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.Comment: 4 pages. To appear in C. R. Math. Acad. Sci. Paris. Preprint posted
  earlier to http://www.mathematik.uni-bielefeld.de/LAG

Berhuy

Brosnan

Deligne

Florence

Patrick Brosnan

Ramesh Sreekantan

Serre

English

arXiv

Crossref

Essential dimension of Abelian varieties over number fields

Brosnan, Patrick

Sreekantan, Ramesh

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 346 (2008) 417–420http://france.elsevier.com/direct/CRASS1/Algebraic Geometry/Number TheoryEssential dimension of Abelian varieties over number fieldsPatrick Brosnan a, Ramesh Sreekantan ba Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2b School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai 400 005, IndiaReceived 25 November 2007; accepted 8 February 2008Available online 7 March 2008Presented by Jean-Pierre SerreAbstractWe show that the essential dimension of a non-trivial Abelian variety over a number field is infinite. To cite this article:P. Brosnan, R. Sreekantan, C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméDimension essentielle d’une variété abélienne sur un corps de nombres. On montre que la dimension essentielle d’une variétéabélienne non-triviale définie sur un corps de nombres est infinie. Pour citer cet article : P. Brosnan, R. Sreekantan, C. R. Acad.Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionLet k be a field and let Fieldsk denote the category whose objects are field extensions L/k and whose morphismsare inclusions M ↪→ L of fields. Let F : Fieldsk → Sets be a covariant functor. A field of definition for an elementa ∈ 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) iseda := inf{trdegk M | M is a field of definition for a}. The essential dimension of the functor F is edF := sup{eda |L ∈ Fieldsk, a ∈ F(L)}.If G is an algebraic group over k, we write edG for the essential dimension of the functor L  H1fppf(L,G). That isedG 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 theessential dimension of a functor is a generalization given later by A. Merkurjev. In [3] (which the reader could consultfor further background), a notion of essential dimension for algebraic stacks was introduced. In the terminology ofthat paper, edG is the essential dimension of the stack BG.The purpose of this Note is to generalize the following result:E-mail addresses: brosnan@math.ubc.ca (P. Brosnan), ramesh@math.tifr.res.in (R. Sreekantan).1631-073X/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2008.02.008418 P. Brosnan, R. Sreekantan / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 417–420Theorem 1.1. (Corollary 10.4 [3].) Let E be an elliptic curve over a number field k. Assume that there is at least oneprime 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 any elliptic curve E overa number field such that j (E) is not an algebraic integer. The result was proved by showing that Tate curves haveinfinite essential dimension. 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 answeredby the following:Theorem 1.2. Let A be a non-trivial Abelian variety over a number field k. Then edA = +∞.Note that if A is an Abelian variety over C, then edA = 2 dimA. This is the main result of [2].The theorem is an easy consequence of the following result whose formulation does not involve essential dimen-sion. To state it, for a positive integer m, let μm denote the group scheme of mth roots of unity; and, for a rationalprime l, let μl∞ denote the union⋃n∈Z+ μln .Theorem 1.3. Let A be a non-trivial Abelian variety over a number field k. Then there is an odd prime  and analgebraic field extension L/k such that(i) Q/Z ⊂ A(L).(ii) 1 < |μ∞(L)| < ∞.In Section 2, we derive Theorem 1.2 from Theorem 1.3. To do this, we use a result of M. Florence concerning theessential dimension of Z/n. In Section 3, we prove Theorem 1.3. Here the main results used are those of Bogomolovand Serre on the action of the absolute Galois group Gal(k) on the Tate module TA.Remark 1.4. The recent preprint [7] of Karpenko and Merkurjev provides another way to show that the essentialdimension of an Abelian variety over a number field is infinite. To be precise, by generalizing the results of that paperslightly, one can use them to compute the essential dimension of the group scheme A[n] of n-torsion points of anAbelian variety. In fact, using this idea one can show that the essential dimension of an Abelian variety over a p-adicfield is also infinite. However, the present proof of Theorem 1.2 is shorter than a proof using [7] would be and wehope that Theorem 1.3 is independently interesting.2. Theorem 1.3 implies Theorem 1.2As mentioned above, we will use the following result [6, Theorem 4.1] of M. Florence:Theorem 2.1. Let  be an odd prime and r a positive integer. Let L/Q be a field such that |μ∞(L)| = r . Then, forany positive integer k,edL Z/k = max{1, k−r}.Corollary 2.2. Let A be an Abelian variety over a field L of characteristic 0. Let  be an odd prime and suppose thatthe statements in the conclusion of Theorem 1.3 are satisfied; i.e.:(i) Q/Z ⊂ A(L).(ii) 1 < |μ∞(L)| < ∞.Then edA = +∞.Proof. Since L satisfies (ii), edL Z/n → ∞ as n → ∞. By (i), there is an injection (Z/n)L → A. Therefore, by[1, Theorem 6.19], edA  edL Z/n − dimA for all n. Letting n tend to ∞, we see that edA = +∞. P. Brosnan, R. Sreekantan / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 417–420 419Proof of Theorem 1.2 assuming Theorem 1.3. Let A be a non-trivial Abelian variety over a number field k. UsingTheorem 1.3 and Corollary 2.2, we can find a field extension L/k such that edAL = +∞. This implies that edA =+∞ (by [1, Proposition 1.5]). 