Endomorphism fields of abelian varieties
aa r X i v : . [ m a t h . N T ] J un ENDOMORPHISM FIELDS OF ABELIAN VARIETIES
ROBERT GURALNICK AND KIRAN S. KEDLAYA
Abstract.
We give a sharp divisibility bound, in terms of g , for the degree of the fieldextension required to realize the endomorphisms of an abelian variety of dimension g overan arbitrary number field; this refines a result of Silverberg. This follows from a strongerresult giving the same bound for the order of the component group of the Sato-Tate groupof the abelian variety, which had been proved for abelian surfaces by Fit´e–Kedlaya–Rotger–Sutherland. The proof uses Minkowski’s reduction method, but with some care required inthe extremal cases when p equals 2 or a Fermat prime. Introduction
For A an abelian variety over a field K , the endomorphism field of A is the minimalalgebraic extension L of K such that End p A L q “ End p A L q . The purpose of this paper is toestablish a bound on the degree r L : K s in terms of the dimension of A ; more precisely, wecompute the LCM of all possible degrees as A, K vary while dim K A remains fixed.Before stating our result, we state a prior result of Silverberg [8] which already containsmany of the main ideas. For g a positive integer and p a prime, define r p g, p q : “ ÿ i “ Z g p p ´ q p i ^ . Theorem 1.1 (Silverberg) . For A an abelian variety of dimension g over a number field K ,the endomorphism field of A is a finite Galois extension of K of degree dividing ˆ ś p p r p g,p q . The proof [8, Theorem 4.1] is elegantly simple: one verifies that for each prime ℓ ą
2, theGalois group of the endomorphism field extension is isomorphic (via its action on ℓ -torsionpoints) to a subquotient of the group Sp p g, F ℓ q . The bound is then obtained by taking thegreatest common divisor of the orders of these finite groups. This echoes the method usedby Minkowski to bound the order of a finite group of integer matrices (as exposed in [3]); wewill return to this analogy a bit later.From this proof, it is not clear whether one should expect the bound of Theorem 1.1 to besharp. However, a moment’s thought shows that it is not sharp for g “
1: the bound is 2 ˆ K . More seriously, for g “
2, the bound of Theorem 1.1 is 2 ˆ ˆ ˆ
3. This raises the question of identifying the discrepancybetween Theorem 1.1 and the optimal bound, and this is achieved by our main result.
Date : May 30, 2017.Thanks to Francesc Fit´e, Ga¨el R´emond, Jean-Pierre Serre, Alice Silverberg, and Andrew Sutherland forfeedback. Guralnick was partially supported by NSF grants DMS-1302886 and DMS-1600056. Kedlaya wassupported by NSF grant DMS-1501214 and UC San Diego (Warschawski Professorship). heorem 1.2. For A an abelian variety of dimension g ą over a number field K , thedegree over K of the endomorphism field of A divides ź p p r p g,p q , r p g, p q : “ $’&’% r p g, p q ´ g ´ if p “ ; max t , r p g, p q ´ u if p is a Fermat prime; r p g, p q otherwise.Moreover, for fixed g, p (but varying over all K ), this value of r p g, p q is best possible. To give one more example, for g “
3, the bound from Theorem 1.1 is 2 ˆ ˆ ˆ ˆ ˆ
7. It is easy to see that the factor of 7 is necessary,e.g., by considering twists of the Klein quartic (see [5, § L of A and the Sato-Tate group of A ; the latter is a compact Lie group whose component groupsurjects canonically onto Gal p L { K q , and the bound we ultimately prove is for the order of thecomponent group (see Theorem 5.4). The Sato-Tate group is constructed as a compact formof a certain linear algebraic group over Q , the algebraic Sato-Tate group , which allows us tobound the order of the component group using a variant of Minkowski’s method. One keypoint is that the extremal cases occur for CM abelian varieties, for which the connected partof the algebraic Sato-Tate group is a torus which splits over a CM field; what distinguishes aFermat prime p in this context is that Q p µ p q contains no proper subfield which is CM. (For p “
