Curves which do not become semi-stable after any solvable extension

Curves which do not Become Semi-Stable After any Solvable Extension

AMBRUSPAÂL(*)

ABSTRACT- We show that there is a fieldFcomplete with respect to a discrete va- luation whose residue field is perfect and there is a finite Galois extensionKjF such that there is no solvable Galois extensionLjFsuch that the extensionKLjK is unramified, whereKLis the composite ofKandL. As an application we deduce that that there is a fieldFas above and there is a smooth, projective, geome- trically irreducible curve overF which does not acquire semi-stable reduction over any solvable extension ofF.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 11G25, 11G10.

KEYWORDS. Local fields, abelian varieties, semistable reduction.

1. Introduction

We say that a finite Galois field extensionKjFis solvable if its Galois group is solvable. In this paper we will show the following:

THEOREM1.1. There is a field F of characteristic zero complete with respect to a discrete valuation whose residue field is perfect and there is a finite Galois extension KjF such that there is no solvable Galois extension LjF such that the extension KLjK is unramified, where KL is the composite of K and L.

This result gives a counterexample to Lemma 6.4 of [9] on page 632. Its proof is also quite intricate, for example it uses some non-trivial facts about

(*) Indirizzo dell'A.: Department of Mathematics, 180 Queen's Gate, Imperial College, London SW7 2AZ, United Kingdom.

E-mail: a.pal@imperial.ac.uk

Drinfeld modular curves. However it is mainly interesting because it has the following

COROLLARY1.2. There is a field F of characteristic zero complete with respect to a discrete valuation whose residue field is perfect and there is a smooth, projective, geometrically irreducible curve over F which does not acquire semi-stable reduction over any solvable extension of F.

This result is significant because it opens up the possibility to go beyond the results of my paper [9] in its quest to construct curves of a given genus without solvable points. In a joint paper [6] Gyula KaÂrolyi and I determined the set of natural numbers for which the original method of [9] can work which uses curves with semi-stable reduction. This set does not contain the number 23, for example, but it looks unlikely that every genus 23 curve has a solvable point. However it might be possible to construct counter-ex- amples with this genus by using a curve which will not acquire semi-stable reduction over any solvable extension.

The contents of this paper are the following. In the next section we prove Theorem 1.1. In the third section we first give an overview of Ger- ritzen's uniformisation theory of abelian varieties by non-archimedean tori, then we derive Corollary 1.2.

2. Extensions whose ramification does not disappear after any solva- ble extension

NOTATION2.1. For every fieldK letK denote its separable closure.

We say that a fieldF is local if it is complete with respect to a discrete valuation. For the sake of simplicity we will assume that every local field in this paper has perfect residue field. Let F be a local field of char- acteristic zero whose residue fieldkF has characteristicp>0. For every suchF letG_F andI_F denote the absolute Galois group ofF and the in- ertia subgroup ofG_F, respectively. For everyF_p[SL₂(F_p)]-moduleMwe say that a groupGisM-type if it is the extension ofSL₂(F_p) byM and the action ofSL₂(F_p) onM via conjugation inGis the given one. Given two Galois extensions KjFand LjFlet KLjF denote their composite.

LEMMA2.2. Assume that p5and let LjF be a finite Galois extension whose Galois group is M-type for anF_p[SL₂(F_p)]-module M. Also suppose that the image of I_F in Gal(LjF) is an F_p[SL₂(F_p)]-submodule NM

which is isomorphic to F²_p with its usual SL2(Fp)-action. Then for every finite solvable Galois extension PjF the image of IP in Gal(LPjP) Gal(LjF)is also N.

PROOF. Note that Gal(LPjP) is a normal subgroup of Gal(LjF) whose quotient subgroup is isomorphic to Gal(PjF), and hence the image of Gal(LPjP) in the quotientSL2(Fp) of Gal(LjF) byMis also a normal sub- group with a solvable quotient. Since the only such subgroup ofSL2(Fp) is itself we get that Gal(LPjP) surjects onto the quotientSL₂(F_p) of Gal(LjF).