3. Galois representations and the proof of Theorem 1.3Let A be a non-trivial Abelian variety over k as in Theorem 1.3. Before proving Theorem 1.3, we fix some (stan-dard) notation. We write Gal(k) := Gal(k̄/k) for the absolute Galois group of the number field k. For a rationalprime , we write TA for the Tate-module lim←− A[n] of the Abelian variety A. We write VA for TA ⊗Z Q. For aninteger n, we write Z/n(1) for μn, and for j ∈ Z, we write Z/n(j) for μ⊗jn . We write Z(j) := lim←− Z/lm(j).For any prime p of k where A has good reduction, write Tp for the corresponding Frobenius torus. (For this notionsee [4, Definition 3.1 and p. 326] or [9].) Since A is non-trivial, Tp contains a rank 1 torus D ∼= Gm such that, forevery rational prime  /∈ p, D(Q) ⊂ GL(VA) is the set of homotheties (i.e. scalar matrices) [4, Proposition 3.2].Lemma 3.1. Let p be a prime of k such that the reduction A/p of A at p is good but not supersingular. Then the rankof Tp is strictly greater than 1.Proof. This follows directly from [4, Proposition 3.3]. The following lemma was suggested to us by N. Fakhruddin:Lemma 3.2. Let V be an n-dimensional vector space over a field F , and let T be an F -split torus in GLV of rank atleast 2 containing the homotheties. Then there is a non-zero vector v ∈ V and a rank 1 subtorus S of T such that(i) S fixes v;(ii) the determinant map det :S → Gm is surjective.Proof. The proof is elementary linear algebra with the character lattice, X∗(T ).We can find a basis e1, . . . , en of V and characters λ1, . . . , λn ∈ X∗(T ) such that tei = λi(t)ei for t ∈ T ,i ∈ {1, . . . , n}. Since T contains the homotheties, det is a non-trivial character of T . Moreover, since T ⊂ GLV ,the λi generate X∗(T ). Since dimX∗(T )⊗ Q  2, it follows that there exists i such that λ⊥i ⊂ det⊥. Thus we can finda cocharacter ν such that 〈ν,λi〉 = 0 but 〈ν,det〉 = 0. Set S equal to the image of ν in T and v = ei . Proof of Theorem 1.3. Let A be a non-trivial Abelian variety over a number field k. We can find a prime p in k suchthat A has good reduction at p but A/p is not supersingular. (This is well known if dimA = 1: the case where A hasCM is standard and otherwise it follows from the exercise on page IV-13 of [8]. When dimA > 1 it can be provedby adapting the exercise as Ogus does in Corollary 2.8 of his notes in [5].) Thus the Frobenius torus Tp has rank atleast 2. Using Tchebotarev density, it is easy to see that Tp ⊗ Q is a split torus for all rational primes  in a set ofpositive density. Thus, we can find an odd rational prime  such that  /∈ p and Tp ⊗ Q is split. Now, set F = k(ζ)where ζ is a primitive th root of unity. Note that Tp is the Frobenius torus for AF as Frobenius tori are invariantunder finite extension of the ground field.Now, using Lemma 3.2, we can find a rank 1 subtorus S ⊂ Tp ⊗ Q and a vector v ∈ TAF such that S fixes v anddet :S → Gm is surjective. Let ρ : Gal(F ) → Aut(VAF ) denote the Galois representation on the Tate module and letH = {g ∈ Gal(F ) | ρ(g)v = v}. By a theorem of Bogomolov [4, Theorem B] (and the fact that S fixes v), it followsthatLie(S) ⊂ 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 -adicgroup. Therefore the intersection of S(Q) with ρ(H) contains an open neighborhood of the identity in S(Q). Inparticular, det(H) contains a neighborhood of the identity in Q∗ . Set L := F̄ H . Then, from the fact that v is fixedby H , it follows that Q/Z ⊂ A(L). On the other hand, since ∧2 dimATA ∼= Z(dimA), the fact that det(H) containsa neighborhood of the identity in Q∗ implies that μ∞(L) is finite. This completes the proof of Theorem 1.3. 420 P. Brosnan, R. Sreekantan / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 417–420AcknowledgementsIt is a pleasure to thank G. Pappas, Z. Reichstein and A. Vistoli for valuable conversations and J.-P. Serre forvaluable editorial comments. We are also extremely grateful to N. Fakhruddin. After seeing a early version of thisNote proving Theorems 1.2 and 1.3 for elliptic curves (using Serre’s results on Galois representations for ordinaryelliptic curves and the theory of complex multiplication for elliptic curves with CM), he pointed out that Bogomolov’sresults could be applied to prove Theorems 1.2 and 1.3 for Abelian varieties of any positive dimension.References[1] G. Berhuy, G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003) 279–330 (electronic).[2] P. Brosnan, The essential dimension of a g-dimensional complex abelian variety is 2g, Transform. Groups 12 (3) (2007) 437–441.[3] P. Brosnan, Z. Reichstein, A. Vistoli, Essential dimension and algebraic stacks, arXiv: org:math/0701903.[4] W.C. Chi, l-adic and λ-adic representations associated to abelian varieties defined over number fields, Amer. J. Math. 114 (2) (1992) 315–353.[5] P. Deligne, J.S. Milne, A. Ogus, K.-y. Shih, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin, 1982.[6] M. Florence, On the essential dimension of cyclic p-groups , Invent. Math. 171 (1) (2008) 175–189.[7] N. Karpenko A. Merkurjev, Essential dimension of finite p-groups, Invent. Math., Online first. DOI: 10.1007/s00222-007-0106-6.[8] J.-P. Serre, Abelian l-adic Representations and Elliptic Curves, Research Notes in Mathematics, vol. 7, A K Peters Ltd., Wellesley, MA, 1998.With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original.[9] J.-P. Serre, Letter to K. Ribet dated 1.1.1981, in: Œuvres. Collected Papers, IV, Springer-Verlag, Berlin, 2000, pp. 1985–1998, pp. viii+657.

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Essential dimension of abelian varieties over number fields

Essential dimension of abelian varieties over number fields

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