ON THE FONTAINE-MAZUR CONJECTURE FOR NUMBER FIELDS AND AN ANALOGUE FOR FUNCTION FIELDS JOSHUA BRANDON HOLDEN

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1 ON THE FONTAINE-MAZUR CONJECTURE FOR NUMBER FIELDS AND AN ANALOGUE FOR FUNCTION FIELDS JOSHUA BRANDON HOLDEN Abstract. The Ftaie-Mazur Cjecture fr umber elds predicts that iite `-adic aalytic rups cat ccur as the Galis rups f uramied `-extesis f umber elds. We ivestiate the aalus questi fr fucti elds f e variable ver ite elds, ad the prve sme special cases f bth the umber eld ad fucti eld questis usi ideas frm class eld thery, `-adic aalytic rups, Lie alebras, arithmetic alebraic emetry, ad Iwasawa thery. 1. Itrducti A cjecture f Ftaie ad Mazur, i [11], says: Cjecture 1 (Ftaie-Mazur). Let k be a umber eld ad ` ay prime. uramied `-adic aalytic `-extesi f k, the M is a ite extesi f k. If M is a (See belw fr deitis.) It seems reasable ad straihtfrward t pse the same cjecture i the fucti eld case. We immediately see that the aive versi must be false, because the extesi f a fucti eld k ver a ite eld F ive by cstat eld `-extesis f F is already iite ad `-adic aalytic. The ext lical questi t ask wuld be smethi like the fllwi: Questi 1. Let k be a fucti eld ver a ite eld F f characteristic p ad rder q, ad ` a prime t equal t p. Let K = kf`1 be btaied frm k by taki the maximal `-extesi f the cstat eld. If M is a uramied `-adic aalytic `-extesi f k, ad M des t ctai K, must M be a ite extesi f k? The aswer is ; i fact Ihara has cstructed examples (fr example, see [18]) f fucti elds k which have, fr all but itely may ` t equal t p, a Galis extesi M with extesi f the cstat eld, ad with Gal(M=k) ismrphic t PSL 2 (Z`). Frey, Kai, ad Vlklei, usi ather cstructi based the e described i [12], have als achieved similar results. Nevertheless, it is pssible t prve that the aswer is yes i sme special cases. These cases are f iterest t ly idepedetly, but als because the methds used t address them, which ivlve basic facts abut abelia varieties ver ite elds, ca be adapted t umber elds thruh the use f Iwasawa Z`-extesis ad `-adic L-fuctis, as we will shw. (It is likely that they ca als be used t address sme cases f a cjecture recetly made by de J, i [8], abut uramied extesis f fucti elds. A future paper will address this issue.) These methds, which stem frm wrk f Bst ([3, 4]) usi rup Date: 21 Auust Mathematics Subject Classicati. Primary 11R37; Secdary 11R58, 11R42, 20E18. Accepted fr publicati i the Jural f Number Thery, as f Auust 25,

2 2 JOSHUA BRANDON HOLDEN autmrphisms, als itrduce `-adic Lie alebras i a crucial way, relati them t stadard cstructis f arithmetic alebraic emetry, such as the Tate mdule. Cjecture 1 cmes frm tw surces; e is a brad cjecture f Ftaie ad Mazur abut irreducible `-adic represetatis f abslute Galis rups, psiti ecessary ad suciet lcal cditis fr the represetati t \cme frm alebraic emetry" i a certai specic sese. Whe cmbied with the Tate Cjecture abut the subspace f etale chmly eerated by alebraic cycles, this brader cjecture yields the mre arrw Cjecture 1. The ther surce is the Gld-Shafarevich prf that iite class eld twers exist; these ive iite uramied `-extesis ad the abve cjecture says that such extesis cat be aalytic. (See [14] fr mre the lik with Gld-Shafarevich.) Fr ather frmulati f Cjecture 1, csider the fllwi deitis frm [10]: Deiti 1.1. Let G be a pr-` rup. G is pwerful if ` is dd ad G=G` is abelia, r if ` = 2 ad G=G 4 is abelia. (G is the