This means that we may reduce to the case when Gal(PjF) is a group whose order is a primelby induction on the order of Gal(PjF). The quotient group IF=IP is isomorphic to Gal(P^unjF^un) whereF^un andP^un are the maximal unramified extensions of F and P, respectively. Since Gal(P^unjF^un) is a subgroup of Gal(PjF) we get that the index ofI_PinI_Fdividesl. Therefore the imageRofI_Pin Gal(LPjP) is a normal subgroup ofNof index dividingl.

MoreoverRis a normal subgroup in Gal(LPjP) so it is invariant under the action ofSL2(Fp). Because the action ofSL2(Fp) onNis irreducible we get

thatRis equal toN. p

NOTATION2.3. For every local fieldFas above letO_Fandk_Fdenote its valuation ring and its residue field, respectively. Let ord_F denote the val- uation ofFnormalised such that ordF(p)ˆ1 for every uniformizerp2 OF. LeteˆordF(p) denote the absolute ramification index ofF. For the sake of simple notation letU_F ˆ O_Fand for everyi2Nlet

U_F⁽ⁱ⁾ˆ fu2U_Fjord_F(1 u)ig:

In the next three lemmas we will assume thatFcontains thep-th roots of unity. Letmˆe=(p 1).

LEMMA2.4. Every extension KjF which we get by adjoining a p-th root of anya2U_F^(mp)is unramified.

PROOF. We may assume without the loss of generality thatais not ap- th root inF. Letb2Kbe ap-th root ofaand fix a uniformizerp2F. Then gˆp ^m(b 1) generates the degreepextensionKjF. The minimal poly- nomial of this element is:

f(x)ˆp ^mp…(p^mx‡1)^p a† ˆp ^mp(1 a)‡X^{p 1}

kˆ1

p

k p^{m(k p)}x^k‡x^p2F[x]:

This is a monic polynomial whose constant termp ^mp(1 a) is inOF by assumption. For every integerkwith 1kp 1 we have:

ord_F( p

k p^{m(k p)})ˆm(p 1)‡m(k p)ˆm(k 1)0;

so f(x)2 O_F[x]. Let LjF be the unique unramified Galois extension such that k_L is the splitting field of f(x) modulo pO_F. Since f⁰(x)pp^{m(1 p)}60 modpO_F the polynomial f(x) splits inLby Hensel's lemma. Therefore K ˆF(g) is contained in L, and hence it is an

unramified extension of F as we claimed. p

LEMMA2.5. We have(F)^p\U^(mp_F ¹⁾U_F^(mp).

PROOF. Fix a uniformizerp2 OFand letx2Fsuch thatx^p2U^(mp_F ¹⁾. Clearly we have x2U⁽¹⁾_F and hence there is a positive integer n and an x₀2 O_Fsuch thatxˆ1‡pⁿx₀. By the binomial theorem:

x^pˆ(1‡pⁿx₀)^pˆ1‡ppⁿx₀‡ ‡ p

k p^nkx^k₀‡ ‡p^npx^p₀: (2:5:1)

Assume first thatn5m. Then for every integerkwith 1kp 1 we have:

ord_F(p^npx^p₀)ˆpn5(p 1)m‡nkˆord_F( p

k p^nkx^k₀);

and hencex^p2U= _F^{(mp 1)}. This is a contradiction, sonm. In this case pmord_F(p^npx^p₀) andpmord_F( p

k p^nkx^k₀);

and hencex^p2U_F^(mp) as we claimed. p

For every groupGand for everyF_p[G]-moduleNletN^_denote its dual Fp[G]-module HomFp(N;Fp). (Recall that for a left Fp[G]-module N we define the leftG-multiplication onN^_by the formula gl(x)ˆl(g ¹x) for everyg2G;l2Hom_Fp(N;F_p) andx2N.)

LEMMA2.6. Let KjF be a finite unramified Galois extension with Galois group SL2(Fp). Let NkK be an irreducible finite Fp[Gal(kKjkF)]ˆ Fp[Gal(KjF)]-module. Then there is a finite Galois extension LjF such that (i) the groupGal(LjF)is M-type for a finiteF_p[SL₂(F_p)]-module M

which contains N^_,

(ii) the field L contains K and MGal(LjK), (iii) the image of I_F in MGal(LjF)is N^_.