2, the same statement about Q p µ q plays an analogous role.)To prove that Theorem 1.2 is sharp, we use the relationship between twisting of abelianvarieties and Sato-Tate groups; this reduces the problem to exhibiting abelian varieties ad-mitting actions by large finite groups, which we achieve using CM abelian varieties and thesame wreath product construction as in Minkowski’s theorem. Note that the fields of defini-tion of the resulting abelian varieties are controlled by class groups of abelian number fields,so we are unable to establish lower bounds over any fixed number field.To conclude this introduction, we comment on the subtler problem of giving bounds bysize, rather than divisibility. As described in [3, § n ą
10, the largestfinite subgroups of GL p n, Q q have order 2 n n ! (and are unique up to conjugacy). For abelianvarieties of sufficiently large dimension g , one would expect that the largest possible endomor-phism field extension, and the largest possible component group of the Sato-Tate group, areobtained by twisting a power of an elliptic curve with j -invariant 0 using an automorphismgroup of order 6 g g ! (note that these examples already occur over Q ). For the endomorphismfield, this expectation has been confirmed by work of R´emond [6]; it is highly likely thata similar analysis applies to the component group (because the cases where the two differtend not to have enough CM to trouble the bounds), but this would require some additionalargument. 2. Group schemes
We start with some notation. Our notation choices are not entirely typical; they are madeto help us distinguish between groups and group schemes.
Definition 2.1.
For G a group scheme (over some base), we write G X for the base extensionof G to the base scheme X , and G p X q for the group of X -valued points of G . We say that is pointful over X if G p X q occupies a Zariski-dense subset of G . By convention, all groupschemes we consider will be smooth and linear; the standard linear groups will be consideredas schemes over Z , and we will write GL p n, X q instead of GL p n qp X q and so on.For G a group scheme over a connected base, let G ˝ denote the identity connected com-ponent, and write π p G q : “ G { G ˝ for the group of connected components, viewed as a finitegroup scheme over the same base. If G is pointful, then so are both G ˝ and G { G ˝ .For L { K a finite extension of fields and G a group scheme over Spec L , write Res LK G forthe Weil restriction of scalars of G to Spec K . Example 2.2.
For n a positive integer, the n -torsion subscheme of the multiplicative groupover Q is the group scheme Spec Q r x s{p x n ´ q which is obviously defined over Q . However,it is only pointful for n “ , Definition 2.3.
For G a group scheme, let Out p G q be the group scheme of outer automor-phisms of G , i.e., the cokernel of the map G Ñ Aut p G q induced by conjugation. Note that for G a group scheme over a field k which is not algebraically closed, an element of Aut p G qp k q may map trivially to Out p G qp k q even though it does not come from the image of G p k q ; thatis, the scheme-theoretic notion of an outer automorphism disagrees with the group-theoreticnotion because the latter is not stable under base extension.3. Minkowski’s method
We next formulate our version of Minkowski’s reduction method. We implicitly follow [3],but see also [7] for another detailed treatment (both considering only finite groups).Throughout §
3, let G be a group scheme over a number field K . Definition 3.1.
By convention G is a scheme of finite type over K , so it can be extendedto a group scheme over o K r { N s for some positive integer N . In particular, it makes senseto form the base extension of G to F q : “ o K { q for all but finitely many prime ideals q of o K .Let H be a finite pointful subquotient group scheme of G . By the previous paragraph, H p K q is isomorphic to a subquotient of G p F q q for all but finitely many q ; in particular, foreach prime p , any p -Sylow subgroup of H p K q is isomorphic to a subquotient of a p -Sylowsubgroup of G p F q q for all but finitely many q .To translate this into a numerical bound, define the nonnegative integers r p G, p q by theformula ź p p r p G,p q “ sup S t gcd q P S G p F q qu where S runs over all cofinite sets of prime ideals of o K . Then the preceding discussion impliesthat H has order dividing ś p p r p G,p q ; in particular, this bound applies to the component groupof any pointful subgroup scheme of G .We collect some remarks related to this construction. Remark 3.2.