subrup f G eerated by the -th pwers f elemets i G, ad G is its clsure.) Deiti 1.2. A pr-` rup is uifrmly pwerful, h r i justhuifrm, i if (i) G is itely eerated, (ii) G is pwerful, ad (iii) fr all i, G`i : G`i+1 = G : G`. (ad fr cmpleteess) Deiti 1.3. Let G be a tplical rup. We call G a `-adic aalytic rup (r `- adic Lie rup) if G has the structure f a `-adic aalytic maifld such that the fucti : G G! G deed by (x; y) 7! xy?1 is aalytic. (This is the same as the deiti f a real Lie rup ver the real umbers. Such rups eter it the Ftaie-Mazur prram as rups f `-adic represetatis, s it may be cveiet t thik f such examples as SL (Z`) ad its subrups.) Ad the fllwi imprtat therems: Therem 1.4 (4.8 f [10]). A pwerful itely eerated pr-` rup is uifrm if ad ly if it is trsi-free. (All f the Galis rups we will be csideri are, i fact, itely eerated.) Therem 1.5 (9.34 f [10]; Lazard, 1965). Let G be a tplical rup. The G has the structure f a `-adic aalytic rup if ad ly if G ctais a pe subrup which is a uifrmly pwerful pr-` rup. Usi these, we see that the Cjecture 1 is equivalet t: Cjecture 2. Let k, ` be as i Cjecture 1. `-extesi f k, the M = k. If M is a uifrmly pwerful uramied (Nte that a ite uifrmly pwerful rup must be trivial, by Therem 1.4.) Similarly, usi the abve ad the fact that K=k is a Z`-extesi, Questi 1 is equivalet t: Questi 2. Let k, F, p, q, ` be as i Questi 1. If M is a uifrmly pwerful uramied `-extesi f k with cstat eld extesi, must we have M = k?

3 ON THE FONTAINE-MAZUR CONJECTURE 3 As special cases where the aswer t Questi 2 is yes, we have the fllwi therems. The rst was riially prved by Bst fr umber elds i [4]; the same prf applies fr fucti elds. Therem 1. If a fucti eld k has a subeld k 0 such that k=k 0 is cyclic f deree prime t ` ad such that ` des t divide the class umber h(k 0 ), the ay everywhere uramied pwerful ( a frtiri uifrm) pr-` extesi f k, Galis ver k 0, with cstat eld extesi, is ite. (This is true eve if ` = p, ulike the fllwi therems.) The fllwi tw therems will be prved i Sectis 2 ad 4, respectively. Therem 2. Let k 0 be a fucti eld ver a ite eld f characteristic p, ad let k be a cstat eld extesi. Let ` be a prime t equal t p. If ` des t divide the class umber f k 0, the ay everywhere uramied pwerful ( a frtiri uifrm) pr-` extesi f k, Galis ver k 0, with cstat eld extesi, is ite. (We will shw hw t prve Therem 2 bth idepedetly ad as a special case f Therem 1.) Therem 3. Let k 0 be a fucti eld ver a ite eld F 0, such that the Jacbia f its crrespdi curve has a cmmutative edmrphism ri. Fr all but itely may primes `, ive ay cstat eld extesi k = k 0 F, ay everywhere uramied pwerful ( a frtiri uifrm) pr-` extesi f k, Galis ver k 0, with cstat eld extesi, is ite. Remark 1.6. (a) The set f ` which eed t be excluded ca be calculated, ad des t deped the class umber f k 0. (b) This therem is extedable t the case where the abelia variety has a edmrphism ri which is t quite abelia, with a sliht elaremet f the set f excluded `. (See Therem 4.9 f Secti 4.) Furthermre, the idea behid Therem 2 ca be applied t umber elds as well, usi the Mai Cjecture f Iwasawa thery as prved by Wiles. Let k 0 w be a ttally real umber eld, ad ` be a dd prime. Let k = k 0 (`) fr > 1 be a -trivial extesi f k 1 = k 0 (`), where ` will dete a primitive `-th S rt f uity. Let K = k. Let = Gal(k 1 =k 0 ), ad let = jj. Let `e be the larest pwer f ` such that `e 2 k 0 (`). We eed rst a ew deiti. Deiti 1.7. Let k0 be the zeta fucti fr k 0. We