PROOF. Fix a uniformizer p2 OF. Then the homomorphism OK! U_K^{(mp 1)} given by the rulex7!1‡xp^{mp 1}mapspOF ontoU^(mp)_F and hence induces an isomorphism:

kK OK=pOKU_K^{(mp 1)}=U_K^(mp):

This isomorphism is Gal(KjF)-equivariant where we consider kK as a Gal(KjF)-module via the natural action of Gal(k_Kjk_F) and we equip U_K^{(mp 1)}=U_K^(mp)with the quotient Gal(KjF)-module structure. By applying Lemma 2.5 to the local field K we get that there is a natural quotient map:

q:U_K^{(mp 1)}=U_K^{(mp 1)}\(K)^p!U^(mp_K ¹⁾=U^(mp)_K :

Choose a finite set NU_K^{(mp 1)}=U_K^{(mp 1)}\(K)^p such that the image of N under qisN. LetM0U_K^{(mp 1)}=U_K^{(mp 1)}\(K)^p be theFp[Gal(KjF)]- module generated by N. Since M₀ is the F_p-span of all the Gal(KjF)- conjugates of N, and since that set is finite, we get that M₀ is a finite Fp[Gal(KjF)]-module. LetMU_K^{(mp 1)}be a set of representatives ofM0

and letLjKbe the finite Galois extension which we get by adjoining the p-th roots ofM. This extension is independent of the choice of M and since M₀ is invariant under the action of Gal(KjF) the extensionLjF is also Galois. By Kummer theory Gal(LjK) is isomorphic toMˆM^_₀ as a Gal(KjF)-module. The image ofM₀with respect toqisNand henceM₀ has a quotientFp[Gal(KjF)]-module isomorphic toN. By dualityMhas a Fp[Gal(KjF)]-submodule isomorphic toN^_. So properties (i) and (ii) hold for LjF.

LetM⁰₀be the kernel of the restriction ofqontoM0and letM⁰U_K^(mp) be a set of representatives of M⁰₀. Let L⁰jK be the the finite Galois ex- tension which we get by adjoining thep-th roots ofM⁰. By Lemma 2.4 the extensionL⁰jK is unramified. Since the extensionKjF is also unramified we get thatL⁰jFis an unramified extension, too. BecauseN^_Gal(LjF) is equal to the Galois group Gal(LjL⁰), the image ofIFin Gal(LjF) is contained inN^_. By Lemma 2.5 there is not any unramified extension ofKwhere any element of U_K^{(mp 1)} U_K^(mp) could be ap-th root. Therefore the extension LjK is ramified and so the image ofI_FinN^_is non-trivial. BecauseN^_is the dual of an irreducible Fp[Gal(KjF)]-module, it is also irreducible and

hence the image ofIF must beN^_itself. p

PROPOSITION2.7. Letf be an infinite perfect field of characteristic p whose Brauer group has trivial 2-torsion and letkjf be a finite Galois

extension. Let N be anFp[Gal(kjf)]-module which has dimension two as a vector space overFp. Then there is aFp[Gal(kjf)]-submodule ofkwhich is isomorphic to N.

PROOF. Note that it will be enough to show that there is an F_p[Gal(fjf)]-submoduleMoffwhich is isomorphic toNas anF_p[Gal(fjf)]- module. In fact in this case the action of Gal(fjf) on the extension of f generated byMfactors through Gal(kjf). ThereforeMwill be a submodule ofkby the fundamental theorem of Galois theory. LetAdenote the poly- nomial ringF_p[t]. We may considerf as an extension of the residue field F_pˆA=(t‡1) of the prime ideal (t‡1)/Aand hence we may talk about DrinfeldA-modules of rank 2 of characteristic (t‡1) with coefficients inf. For such a DrinfeldA-modulefitst-torsion group schemef[t] is eÂtale and hencef[t](f) is aF_p[Gal(fjf)]-submodule off which has dimension two as a vector space overF_p.

