ESSENTIAL DIMENSION OF ABELIAN VARIETIES OVER NUMBER FIELDS

PATRICK BROSNAN AND RAMESH SREEKANTAN

Abstract. We affirmatively answer a conjecture in the preprint “Essential dimension and algebraic stacks,” proving that the essential dimension of an abelian variety over a number field is infinite.

Let k be a field and let Fields_{k} denote the category whose objects are field extensions L/k and whose morphisms are inclusions L ↪ M of fields. Let F : Fields_{k} → Sets be a covariant functor. A field of definition for an element a ∈ F (L) is a subfield M of L over k such that a ∈ im(F (M) → F (L)). The essential dimension of a ∈ F (L) is eda:= inf{trdeg_{k}M |M is a field of definition for a}. The essential dimension of the functor F is edF := sup{eda|L ∈ Fields_{k},a ∈ F (L)}.

If G is an algebraic group over k, we write edG for the essential dimension of the functor L ↦ H_{fppf}^{1}(L,G). That is edG is the essential dimension of the functor sending a field L to the set of isomorphism classes of G-torsors over L. The notion of essential dimension of a finite group was introduced by J. Buhler and Z. Reichstein. The definition of the essential dimension of a functor is a generalization given later by A. Merkurjev. In [3] (which the reader could consult for further background), a notion of essential dimension for algebraic stacks was introduced. In the terminology of that paper, edG is the essential dimension of the stack BG.

The purpose of this paper is to generalize the following result.

Theorem 1 (Corollary 10.4 [3]). Let E be an elliptic curve over a number field k. Assume that there is at least one prime 𝔭 of k where E has semistable bad reduction. Then edE = +∞.

Note that another equivalent way of stating the theorem is to say that edE = +∞ for all elliptic curves E such that j(E) ∈ \overline{ℚ}\\{algebraic integers \}. The result was proved by showing that Tate curves have infinite essential dimension. However, this method does not apply to elliptic curves with integral j invariants. Nonetheless, Conjecture 10.5 of [3] guesses that edE = +∞ for all elliptic curves over number fields. This conjecture is answered by the following.

Theorem 2. Let A be an abelian variety over a number field k. Then edA = +∞.

Note that if A is an abelian variety over ℂ, then edA = 2dimA. This is the main result of [2].

The theorem is an easy consequence of the following result whose formulation does not involve essential dimension. To state it, for a positive integer m, let μ_{m} denote the group scheme of m-th roots of unity; and, for a rational prime l, let μ_{l^{∞}} denote the union ∪_{n∈ℤ_{+}} μ_{l^{n}}.

Theorem 3. Let A be an abelian variety over a number field k. Then there is an odd prime ℓ and an algebraic field extension L/k such that

(1) ℚ_{ℓ}/ℤ_{ℓ} ⊂ A(L).

(2) 1 < |μ_{ℓ^{∞}}(L)| < ∞.

In the first section, we derive Theorem 2 from Theorem 3. The technique used is a result of M. Florence concerning the essential dimension of ℤ/ℓ^{n}. In the section 2, we prove Theorem 3. Here the main results used are those of Bogomolov and Serre on the action of the absolute Galois group Gal(k) on the Tate module T_{ℓ}A.

Remark 4. The recent preprint [7] of Karpenko and Merkurjev provides another way to show that the essential dimension of an abelian variety over a number field is infinite. To be precise, by generalizing the results of that paper slightly, one can use them to compute the essential dimension of the group scheme A[n] of n-torsion points of an abelian variety. In fact, using this idea one can show that the essential dimension of an abelian variety over a p-adic field is also infinite. However, the present proof of Theorem 2 is shorter than a proof using [7] would be and we hope that Theorem 3 is independently interesting.

Acknowledgments. It is a pleasure to thank Z. Reichstein and A. Vistoli for useful conversations. We are also extremely grateful to N. Fakhruddin. After seeing a primitive version of this paper proving Theorems 2 and 3 for elliptic curves (using Serre’s results on Galois representations for ordinary elliptic curves and the theory of complex multiplication for elliptic curves with CM), he pointed out that Bogomolov’s results could be applied to prove the same results for abelian varieties.

1. Theorem 3 implies Theorem 2

As mentioned above, we will use the following result [6, Theorem 4.1] of M. Florence.

Theorem 5. Let ℓ be an odd prime and r a positive integer. Let L/ℚ be a field such that |μ_{ℓ^{∞}}(L)| = ℓ^{r} < ∞. Then, for any positive integer k,

ed_{L}ℤ/ℓ^{k} = max{1,k − r}.

Corollary 6. Let A be an abelian variety over a field L of characteristic 0. Let ℓ be an odd prime and suppose that the statements in the conclusion of Theorem 3 are satisfied; i.e:

(1) ℚ_{ℓ}/ℤ_{ℓ} ⊂ A(L).

(2) 1 < |μ_{ℓ^{∞}}(L)| < ∞.

Then edA = +∞.

Proof. Since L satisfies (2), ed_{L}ℤ/ℓ^{n} → ∞ as n → ∞. By (1), there is an injection (ℤ/ℓ^{n})_{L} → A. Therefore, by [1, Theorem 6.19], edA ≥ ed_{L}ℤ/ℓ^{n}− dimA for all n. Letting n tend to ∞, we see that edA = +∞. □

Proof of Theorem 2 assuming Theorem 3. Let A be our abelian variety over a number field k. Using Theorem 3 and Corollary 6, we can find a field extension L/k such that edA_{L} = +∞. This implies that edA = +∞ (by [1, Proposition 1.5].)