say that ` is k 0 -reular if ` is relatively prime t k0 (1? m) fr all eve m such that 2 m? 2 ad als ` is relatively prime t `e k0 (1? ). (It is kw that these umbers are ratial; we will prve that they are `-iteral als.) The i Secti 6 we will prve, as a special case f Cjecture 2: Therem 4. Suppse ` is k 0 -reular. The there are uramied iite pwerful pr-` extesis M f k, Galis ver k 0, such that K \M = k ad Gal(M el =k ) = Gal(M el =k )? accrdi t the acti f, where M el is the maximal elemetary abelia subextesi f M=k. A shrt utlie f the paper is perhaps i rder. Therem 1 ad Therem 2 are discussed i Secti 2, with sme brief tes hw they may be prved. Secti 3 describes a Lie

4 4 JOSHUA BRANDON HOLDEN alebra which we ca assciate t ur uifrm rups ad sme aspects f the relatiship betwee this Lie alebra ad the riial rup, ad the Secti 4 uses this Lie alebra t treat a situati which is related t thse f Secti 2, but which des t quite fllw the the same pla f attack. We prve a lemma relati eiespaces ad ideals f Lie alebras, ad the prve Therem 3 ad a extesi theref. Examples f elds t which the therems discussed s far ca be applied are ive i Secti 5. After that, we tur back t umber elds. Secti 6 itrduces the Mai Cjecture ad uses it t prve Therem 4. Secti 7 presets a therem f Greeber which als ivlves k 0 -reular primes ad explres its cecti t Therem 4. Fially, we clse with sme umerical data k 0 -reular primes i Secti Bst's therem ad sme clse relatives The fllwi therem is frm Bst [4]: Therem 2.1 (Bst, 1994). If a umber eld k has a (prper) subeld k 0 such that k=k 0 is cyclic f deree prime t ` ad such that ` des t divide the class umber h(k 0 ), the ay everywhere uramied pwerful pr-` extesi f k, Galis ver k 0, is ite. Therem 2.1 is prved by usi the Schur-Zassehaus lemma t et a reular autmrphism (i.e. e with -trivial xed pits) the Galis rup, ad the usi Shalev's therem frm [28] the derived leth f a rup with few xed pits. We have als a equivalet therem fr fucti elds, prved the same way: Therem 2.2. If a fucti eld k has a (prper) subeld k 0 such that k=k 0 is cyclic f deree prime t ` ad such that ` des t divide the class umber h(k 0 ), the ay everywhere uramied pwerful pr-` extesi f k, Galis ver k 0, with cstat eld extesi is ite. Remark 2.3. This hlds als fr ` = p. We w address the case where k is a -trivial extesi f sme eld k 0, such that k is btaied frm k 0 by cstat eld extesi. Let k 0 be a fucti eld ver a ite eld F 0 f characteristic p ad rder q 0, ad let k = k 0 F where F is the extesi f F 0 with deree > 1 ad rder q = (q 0 ). Therem 2.4. If ` des t divide the class umber f k 0, the there are uramied iite pwerful pr-` extesis f k, Galis ver k 0, with cstat eld extesi. This therem ca be prved i much the same way as the previus es. The imprtat dierece is that the reular autmrphism w cmes frm a Frbeius acti, rather tha frm the Schur-Zassehaus lemma as i Bst's result. Remark 2.5. Istead f prvi Therem 2.4 directly, we culd als prve it usi Therem 2.2 as fllws: Let `m 0 be the deree f k =k 0, where 0 is prime t `. Let k 0 0 be the cstat eld extesi f k 0 f deree `m; the k is a cstat eld extesi f k 0 f 0 deree 0, ad thus a cyclic extesi f deree prime t `. If ` des t divide the class umber f k 0, the it ca be prved that ` des t divide the class umber f the cstat eld extesi k0, 0 s we may apply Therem 2.2 t the extesi k =k0. 0 The there are uramied iite pwerful pr-` extesis f k, Galis ver k0, 0 with cstat eld extesi, ad a frteriri e Galis ver k 0.