Suppose that there exists a finite pointful subquotient group scheme H of G such that the p -part of H equals the upper bound p r p G,p q . Let P be a p -Sylow subgroup of H ; then for infinitely many q , P has the same cardinality as a p -Sylow subgroup of G p F q q ,so by Sylow’s theorem the two must be isomorphic. That is, the isomorphism type of P isuniquely determined by G . emark 3.3. Let H be a pointful subgroup scheme of G . Then π p H q divides ś p p r p G,p q´ δ p G,p q for ź p p δ p G,p q “ inf S t gcd q P S H ˝ p F q qu . Remark 3.4.
In light of Wedderburn’s theorem, for any number field K and any twistedform G of GL p n q K one has r p G, p q “ r p GL p n q K , p q . Remark 3.5.
For each prime p , the group C p ≀ S t n {p p ´ q u embeds into GL p n, Q q , as then doits p -Sylow subgroups. The original theorem of Minkowski asserts that the conjugates of thelatter are the largest possible p -subgroups of GL p n, Q q . However, if we compare this to thevalues r p GL p n q Q , p q “ Z np ´ ^ ` Z np p p ´ q ^ ` Z np p p ´ q ^ ` ¨ ¨ ¨ p p ą q r p GL p n q Q , q “ n ` Y n ] ` Y n ] ` ¨ ¨ ¨ , we see that the Minkowski bound is only sharp for p ą
2; for p “
2, the Minkowski boundis too large by X n \ , and one must supplement using some extra analysis involving quadraticforms over finite fields [3, § C p ≀ S t n {p p ´ q u is optimal also for subquotients when p “ Remark 3.6.
For K a number field, the Chebotarev density theorem implies that r p GL p n q K , p q depends only on K X Q p µ p q . For m p K, p q : “ min t m ě K X Q p µ p m q “ K X Q p µ p qu t p K, p q : “ r Q p µ p m p K,p q q : K X Q p µ p m p K,p q qs , for p ą r p GL p n q K , p q “ m p K, p q Z nt p K, p q ^ ` Z npt p K, p q ^ ` Z np t p K, p q ^ ` ¨ ¨ ¨ and this bound is again optimal (see [3, § p “
2, the situation depends crucially on whether Q p ζ q Ď K . If so, then t p K, q “ r p GL p n q K , q “ m p K, q n ` Y n ] ` Y n ] ` ¨ ¨ ¨ , and this bound is optimal, achieved by C m p K, q ≀ S n . If not, the situation is more complicated;we limit ourselves to observing that for K “ Q p?´ q we have r p GL p n q K , q “ r p GL p n q Q , q ,while for K “ Q p? q we have r p GL p n q K , q “ r p GL p n q Q , q ` X n \ . Remark 3.7.
Although we do not need this for our main result, we note in passing thefollowing corollary of Remark 3.6 (which only affects the prime p “ p g, Q q is 2 r p GL p g q Q p i q , q ś p ą p r p GL p g q ,p q . Thiscomes down to the fact that any irreducible finite subgroup of Sp p g, Q q is centralized bysome totally imaginary number field [4, Lemma 2.3]. Remark 3.8.
It is natural to use Minkowski’s method as a starting point for boundingthe order of finite subgroups of any reductive group over any field. This has been discussedextensively by Serre [7]. . Comparison of Minkowski bounds
For our purposes, it will be important to compare the Minkowski bounds for variousgroup/subgroup pairs. The key points will be to identify discrepancies for p “
2, and toisolate cases for p ą Remark 4.1.
A trivial but useful observation along these lines is that for n ě p ą d ą r p GL p dn q Q , p q ą r p GL p n q Q , p q whenever r p GL p dn q Q , p q ą . A slightly less trivial observation is that for n ě r p GL p n q Q p i q , q ą r p GL p n q Q , q . Lemma 4.2.
