dividing n, and n ≥ 3, then the abelian variety has semistable reduction (and acquires semistable reduction in a (Z/2Z)^{r}-extension, if n = 2, the valuation ring is henselian, and the residue field is separably closed). One may prove (compare with [2], [1]; see also 3.6 of [8]) that if an abelian variety and all its n-torsion points are defined over a finitely generated extension of Q, and n ≥ 3, then the Zariski closure of the image of the absolute Galois group under every ℓ-adic representation is connected. Another consequence of the Minkowski idea (see [5]) is that if two abelian varieties are defined over a field F, then every homomorphism between them is defined over every extension of F over which the n-torsion points of the abelian varieties are defined, as long as n is not divisible by the characteristic of F and is at least 3.

In Theorem 1 below we state a generalization of the Minkowski idea. In this generalization, we replace the assumption that the root of unity α is congruent to 1 modulo n by the assumption that (α − 1)^{k} is congruent to 0 modulo n. In the remainder of the paper we state generalizations of the above applications of the Minkowski idea.

2 YU. G. ZARHIN AND A. SILVERBERG

Definition 1. If k is a positive integer, define a finite set N(k) by

N(k) = {prime powers ℓ^{m} : 0 ≤ m(ℓ − 1) ≤ k}.

Let R(k, 1) = 0; if n is a positive integer which is not in N(k), let R(k, n) = 1; if 1 ≠ n = ℓ^{m} ∈ N(k) with ℓ a prime, let

R(k, n) = ℓ^{r(k,n)} where r(k, n) = max{r ∈ Z^{+} : m(ℓ − 1)ℓ^{r−1} ≤ k}.

For example,

N(1) = {1, 2}, N(2) = {1, 2, 3, 4},
N(3) = {1, 2, 3, 4, 8}, N(4) = {1, 2, 3, 4, 5, 8, 9, 16};
R(1, 2) = 2, R(1, n) = 1 if n ≥ 3,
R(2, 2) = 4, R(2, 3) = 3, R(2, 4) = 2, and R(2, n) = 1 if n ≥ 5.

Definition 2. If Δ is a subset of a ring O, we say Δ has no O-zero divisors if there do not exist x ∈ Δ and 0 ≠ y ∈ O such that xy = 0 = yx. In particular, if Δ has no O-zero-divisors then 0 ∉ Δ.

If F is a field, let F^{s} denote a separable closure and let char(F) denote the characteristic of F. Suppose (X, λ) is a polarized abelian variety defined over F. Write X_{n} for the kernel of multiplication by n in X(F^{s}). If n is a positive integer not divisible by char(F), and μ_{n} is the Gal(F^{s}/F)-module of n-th roots of unity in F^{s}, then define a skew-symmetric Galois-equivariant bilinear map

e_{λ,n} : X_{n} × X_{n} → μ_{n}, e_{λ,n}(x_{1}, x_{2}) = e_{n}(x_{1}, λ(x_{2})),

where e_{n} is the Weil pairing. If n is relatively prime to the degree of the polarization λ, then the pairing e_{λ,n} is nondegenerate. Suppose ℓ is a prime number not equal to char(F). Let T_{ℓ}(X) = lim_{←} X_{ℓ^{r}} (the Tate module), let V_{ℓ}(X) = T_{ℓ}(X) ⊗_{Z_{ℓ}} Q_{ℓ}, let d = dim(X), and let ρ_{X,ℓ} denote the ℓ-adic representation

ρ_{X,ℓ} : Gal(F^{s}/F) → Aut(T_{ℓ}(X)) ⊆ Aut(V_{ℓ}(X)) ≅ GL_{2d}(Q_{ℓ}).

Let G_{F,X} denote the image of Gal(F^{s}/F) under ρ_{X,ℓ}. Let 𝔊_{ℓ}(F, X) denote the algebraic envelope of the image of ρ_{X,ℓ}, i.e., the Zariski closure of G_{F,X} in GL_{2d}(Q_{ℓ}).

Theorem 1. Suppose O is a ring such that the natural map O → O ⊗_{Z} Q is injective, and suppose k and n are positive integers. Suppose that for every rational prime divisor ℓ of n, 1 + ℓO has no O-zero-divisors. Suppose α is an element of O of finite multiplicative order such that (α − 1)^{k} ∈ nO. Then α^{R(k,n)} = 1. In particular, if n ∉ N(k) then α = 1.