Let Y(t) be the Drinfeld modular curve over F_pˆA=(t‡1) which parameterizes DrinfeldA-modules of rank 2 of characteristic (t‡1) with a fullt-level structure. By fixing anFp-basis ofNwe get a continuous Galois representation r:Gal(fjf)!GL2(Fp). Let Yr be the twist of Y(t) with respect torvia the natural action ofGL₂(F_p) onY(t). Thef-valued rational points ofY_r correspond to DrinfeldA-modules of rank 2 of characteristic (t‡1) with coefficients in f such that the Gal(fjf)-module f[t](f) is iso- morphic toN. Since the smooth compatification of the affine curveY(t) is geometrically irreducible of genus zero, the same holds forYr. As the 2- torsion of the Brauer group offis zero we get thatY_rhas a Zariski-dense

set off-valued points. p

PROOF OF THEOREM 1.1. Let p5 and let f be the perfection of a function field of transcendence degree one over an algebraically closed field of characteristicp. Then there is a finite Galois extensionkjf with Galois group SL₂(F_p) by Harbater's theorem (see Corollary 1.5 of [5] on page 284). LetF be the field which we get by adjoining thep-th roots of unity to the fraction field of the ring of Witt vectors off and letKbe the unique unramified extension ofFwith residue fieldk. By Tsen's theorem the Brauer group of f is trivial and hence there is a Gal(kjf)-submodule Nk_K which is isomorphic to F²_p with its usual SL₂(F_p)-action by Pro- position 2.7. Because F²_p is self-dual as an F_p[SL₂(F_p)]-module we may apply Lemma 2.6 to the extensionKjF and the moduleN to get a finite Galois extension LjF which satisfies the conditions in Lemma 2.2. The

claim now follows from this lemma. p

3. Abelian varieties which do not become semi-stable after any solvable extension

DEFINITION3.1. For every algebraic groupTover a fieldFwhich is a split torus, letC(T) denote its group of cocharacters. ThenC(T) is a free and finitely generated abelian group whose rank is equal to the dimension ofT overF. The groupT(F) ofF-valued points ofTis canonically isomorphic to FC(T). Now letFbe a field complete with respect to a discrete valuation v:F!Z. A subgroupLofT(F)ˆFC(T) is called a discrete lattice if the restriction of the homomorphismv1:FC(T)!ZC(T) toLis injective and it has finite cokernel. In this case the quotientT=Lexists in the category of rigid analytic spaces and it is a proper rigid analytic group such that the quotient mapT!T=Lis a homomorphism of rigid analytic groups (see pages 324-325 of [3]). Let End(T;G) denote the ring of endomorphisms of the algebraic groupToverFsuch thatf(G)G. The operation of forming quotients induces an injective homomorphism

h_T;L:End(T;G) !End(T=G)

where the latter is the ring of rigid analytic endomorphisms of the rigid analytic groupT=L.

THEOREM3.2. The homomorphism hT;Lis an isomorphism.

PROOF. This is Satz 5 of [2] on page 33. p

THEOREM3.3. Let F be a local field, let T be a split torus over F and letLT(F)be a discrete lattice. Then the quotient T=Lis isomorphic to the rigid analytic variety underlying an abelian variety over F if and only if there is a homomorphism l from L to the character group Hom(T;G_m)of T such that the bilinear map:

(a;b)7!l(a)(b):LL!F is symmetric andordF((a;a))>0whenever16ˆa2L.

PROOF. This is Theorem 5 of [3] on page 338. p NOTATION3.4. For every field F and for every commutative group schemeBover Flet End(B) denote the ring of endomorphisms ofBas a group scheme overF. Moreover let Aut(B)End(B) denote the group of automorphisms ofBoverF. LetFbe a local field, letTbe a split torus over

F and let LT(F) be a discrete lattice such that the quotientT=Liso- morphic to the rigid analytic variety underlying an abelian varietyBoverF.