Let K be a number field of degree d over Q . For each integer n ě and eachodd prime p , r p GL p dn q Q , p q ě r p GL p n q K , p q ` r p S d , p q ; moreover, if n ą and m p K, p q ą , or if d ě p p p ´ q , then the inequality is strict.Proof. There is nothing to check when d “
1, so we may assume d ě
2. Put m “ m p K, p q , t “ t p K, p q ; then the desired inequality is Z dnp ´ ^ ` Z dnp p p ´ q ^ ` ¨ ¨ ¨ ě m Y nt ] ` Z npt ^ ` ¨ ¨ ¨ ` Z dp ^ ` Z dp ^ ` ¨ ¨ ¨ . From the equality dt “ p m ´ p p ´ q , we see that dp ´ equals t times the integer p m ´ which isno less than m (and strictly greater than m if m ą Z dnp ´ ^ ´ m Y nt ] ` Z dnp p p ´ q ^ ´ Z npt ^ ` ¨ ¨ ¨ ` Z dp ^ ` Z dp ^ ` ¨ ¨ ¨ , we see that this difference does not decrease if we increase n by 1 (and strictly increases if m ą n “
1, then the desired inequality becomes r p GL p d q Q , p q ě r p S d , p q , which holdsbecause S d embeds into GL p d, Q q ; this equality is strict whenever d ě p p p ´ q . This provesthe claim. (cid:3) Corollary 4.3.
For K a number field of degree d , for p ą we have (4.1) r p GL p dn q Q , p q ě r p Aut p K { Q q ˙ Res K Q GL p n q K , p q . Moreover, if K Ę Q p µ p q , K is not the degree- p subextension of Q p µ p q , and r p GL p dn q Q , p q ‰ , then the equality is strict.Proof. The inequality (4.1) holds because Aut p K { Q q ˙ Res K Q GL p n q K embeds into GL p dn q Q .We thus only need to obtain a contradiction assuming that K Ę Q p µ p q , K is not the degree- p subextension of Q p µ p q , r p GL p dn q Q , p q ą
0, and equality holds in (4.1).Note that (4.1) also follows from Lemma 4.2, so equality must also hold in the latter. ByRemark 3.6, if K : “ K X Q p µ p q has degree d ‰ d , then r p GL p n q K , p q “ r p GL p n q K , p q ; wethen get the strict inequality by applying Lemma 4.2 to the field K and invoking Remark 4.1(using the condition that r p GL p dn q Q , p q ą K “ K and hence K Ď Q p µ p q ; since we assumed K Ę Q p µ p q , this implies that m p K, p q ą
1. To have equalityin Lemma 4.2, we must then have n “ d ă p p p ´ q . ince m p K, p q ą K must contain the degree- p subextension of Q p µ p q , necessarilystrictly by hypothesis; hence d { p is an integer strictly greater than 1. By the bound on d , wecannot then have Q p µ p q Ď K , so r p GL p q K , p q “
0. Meanwhile, K is an abelian extensionof Q , so r p Aut p K { Q q , p q “
1. However, since d ě p , r p GL p d q Q , p q ě
2, yielding the desiredcontradiction. (cid:3)
Corollary 4.4.
For any integers n, d ą , for each odd prime p for which r p GL p nd q Q , p q ą ,we have r p GL p dn q Q , p q ą r p GL p n q Q , p q ` r p S d , p q . Proof.
We first reduce to the case where d is even. Namely, we may do this by replacing d with 2 X d \ except if d is odd, r p GL p dn q Q , p q ą
0, and r p GL pp d ´ q n q Q , p q “
0. This implies p d ´ q n ă p ´ ď dn , which implies on one hand that n ă p ´ r p GL p n q Q , p q “
0, andon the other hand that d ´ ă p ´ r p S d , p q “
0; this yields the claimed inequality.For any number field K of even degree d , by Lemma 4.2 we have r p GL p dn q Q , p q ě r p GL p n q K , p q ` r p S d , p q ě r p GL p n q Q , p q ` r p S d , p q , so it suffices to confirm that both equalities cannot hold simultaneously. Since we are freeto choose K , we take it to contain the quadratic subfield of Q p µ p q ; then by Remark 3.6, wehave r p GL p n q K , p q ą r p GL p n q Q , p q unless both quantities equal zero. It thus suffices to ruleout the equality r p GL p dn q Q , p q “ r p S d , p q ; since r p GL p d q Q , p q ě r p S d , p q , this follows fromRemark 4.1. (cid:3) For p “
2, we have the following analogue of Corollary 4.3.