2. Galois representations and the proof of Theorem 3

Before proving Theorem 3, we fix some (standard) notation. We write G:= Gal(\overline{k}/k) for the absolute Galois group of the number field k. For a rational prime ℓ, we write T_{ℓ}A for the Tate-module lim_{←}A[ℓ^{n}] of the abelian variety A. We write V_{ℓ}A for T_{ℓ}A ⊗_{ℤ_{ℓ}}ℚ_{ℓ}. For an integer n, we write ℤ/n(1) for μ_{n}, and for j ∈ ℤ, ℤ/n(j) for μ_{n}^{⊗j} . We write ℤ_{ℓ}(j) := lim_{←}ℤ/l^{m}(j).

For any prime 𝔭 of k where A has good reduction, write T_{𝔭} for the corresponding Frobenius torus [4, Definition 3.1 and p. 326]. By [4, Proposition 3.2], T_{𝔭} contains a rank 1 torus D ≅ 𝔾_{m} such that, for every rational prime ℓ ∉ 𝔭, D(ℚ_{ℓ}) ⊂ Gl(V_{ℓ}A) is the set of homotheties (i.e. scalar matrices).

Lemma 7. Let 𝔭 be a prime of k such that the reduction A/𝔭 of A at 𝔭 is good but not supersingular. Then the rank of T_{𝔭} is strictly greater than 1.

Proof. This follows directly from [4, Proposition 3.3]. □

The following proposition was suggested to us by N. Fakhruddin.

Proposition 8. Let V be an n-dimensional vector space over a field F, and let T be an F -split torus in Gl_{n}(V) of rank at least 2 containing the homotheties. Then there is a non-zero vector v ∈ V and a rank 1 subtorus S of T such that

(1) S fixes v;

(2) the determinant map det: S → 𝔾_{m} is surjective.

Proof. First note that it suffices to prove the proposition when T is rank 2. Pick a basis e_{1},...,e_{d} for V. Then we can assume that the injection T → Gl(V) is given by (λ,μ) ↦ T (λ,μ) where T (λ,μ)e_{i} = λμ^{r_{i}}e_{i} for some integers r_{i} with (r_{1},...,r_{d}) = 1.

Now, each e_{i} is fixed by the group T_{i} of matrices of the form T (μ^{−r_{i}} ,μ). We have

detT (μ^{−r_{i}} ,μ) = μ^{−nr_{i}+∑ _{j=1}^{n} r_{j}} .

This determinant is trivial if and only if nr_{i} = ∑ _{j=1}^{n} r_{j}, and this can happen for all i only if all the r_{i} are the same. However, since T has rank at least 2, at least two of the r_{i} must differ. Thus, we can set v equal to one of the e_{i} and S = T_{i}. □

Proof of Theorem 3. Let A be our abelian variety over a number field k. We can find a prime 𝔭 in k such that A has good reduction at 𝔭 but A/𝔭 is not supersingular. (This is well-known if dimA = 1: the case where A has CM is standard and otherwise it follows from the exercise on page IV-13 of [8]. When dimA > 1 it can be proved by adapting the exercise as Ogus does in Corollary 2.8 of his notes in [5].) Thus the Frobenius torus T_{𝔭} has rank at least 2. Using Tchebotarev density, it is easy to see that T_{𝔭} ⊗ℚ_{ℓ} is a split torus for all rational primes ℓ in a set of positive density. Thus, we can find an odd rational prime ℓ such that ℓ |̸ 𝔭 and T_{𝔭} ⊗ ℚ_{ℓ} is split. Now, set F = k(ζ_{ℓ}) where ζ_{ℓ} is a primitive ℓ-th root of unity. Note that T_{𝔭} is the Frobenius torus for A_{F} as Frobenius tori are invariant under finite extension of the ground field.

Now, using Proposition 8, we can can find a rank 1 subtorus S ⊂ T_{𝔭} and a vector v ∈ T_{ℓ}A_{F} such that S fixes v and det: S → 𝔾_{m} is surjective. Let ρ : Gal(F) → Aut(V_{ℓ}A_{F}) denote the Galois representation on the Tate module and let H = {g ∈ Gal(F)|ρ(g)v = v}. By the theorem of Bogomolov [4, Theorem B] (and the fact that S fixes v), it follows that the

Lie(S) ⊗_{ℚ} ℚ_{ℓ} ⊂ Lie(ρ(H))

where Lie(S) denotes the Lie algebra of S as an algebraic group and Lie(ρ(H)) denotes the Lie algebra as an ℓ-adic group. Therefore the intersection of S(ℚ_{ℓ}) with ρ(H) contains an open neighborhood of the identity in S(ℚ_{ℓ}). In particular, det(H) contains an open subset of the identity in ℚ_{ℓ}^{∗}. Set L := \overline{F}^{H}. Then, from the fact that v is fixed by H, it follows that ℚ_{ℓ}/ℤ_{ℓ} ⊂ A(L). On the other hand, since ∧^{2dimA}T_{ℓ}A ≅ ℤ_{ℓ}(dimA), the fact that det(H) contains an open subset of the identity in ℚ_{ℓ}^{∗} implies that μ_{ℓ^{∞}}(L) is finite. This completes the proof of Theorem 3.

References

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(Brosnan) Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2

(Sreekantan) School of Mathematics Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai 400 005, India

E-mail address, Brosnan: brosnan@math.ubc.ca

E-mail address, Sreekantan: ramesh@math.tifr.res.in