5 ON THE FONTAINE-MAZUR CONJECTURE 5 We will cme back t this idea i Secti 7 ad cmpare it t the crrespdi situati fr umber elds. 3. The Lie alebra assciated t a uifrm rup I this secti we state sme facts ad therems abut itely eerated pwerful rups ad uifrm rups. May f these are take frm [10], perhaps the mst deitive wrk pwerful ad `-adic aalytic pr-` rups t date. (My thaks t Da Seal fr shari material frm the upcmi secd editi f [10].) Ather vitally imprtat surce is Lazard's paper, [23], which laid the rudwrk fr the whle subject. Secti 3.4 treats a smewhat amalus situati which we will ecuter; this material has t, as far as I kw, appeared elsewhere Deitis. I additi t the deitis ad therems frm the itrducti, we als have a related deiti frm [10] which will be useful: Deiti 3.1. If G is a pr-` rup ad N is a clsed subrup f G, we say N is pwerfully embedded i G (als writte N p:e: G) if ell is dd ad [N; G] N `, r ` = 2 ad [N; G] N 4. Remark 3.2. Nte that i particular such a N is a pwerful rup, ad a rmal subrup f G. We ca als dee aalus prperties fr Lie alebras. I the rest f this secti a \Z`-Lie alebra", r simply a \Lie alebra", refers t a mdule ver Z` with a Lie bracket. A \Z`-Lie ideal" r \Lie ideal" refers t a Z`-subalebra f a Lie alebra which is clsed uder bracket peratis with elemets f the alebra. Deiti 3.3. We say that a Z`-Lie alebra L is pwerful if ` is dd ad (L; L) `L, r ` = 2 ad (L; L) 4L. It is uifrm if it is pwerful, itely eerated, ad trsi-free. (I this case clearly `L = `L, ad s.) We say that a clsed Z`-Lie ideal I is pwerfully embedded i L if ` is dd ad (I; L) `I, r ` = 2 ad (I; L) 4I. Remark 3.4. (a) Nte that if I is pwerfully embedded as a Lie ideal, the it is pwerful as a Lie alebra. (b) Nte that if L is ay itely eerated Lie alebra, the it ctais a characteristic pe uifrm Lie alebra. This is perhaps aalus t the fact that ay pr-` rup f ite rak ctais a uifrm rup as a characteristic pe subrup. I the discussis that fllw, I will assume ` is dd; uless therwise ted everythi is the same fr ` = 2 except where ` shuld be replaced by The equivalece f cateries. Let G be a uifrm pr-` rup. The prperties f uifrm rups shw that each elemet x f G` has a uique `-th rt i G, which we dete x`?, fllwi [10]. The we ca dee a ew rup structure G by trasferri the rup structures frm successive G`'s, ad take the limit. Deiti 3.5 (ad Prpsiti). Fr x; y 2 G, dee x + y = (x` y` )`?:

6 6 JOSHUA BRANDON HOLDEN (Nte that x` y` exists. is a `-th pwer with a uique `-th rt.) The the limit x + y = lim!1 x + y It may be veried that this ives G the structure f a abelia rup, with the same idetity elemet ad iverses as befre. It ca als be shw that if xy = yx, the x +y = xy; s that pwers f a elemet are als the same as befre. This structure will be the rup structure fr ur Lie alebra. We als eed a Lie bracket, which will be deed similarly, as fllws. Deiti 3.6 (ad Prpsiti). Fr x; y 2 G, dee (x; y) = [x` ; y` ]`?2: (The brackets idicate cmmutatrs; te that [G` ; G`] G`2 frm prperties f pwerful rups. See, e.., Prpsitis 1.16 ad 3.6 f [10].) The the limit exists. (x; y) = lim!1 (x; y) Thus the Lie bracket is