Remark 4.5.
For K { F an extension of number fields of degree d , we obviously have r p GL p dn q F , q ě r p Aut p K { F q ˙ Res KF GL p n q K , q . In case F “ Q p i q , one can show using Remark 3.2 that equality holds only when d “ p “
2, we have the following analogue of Corollary 4.4.
Lemma 4.6.
For any integers n, d ą such that g “ dn { is an integer, r p GL p g q Q p i q , q ě r p GL p n q Q , q ` r p S d , q with equality only for p n, d q “ p , q .Proof. We are claiming that dn ` Z dn ^ ` Z dn ^ ` ¨ ¨ ¨ ě n ` Y n ] ` Y n ] ` ¨ ¨ ¨ ` Z d ^ ` Z d ^ ` ¨ ¨ ¨ with equality only for p n, d q “ p , q . For d “
2, this inequality becomes n ě X n \ `
1; for n “
2, it becomes 2 d ě
4. It thus remains to check the cases where n, d ě p d ´ q n ` ˆZ dn ^ ´ Z d ^˙ ` ˆZ dn ^ ´ Z d ^˙ ` ¨ ¨ ¨ ´ Y n ] ´ Y n ] ´ ¨ ¨ ¨ ;in particular, for n ě d by 2. Using the previousparagraph, we deduce the claim when d is even. For d odd, we may argue that r p GL p g q Q p i q , q ě r p GL p g ´ n { q Q p i q , q ě r p GL p n q Q , q ` r p S d ´ , q “ r p GL p n q Q , q ` r p S d , q ith strict inequality if n ą (cid:3) Abelian varieties and Sato-Tate groups
We now specialize to the cases of interest for abelian varieties.
Definition 5.1.
For A an abelian variety over a number field K , let AST p A q denote the algebraic Sato-Tate group of A in the sense of [1, Definition 9.5]. The key properties that weneed are the following. ‚ The group scheme AST p A q is a pointful subgroup scheme of Sp p g q Q whose connectedpart is reductive. The connected part is closely related to the Mumford-Tate group of A . ‚ There exists a torus T Ă AST p A q ˝ C which acts on C g with weights 1 , ´ g . In particular, the fixed space of AST p A q is the zero subspace. ‚ The component group π p A q surjects onto Gal p L { K q for L the endomorphism field of A ; this map is a bijection whenever the Mumford-Tate group is completely explainedby endomorphisms (which holds in all cases when g ď g ě
4, asoriginally shown by Mumford). Moreover, for K a finite extension of K , AST p A K q is the inverse image of Gal p LK { K q Ď Gal p L { K q in AST p A q . ‚ Any decomposition of Q g into indecomposable AST p A q -modules corresponds to aproduct-up-to-isogeny decomposition of A . ‚ The group AST p A q is a torus if and only if A is isogenous to a product of abelianvarieties with CM defined over K . Example 5.2.
Put M : “ Q p i q , M : “ Q p?´ q , M : “ Q p?´ q , M : “ Q p?´ q and let K be the compositum of these four fields. Let A be the product of four elliptic curves E , . . . , E with CM by M , . . . , M , respectively. Then AST p A q is a torus of dimension 3. Remark 5.3.
The
Sato-Tate group of A , as studied for abelian surfaces in [2], is a maximalcompact subgroup of AST p A, C q ; it therefore has the same component group as AST p A q .On one hand, this means that the argument using algebraic Sato-Tate groups in the proof ofTheorem 5.4 directly applies also to the component groups of Sato-Tate groups; on the otherhand, the conclusion of Theorem 5.4 in the case g “ Theorem 5.4.