Corollary . Suppose n is a positive integer, O is an integral domain of characteristic zero such that no rational prime which divides n is a unit in O, α ∈ O, α has finite multiplicative order, and (α − 1)^{2} ∈ nO. If n ≥ 5 then α = 1, if n = 4 then α^{2} = 1, if n = 3 then α^{3} = 1, and if n = 2 then α^{4} = 1.

Theorem 2. Suppose G is a commutative group scheme over a field, which is an extension of an abelian variety by a torus, n and k are positive integers, α is an endomorphism of G of finite multiplicative order, and (α − 1)^{k} is 0 on the schemetheoretic kernel of multiplication by n on G. Then α^{R(k,n)} = 1.

The following result is an analogue of the result of Raynaud stated in the introduction, and generalizes to higher dimensional abelian varieties earlier work on elliptic curves due to Frey [7] and to Flexor and Oesterlé [6].

Theorem 3. Suppose (X, λ) is a polarized abelian variety defined over a field F, v is a discrete valuation on F, n is an integer not divisible by the residue characteristic of v, X̃_{n} is a maximal isotropic subgroup of X_{n} with respect to the pairing e_{λ,n}, and the points of X̃_{n} are defined over F. If n ≥ 5 then X has semistable reduction at v. If n = 2, 3, or 4, then X has semistable reduction above v over every totally ramified Galois (necessarily cyclic) extension of F of degree 4, 3, or 2, respectively.

Theorem 4. Suppose (X, λ) is a polarized abelian variety defined over a field F which is either a finitely generated extension of Q or a global field of positive characteristic, and let p = char(F). Suppose ℓ is a prime number, n is an integer, n ≥ 5, p does not divide ℓn, and X̃_{n} is a maximal isotropic subgroup of X_{n} with respect to e_{λ,n}. Suppose the points of X̃_{n} are defined over F, and the roots of unity in e_{λ,n}(X_{n}, X̃_{n}) are contained in F. Then 𝔊_{ℓ}(F, X) is connected.

Theorem 5. Suppose (X, λ) and (Y, μ) are polarized abelian varieties defined over a field F, n is a positive integer which is not divisible by the characteristic of F, and n ≥ 5. Suppose X̃_{n}, respectively Ỹ_{n}, is a maximal isotropic subgroup of X_{n}, respectively Y_{n}, with respect to the pairing e_{λ,n}, respectively e_{μ,n}. Suppose the points of X̃_{n} and Ỹ_{n} are defined over F, and the roots of unity in e_{λ,n}(X_{n}, X̃_{n}) and e_{μ,n}(Y_{n}, Ỹ_{n}) are contained in F. Then every homomorphism between X and Y is

defined over F.

Note that the two preceding results are false if one drops the hypothesis that the

roots of unity in e_{λ,n}(X_{n}, X̃_{n}) and e_{μ,n}(Y_{n}, Ỹ_{n}) are contained in F (for example,

consider CM elliptic curves defined over Q).

References

[1] M. Borovoi, Mat. Sbornik (N. S.) 94 (136) (1974), 649–652; English transl. in Math. USSR Sbornik 23 (1974), 613–616.

[2] M. Borovoi, Problems of group theory and homological algebra, No. 1 (Russian), pp. 3–53, Yaroslav. Gos. Univ., Yaroslavl’, 1977.

[3] A. Grothendieck, Groupes de monodromie en géometrie algébrique, SGA7 I, Lecture Notes in Math. vol. 288, 1972, pp. 313–523.

[4] H. Minkowski, J. reine angew. Math. 101 (1887), 196–202.

[5] A. Silverberg, J. Pure and Applied Algebra 77 (1992), 253-262.

[6] M. Flexor and J. Oesterlé, Astérisque 183 (1990), 25–36.

[7] G. Frey, Ark. Mat. 15 (1977), 1–19.

[8] W. Chi, Amer. J. Math. 114 (1992), 315–353.

Institute for Mathematical Problems in Biology, Russian Academy of Sci-

ences, Pushchino, Moscow Region

Pennsylvania State University

Ohio State University