By GAGA (see Theorem 2.8 of [7] on page 349) the ring of endomorphisms of Bas a group scheme overFand the ring of rigid analytic endomorphisms of the rigid analytic groupT=Lare the same. Letg_B:End(B)!End(C(T)) denote the ring homomorphism which is the composition of

h_T;L¹ :End(B)ˆEnd(T=L) !End(T;L) and the forgetful map:

End(T;L) !End(T)ˆEnd(C(T)):

Finally letf_Bdenote theQ-linear representation

(gB)j_Aut(B)idQ:Aut(B)!GL(C(T)Q):

LEMMA 3.5. Let F be a local field and let r:G!GL(V) be a re- presentation of the finite group G on a finite dimensional vector space V overQ. Then there is a split torus T over F and a discrete latticeLT(F) such that the following holds:

(i) the quotient T=Lisomorphic to the rigid analytic variety un- derlying an abelian variety B over F,

(ii) there is an injective homomorphisms:G!Aut(B), (iii) the representation(fB)sis isomorphic tor.

PROOF. Note that there is a G-submodule G5V which is a finitely generated, freeZ-module of rank dim(V) and spansV as a vector space overQ. LetTbe a split torus overFsuch thatC(T) is equal toG. Choose a uniformizerp2Fand letL5FC(T) be the abelian group generated by the setfpgjg2C(T)g. Letv:F!Zbe the valuation onFnormalised such thatv(p)ˆ1. Under the homomorphismvid_C(T)the groupLmaps isomorphically onto its image which isZC(T). ThereforeLis a discrete lattice inT(F).

Let h;i:VV !Qbe a symmetric positive definite bilinear form.

We may assume, by multiplying by a positive integer, if it is necessary, that h;i takes integer values on C(T)ˆG. Let u:C(T)!Hom(C(T);Z) be the homomorphism given by the ruleu(a)(b)ˆ hb;aifor everya;b2C(T).

Letl:L!Hom(T;G_m) denote the composition:

L^vid!^C(T)ZC(T)C(T) !^u Hom(C(T);Z)Hom(T;Gm):

The maplsatisfies the conditions of Theorem 3.3, and hence the quotient

T=L is isomorphic to the rigid analytic variety underlying an abelian varietyBoverF. Note that the natural action ofGonFC(T) induced by the G-module structure on C(T)ˆG leaves the subgroup L invariant.

Therefore the natural action ofGonTdescends down to the quotientT=L, that is, there is an injective homomorphisms:G!Aut(B) by GAGA. The objectsT;L;sobviously satisfy condition (iii) above. p For every abelian varietyAdefined over a fieldFand for every prime number l different from the characteristic of F let T_l(A) be the l-adic Tate module of A and let V_l(A) denote the Galois representation T_l(A)_ZlQ_l over F. For every group G, for every representation r:G!GL(V) on a vector spaceVoverQ, and for every prime numberl let r_l:G!GL(VQQ_l) denote theQ_l-linear extension of r.

PROPOSITION3.6. Let F be a local field and letr:Gal(FjF)!GL(V) be a representation on a finite dimensional vector space V overQwhich is continuous with respect to the discrete topology on GL(V). Then there is an abelian variety A defined over F such that for every prime number l different from the characteristic of F the Gal(FjF)-representation V_l(A) has a quotient isomorphic tor_l.

PROOF. LetGbe the image of Gal(FjF) with respect torand by slight abuse of notation letrdenote the representation furnished by the inclusion G!GL(V). LetTbe a split torus overFand letLT(F) be a discrete lattice which satisfies the properties of Lemma 3.5 with respect to the groupGand the representationr. As above letBbe an abelian variety over K which is isomorphic to T=L as a rigid analytic variety and let s:G!Aut(B) be a homomorphism which satisfies conditions (ii) and (iii) of Lemma 3.5.

Now letlbe a prime number different from the characteristic ofF. The quotient map i:T!B induces an injective homomorphism i_l:T_l(T)! Tl(B) where Tl(T) is the Tate module of the torusT. The cokernel ofilis isomorphic to C(T)ZZl equipped with the trivial Galois-action. There- fore there is a short exact sequence:

0 !C(T)_ZQ_l(1) !V_l(B) !C(T)_ZQ_l !0 (3:6:1)

of Gal(FjF)-representations, where for everyQ_l[Gal(FjF)]-moduleW we letW(1) denote the Tate twist ofW, and we let Gal(FjF) act onC(T) triv- ially. The composition of r:Gal(FjF)!Gandsfurnishes a cohomology class in the pointed Galois cohomology set H¹(F;Aut(B_F)). LetAdenote

the twist ofBwith respect to this class. Because the action ofGonBlifts to T, its action onVl(B) respects the filtration of (3.6.1). Therefore there is a short exact sequence:

0 !V_QQ_l(1) !V_l(A) !V_QQ_l !0 (3:6:2)

where Gal(FjF) acts onV viar. The claim is now clear. p For every varietyVdefined over a fieldFand for every extensionKof FletV_Kdenote the base change ofVto Spec(K). Recall that we say that an abelian varietyAdefined over a local fieldF has semi-stable reduction if the fibre over Spec(k_F) of its NeÂron model over Spec(O_F) is a semi-abelian variety. We will need the following fundamental result.

THEOREM3.7. Let F be a local field and let A be an abelian variety defined over F. Let l be a prime number different from the characteristic of k_F. Then the following conditions are equivalent:

(i) the action of I_Fon V_l(A)is unipotent,

(ii) the abelian variety A has semi-stable reduction over F.

PROOF. This follows from Proposition 3.5 of [4] on page 350 and Cor-

ollaire 3.8 of [4] on page 353. p

PROPOSITION3.8. There is a local field F of characteristic zero with a perfect residue field and there is an abelian variety over F which does not acquire semi-stable reduction over any solvable extension of F.

PROOF. By Theorem 1.1 there is a local fieldF of characteristic zero with a perfect residue field and there is a finite Galois extensionKjFsuch that there is no solvable extension LjF such that the extensionKLjK is unramified, whereKLis the composite ofKandL. Choose a faithful finite- dimensionalQ-linear representationr:Gal(KjF)!GL(V) and by slight abuse of notation let r also denote the composition of the quotient map Gal(FjF)!Gal(KjF) and r. Choose a prime numberldifferent from the characteristic ofk_F. By Proposition 3.6 there is an abelian varietyAoverF such that Vl(A) has a quotient W isomorphic to r_l as a Gal(FjF)-re- presentation.

We claim thatAdoes not acquire semi-stable reduction over any solv- able extension of F. Assume the contrary and let LjF a finite solvable Galois extension such thatA_Lhas semi-stable reduction. By Theorem 3.7 the action ofILonVl(A) is unipotent. Therefore the the action ofILonWis

unipotent, too. In particular the image ofILunderr_lis a pro-lgroup, and hence the extension Gal(KLjL) is tamely ramified. Therefore there is a finite solvable Galois extensionL⁰jLsuch that Gal(KL⁰jL⁰) is unramified by Lemma 4.6 of [10]. The extensionL⁰jFis solvable as it is a tower of solvable

extensions. This is a contradiction. p

PROOF OFCOROLLARY1.2. By the proposition above there is a fieldFof characteristic zero complete with respect to a discrete valuation whose residue field is perfect and there is an abelian varietyAoverFwhich does not acquire semi-stable reduction over any solvable extension ofF. For any geometrically irreducible, smooth, projective curveDwe let Jac(D) denote the Jacobian ofD. By Faltings's trick (see Theorem 10.1 of [8] on page 198) there is a smooth, projective, geometrically irreducible curve C over F such that there is a surjective homomorphism f:Jac(C)!A of abelian varieties over F. We claim thatCdoes not acquire semi-stable reduction over any solvable extension ofF. Assume the contrary and letKjFa finite solvable Galois extension such that CK has semi-stable reduction. Then Jac(C)_KˆJac(CK) also has semi-stable reduction by Theorem 2.4 of [1] on page 89. Therefore the action ofI_KonV_l(Jac(C)_K) is unipotent by Theorem 3.7 where l is a prime number different from the characteristic of k_F. Because the homomorphism f_l:V_l(Jac(C)_K)!V_l(A_K) induced by f is surjective we get that the action of IK on Vl(AK) is unipotent, too.

Therefore AK has semi-stable reduction by Theorem 3.7. This is a con-

tradiction. p

Acknowledgement. The author was partially supported by the EPSRC grant P19164.

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Manoscritto pervenuto in redazione il 5 Marzo 2012.