For A an abelian variety of dimension g over a number field K , the compo-nent group of the algebraic Sato-Tate group of A (or equivalently, the Sato-Tate group of A )has order dividing ś p p r p g,p q .Proof. Put G “ AST p A q . It suffices to check the claimed divisibility for the p -part of π p G q ;this is immediate from Remark 3.5 (applied with n “ g ) unless p “ p is a Fermat prime.In light of Remark 5.3, we may assume further that g ě p , r p g, p q is superadditive in g ; we may thus reduce to the casewhere A is indecomposable, which as noted above implies that G acts indecomposably on V “ Q g . Using Corollary 4.4 and Lemma 4.6, we may also deduce the claim in case G ˝ does ot act isotypically on V (the exceptional case of Lemma 4.6 cannot occur for g ě G ˝ acts isotypically on V .Let W be an irreducible G ˝ -representation occurring in V , and put D : “ End G ˝ p W q , M : “ End G ˝ p V q , F : “ Z p D q “ Z p M q . By Schur’s lemma, D is a division algebra, M is a matrix ring over D , and T : “ image p G ˝ Ñ M ˆ q is a torus which splits over F . If we define H : “ ker p G Ñ Out p G ˝ qq , we obtain aninduced injective morphism H { G ˝ ã Ñ M ˆ { T . Meanwhile, since V is G ˝ -isotypical, G { H actsfaithfully on the set of isomorphism classes of G ˝ -constituents of W b Q Q ; this implies thatthe map G Ñ Aut p F { Q q induces an injective morphism G { H ã Ñ Aut p F { Q q .Define the following positive integers: a : “ r F : Q s ; b : “ rank F D “ the G ˝ -multiplicity of W b F F ; c : “ rank D M “ the G ˝ -multiplicity of W in V ; d : “ gabc “ the Q -dimension of a G ˝ -constituent of V b Q Q .In this notation, G { H injects into S a while H { G ˝ injects into a subgroup of a twisted formof GL p bc q F . In light of Remark 3.4, it follows that the p -adic valuation of π p G q is at most(5.1) r p Aut p F { Q q , p q ` r p GL p bc q F , p q ď r p GL p abc q Q , p q . If d ą
1, then Remark 4.1 immediately yields the desired bound (even for p “ d “
1, which implies that G ˝ is abelian and hence a torus. In thiscase, F must be totally imaginary; it is in fact the CM field associated to the unique simpleisogeny factor of A .If p ą F Ď Q p µ p q in light of Corollary 4.3 (thedegree- p subextension of Q p µ p q cannot be a compositum of CM fields because its degreeis odd). Since F is totally imaginary, this would force F “ Q p µ p q . However, under theseconditions, we may invoke Remark 3.3: the reduction of G ˝ itself always has order divisibleby p , yielding exactly the correct bound.If p “ F X Q p µ q “ Q , then r p GL p n q F , q “ r p GL p n q Q , q (see Remark 3.6) andso we may invoke Lemma 4.6 to deduce the desired result (again assuming that g ě Q p µ q Ď F , then Remark 4.5 gives a valuation bound which is only off by 2, andagain this discrepancy is accounted for by Remark 3.3 (the reduction of G ˝ itself always hasorder divisible by 4). Otherwise, F : “ F X Q p µ q must equal one of Q p? q or Q p?´ q . Incase F “ Q p?´ q , Remark 3.6 implies that r p GL p n q F , q “ r p GL p n q Q , q ; we thus obtainan upper bound of r p Aut p F { Q q , q ` r p GL p abc { q F , q “ ` r p GL p g q Q , q and again Lemma 4.6 settles the question (for g ě F “ Q p? q , we must argue a bit more carefully. Since F is Galois and totallyreal, we must have F ‰ F and r p Aut p F { Q q , q “ ` r p Aut p F { F q , q . By Remark 3.6, wehave r p GL p n q F , q “ r p GL p n q F , q “ r p GL p n q Q , q ` Y n ] . ote finally that the reduction of G ˝ always has order divisible by 2. From (5.1), we nowobtain an upper bound of r p Aut p F { F q , q ` r p GL p bc q Q , q ` Z bc ^ which by Lemma 4.6 (and the fact that a ě
4) is itself bounded above by r p GL p g q Q , q ` Y ga ] ď r p GL p g q Q , q ` Y g ] . This gives the desired bound once more. (cid:3) Lower bounds
To conclude, we establish the lower bound assertion of Theorem 1.2 by twisting powers ofCM abelian varieties.