derived frm the cmmutatrs; it may be veried that these peratis make the elemets f G it a Lie alebra ver Z`, which we will i the future dete L(G). It is als pssible t dee a rup structure the elemets f a uifrm Lie alebra. Secti 10.5 f [10] ad Secti 4 f [19] ive the essetial irediets. Let L be a uifrm Lie alebra. Deiti 3.7. Fr x; y 2 L, we dee the rup perati xy by xy = (x; y) = x + y + 1 (x; y) + ; 2 fr (X; Y ) the Campbell-Hausdr series i tw -cmmuti idetermiates (discussed i Chapter 7 f [10], ad als i [22]). A priri, this series ly cveres i L Q`, if at all. Hwever, [10] shws that i fact the series cveres, with sum i L. This map des ive a rup structure; the prf is essetially that f Lemma f [10]. Nw let G = G(L) dete L with the rup structure we have just deed. Sice L is trsi free ad itely eerated, it fllws that G is, als. Prpsiti 3.8 (see Lemma 4.1 f [19]). (a) If G is a uifrm rup, the L(G) is a uifrm Lie alebra. (b) If L is a uifrm Lie alebra, the G(L) is a uifrm rup. Fially, we shuld te that these peratis dee a equivalece f cateries frm the catery f uifrm rups with rup hmmrphisms t the catery f uifrm Lie alebras with Lie alebra hmmrphisms. (See, fr example, Secti IV.3.2 f [23].) I particular, we will use the fact that a rup autmrphism f G iduces a Lie alebra autmrphism f L(G).

7 ON THE FONTAINE-MAZUR CONJECTURE Substructures. Let G be a uifrm rup ad L be the uifrm Lie alebra L(G). Prpsiti 3.9 (see Lemma 4.1 f [19]). (a) If B is a pwerful Z`-subalebra f L, the the elemets f B frm a subrup f G. (b) If I is a pwerfully embedded Z`-Lie ideal f L, the the elemets f I frm a rmal subrup f G. Remark It will be useful i the future t te that if (B Q`) \ L = B (i.e. B is \islated"), the B is pwerful: suppse (B Q`) \ L = B, ad let x be i (B; B): Clearly, x 2 (L; L), ad by Prpsiti 3.8 we have (L; L) `L. S x 2 `L, implyi `?1 x 2 L. Hwever, `?1 x 2 B Q` als, s it is i B. Thus x 2 `B, as desired. Similarly, if I is a islated ideal the I is pwerfully embedded. Cversely, we have: Prpsiti 3.11 (see Lemma 4.1 f [19]). (a) If H is a pwerful clsed subrup f G, the the elemets f H frm a Lie subalebra f L. (b) If N is a pwerfully embedded subrup f G, the the elemets f N frm a Lie ideal f L. Althuh we will t use it i this paper, the reader may als be iterested t te the fact that the equivalece f cateries takes cmmutatrs f pwerfully embedded subrups t Lie brackets f pwerfully embedded ideals, ad vice versa. It thus preserves the derived series ad the lwer cetral series, ad thus the prperties f slvability ad ilptecy Qutiets. The qutiet f uifrm structures may have trsi, ad s may t be uifrm. Thus it is ecessary t treat qutiets f uifrm structures as special bjects i sme cases. Prpsiti Let I be a Z`-ideal f L which is als a rmal subrup f G, e.. I is pwerfully embedded. The G=I has the same umber f eeratrs as L=I, ad G=I is trsi free if ad ly if L=I is. Prf. The umber f eeratrs f G=I is the dimesi f (G=I)= Frat(G=I) as a vectr space ver Z=`Z, where Frat(H) is the Frattii subrup f H, amely H `[H; H]. Nw (G=I)= Frat(G=I) is ismrphic t G=I Frat(G)I=I = G Frat(G)I = G= Frat(G) Frat(G)I= Frat(G) ; sice Frat(H=N) = Frat(H)N=N fr rmal subrups N f H. But mdul Frat(G) (which equals `L sice G is uifrm), the structures f G ad L are exactly the same, s this is ismrphic t L=`L (`L + = L I)=`L `L + L=I = I (`L + = L=I I)=I `(L=I) the dimesi f which is the umber f eeratrs f L=I. Pwers i G are the same as pwers i L, s x = 1 2 G=I if ad ly if x 2 I G if ad ly if x 2 I L if ad ly if x = 1 2 L=I. S trsi is the same. Remark If I is a Z`-ideal f L such that L=I is trsi free, the I is islated i the sese f Remark 3.10: suppse x q = y 1 fr x 2 I, q 2 Q`, ad y 2 L. The by cleari the demiatr f q, we et rx 1 = sy 1 fr r; s 2 Z`, ad thus rx = sy. Nw

8 8 JOSHUA BRANDON HOLDEN sy 2 I, ad L=I is trsi-free, s y 2 I. Thus (I Q`) \ L = I. The by Remark 3.10 I is pwerfully embedded, s I is a rmal subrup f G. As a crllary f the prpsiti ad the remark, we et: Prpsiti If I is a Z`-ideal f L such that L=I = Z`, the I is a rmal subrup f G ad G=I = Z`. Prpsiti Suppse ' is a autmrphism f G, ad thus als f L. Further suppse I is a Z`-ideal f L preserved by ', such that ' acts trivially L=I. If I is als a rmal subrup f G, the ' als acts trivially G=I. Prf. We kw that the elemets f I frm a rmal subrup f G preserved by ', s ' acts G=I. Suppse there exists x 2 G such that '(x) 6 x mdul I, i.e. '(x)x?1 =2 I. We ca ivke ur equivalece f cateries, which implies: '(x) x?1 = '(x) (?x) = '(x) + (?x) ('(x);?x) L + : But ('(x); '(x)) L = (x; x) L = 0, ad ('(x);?x) L = ('(x); '(x)? x) L + ('(x);?'(x)) L : The rst term f this is i the ideal I because '(x)? x 2 I, sice ' acts trivially L=I by hypthesis. The secd term is 0, s ('(x);?x) L 2 I. The ther Lie bracket terms f '(x) x?1 are als i I, sice they are Lie brackets f this with ('(x);?x) L. Als '(x) + (?x) is i I, but this ctradicts '(x)x?1 =2 I. S ' must act trivially G=I. 4. A rdiary therem The prf f the fllwi therem is smewhat dieret frm the thers i this paper, as it des t use Shalev's therem. Istead, we allw a small space f xed pits the Lie alebra, ad use them t establish a Z` extesi f sme fucti eld, which we kw is impssible. First, hwever, we eed a lemma abut Lie alebras. Lemma 4.1. Suppse ' is a autmrphism f a Lie alebra L, acti diaalizably the uderlyi vectr space. Let 1 ; : : : ; be the distict eievalues f ', ad suppse fr all i 6= j, i j 6= 1. Suppse further that if ay i =?1, the eiespace crrespdi t i has dimesi 1. Let L 0 be the subalebra f L csisti f the xed pits f '. The there is a ideal L 1 preserved by ' such that L 0 L 1 = L. Prf. If 1 is t a eievalue the there are xed pits, s we are de. Otherwise, let 1 = 1, ad let v i be the eievectrs i L crrespdi t eievalue i. Let L 1 be spaed by fv i ; i 6= 1. Clearly L = L 0 + L 1. Csider the Lie bracket f v i with v j. We have '([v i ; v j ]) = ['(v i ); '(v j )] = [ i v i ; j v j ] = i j [v i ; v j ]: Thus [v i ; v j ] is a eievectr (pssibly 0), crrespdi t the eievalue i j. But i j is t 1 uless i = j = 1 r i = j ad i =?1. If the latter, the [v i ; v j ] = [v i ; v i ] = 0, sice the eiespace crrespdi t i has dimesi 1. Therefre, either [v i ; v j ] = 0, r [v i ; v j ] is sme eievectr i L 1 uless i = j = 1.