Definition 6.1.
We briefly recall [2, Definition 2.20]. Let A be an abelian variety over anumber field K , let L { K be a finite Galois extension, and let f : Gal p L { K q Ñ Aut p A L q bea 1-cocycle. Then there exists an abelian variety A f over K equipped with an isomorphism A fL – A L such that the action of τ P G K on A f p K q – A fL p K q corresponds to the action of f p τ q τ on A p K q – A L p K q . The isomorphism A fL – A L induces an isomorphism End p A fL q – End p A L q in which corresponding elements α P End p A fL q , β P End p A L q satisfy the relation(6.1) τ p α q “ f p τ q τ p β q f p τ q ´ . We use the twisting setup in the following setting.
Definition 6.2.
Fix a prime p and a positive integer m . Let A be an abelian variety ofdimension g over some number field K , such that A has complex multiplication by thering of integers o M of a subfield M of Q p µ p m q ; put d “ r Q p µ p m q : M s . (Note that we cannothope to fix the field K , because its degree over Q is related to the class number of M .) Let G Ď GL p d, M q be a subgroup of order p m stable under Gal p M { Q q and identify G with asubgroup of Aut p A d ,KM q . Put A “ A dn for some positive integer n ; then G “ G ≀ S n maybe identified with a subgroup of Aut p A ,KM q stable under Gal p KM { K q . Let G be the imageof G under the map GL p dn, M q Ñ PGL p dn, M q .Choose an S n -extension L { K linearly disjoint from KM , so that L M { KM is again an S n -extension. Note that for a “generic” C p m -extension L { L M , the Galois closure L of L over M will have Galois group G ≀ S n . Using class field theory, we may further ensure that L is also Galois over K and that there exists a 1-cocycle f : Gal p L { K q Ñ Aut p A ,L q whose restriction to Gal p L { KM q is the preceding identification of the latter with G ≀ S n Ď Aut p A ,KM q .Put A “ A f and let L be the endomorphism field of A . Then KM Ď L and (6.1) impliesthe existence of a surjective morphism from Gal p L { KM q to G , but not in general to G . Theorem 6.3.
For each prime p , there exists an abelian variety A of dimension g over somenumber field K such that the p -part of r L : K s is at least p r g,p .Proof. Suppose first that p ´ p is odd and not a Fermat prime).Then there exists a subfield M of Q p µ p q whose index ℓ is an odd prime divisor of p ´ pplying Definition 6.2 with m “ n “ Y gp ´ ] then yields the desired result; note that inthis case, there is no harm to take K to contain M , which simplifies the analysis somewhat.Suppose next that p is a Fermat prime; we may assume that r g,p ě
1. The previousconstruction breaks down because p ´ m “ M “ Q p µ p q , n “ Y gp ´ ] . Note that we now lose one factor of p tothe quotient map G Ñ G , so again we get the desired result.Suppose finally that p “
2. Apply Definition 6.2 with m “ M “ Q p i q , n “ g ; note thatin this case we may even take K “ Q . This time, we lose two factors of 2 to the quotient map G Ñ G , but gain one back from the extension M { Q . This proves the claim once more. (cid:3) References [1] G. Banaszak and K.S. Kedlaya, An algebraic Sato-Tate group and Sato-Tate conjecture,
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