9 ON THE FONTAINE-MAZUR CONJECTURE 9 Let a = P a i v i 2 L ad b = P j6=1 b jv j 2 L 1 : The [a; b] = [ X a i v i ; X j6=1 b j v j ] = X j6=1 a i b j [v i ; v j ] which is i L 1 by the abve. Thus L 1 is a ideal. Fially, L 1 is eerated by eievectrs, s it is preserved by '. Crllary Suppse ' is a autmrphism f a Z`-Lie alebra L, acti semi-simply the uderlyi vectr space f L Q`. Let 1 ; : : : ; be the distict rts i Q` f the characteristic plymial f ', ad suppse fr all i 6= j, i j 6= 1. Suppse further that if ay i =?1, i is ly a simple rt f the characteristic plymial. The the cclusis f the lemma hld, i.e.: let L 0 be the subalebra f L csisti f the xed pits f '. The there is a ideal L 1 preserved by ' such that L 0 L 1 = L. Prf. Let L 0 = L Z` E, where E = Q`( 1 ; : : : ; ), s that ' acts diaalizably L 0, by extesi f base ri. The the are the eievalues fr the acti, ad satisfy the hyptheses f the lemma, s we have L 0 = L 0 0 L 0 1. By extesi f base ri, L 0 = L 0 0 \ L, ad if we let L 1 = L 0 1 \ L the L 1 is a ideal satisfyi the desired prperties. Let k 0 be a fucti eld ver a ite eld F 0 f characteristic p ad rder q 0, ad let k = k 0 F where F is the extesi f F 0 with deree > 1 ad rder q = (q 0 ). The elds k ad k 0 have the same eus, ad crrespd t sme abelia variety A. Let ` be a prime t equal t p. Let K = k 0 F 0 ad let K urab be the maximal abelia uramied `-extesi f K. We kw that Gal(K=k 0 ) is eerated by the Frbeius elemet ', ad we kw Gal(K urab =K) = T`(A) = (Z`) 2. The Frbeius elemet has a semi-simple acti T`(A) = (Z`) 2 with characteristic plymial P (x) ad miimal plymial m(x), each with iteer ceciets. This P (x) ad this m(x) are idepedet f `. The discrimiat f m(x) is t zer because ' is semi-simple, s m(x) is a prduct f distict irreducibles. Therem 4.2. Suppse that the rts f m(x) mdul ` (pssibly i sme extesi f Z=`Z) are all distict, ad csist f 0 ; 1 ; : : : ; such that fr all i 6= j, i j 6= 1. If the variety A has a cmmutative edmrphism ri, the there are uramied iite pwerful pr-` extesis f k, Galis ver k 0, with cstat eld extesi. The prf beis with a reducti step, which we islate as a lemma s that we ca use it i subsequet prfs. Lemma 4.3 (see [4]). Let M be a everywhere uramied pr-` extesi f k that is iite ad pwerful. The M ctais a subextesi N f k which is iite ad uifrm. If k=k 0 is a Galis extesi f elds, ad M is Galis ver k 0, the s is N. Prf. Let T dete the set f trsi elemets f Gal(M=k): By Therem 4.20 f [10], T is a ite characteristic subrup, such that Gal(M=k)=T is uifrm. Let N be the crrespdi itermediate eld. The N is a iite uifrm uramied pr-` extesi. Fially, N is Galis ver k 0 sice T is a characteristic subrup f Gal(M=k); which is rmal i Gal(M=k 0 ): Prf f Therem. If 1 is t a rt f m(x) the ` des t divide m(1), which divides P (1), s we may use Therem 2.4. Thus we may assume 0 = 1.

10 10 JOSHUA BRANDON HOLDEN Let M be a uramied iite pwerful pr-` extesis f k, Galis ver k 0, with cstat eld extesi. By Lemma 4.3 we may assume M is a uifrm extesi f k. Let G = Gal(M=k ), ad let d be the dimesi f G. Sice M is Galis ver k 0, ' acts G. Let K urel be the maximal elemetary abelia subextesi f K urab =K ad let M el be the maximal elemetary abelia subextesi f M=k. (See Fiure 1.) Fiure 1. The relatiships betwee the subelds f M ad K i the fucti eld case. K urab K (Z=`) d (Z`) 2 (Z=`) 2 M el K K urel M K h'i k k 0 h' i (Z=`) d M el G` G M Sice M has cstat eld extesi, M \ K = k. Let Frat(G) = [G; G]G` be the Frattii subrup f G, which is equal t G` sice G is pwerful ad itely eerated. The G= Frat(G) = G=G` = Gal(M el =k ) = Gal(M el K=K); which is a qutiet f Gal(K urel =K) = (Z=`) 2 ad is thus a vectr space f dimesi d ver Z=`Z. Let R be the semi-simple 2 by 2 matrix, with Z`-ceciets, represeti the acti f ' Gal(K urab =K). Sice G is uifrm, we have a Z`-Lie ri structure G; let L dete G with this structure. By the equivalece f cateries i Secti 3.2, a autmrphism f G iduces a autmrphism f L, s the acti f ' G ives a acti L als. The fact that A has a cmmutative edmrphism ri meas that the deree f m(x) is equal t 2, ad thus it is als equal t the characteristic plymial. Nw ' acts Gal(M el =k ) = L=`L accrdi t sme factr represetati R 0 f the reducti f R mdul `, ad this acti has characteristic plymial equal t a factr f m(x) mdul `. The acti f ' L is als represeted by a matrix whse reducti mdul ` is equal t R 0. The ur cditis the rts f m(x) mdul ` imply that the acti f ' L is as i the crllary (te all f the rts are simple, sice the characteristic plymial equals the miimal plymial), s L = L 0 L 1 fr L 1 a ideal as i the lemma. But all f the eievalues have multiplicity 1, s L 0, the eiespace f L crrespdi t eievalue 1, is e-dimesial. S L 1 is a ideal f cdimesi 1, ad by the crrespdece f G with L, specically Prpsiti 3.14, we have a rmal subrup N = L 1 f G such that G=N = Z`. But this meas there is a uramied iite abelia extesi f k with cstat eld extesi, which is impssible.

11 Here is ather way f lki at this result: ON THE FONTAINE-MAZUR CONJECTURE 11 Therem 4.4. Let k 0 be a fucti eld ver a ite eld F 0, such that its crrespdi abelia variety has a cmmutative edmrphism ri. Fr all but itely may primes `, ive ay cstat eld extesi k = k 0 F, k has uramied iite pwerful pr-` extesis, Galis ver k 0, with cstat eld extesi. Prf. As i the previus prf, the miimal plymial m(x) ad the characteristic plymial P (x) are the same, ad idepedet f `. The discrimiat Q f m(x) is t zer i Z, s the rts are all distict. Let i be the rts, ad set = ( i<j i j? 1). We kw fr all i, j i j = p q 0, s is t zer. Hwever, 2 Z fr the same reas is. Nw we ca say that the primes ` that eed t be excluded are ` = p (f curse), the primes dividi, ad the primes dividi. Fr every ther `, the hyptheses f the previus therem are satised. Remark 4.5. A has a cmmutative edmrphism ri if, fr example, it is the prduct f iseus elemetary abelia varieties which are rdiary. I fact, Therem 4.2 ca be imprved, usi the fllwi result abut uifrm rups. Prpsiti 4.6 (Exercise 4.11.(i) f [10]). Let G be a uifrmly pwerful pr-p rup with 2 (tplical) eeratrs. The G ctais a uique rmal prcyclic subrup H such that G=H is prcyclic, ad H has a cmplemet i G. Therem 4.7. Suppse that the rts f m(x) mdul ` (pssibly i sme extesi f Z=`Z) are all distict, ad csist f 0 ; 1 ; : : : ; such that fr all i 6= j, i j 6= 1. Further, suppse that i 6=?1 fr all i. If the characteristic plymial P (x) assciated t the variety A has ly rts f multiplicity 2 r less, the there are uramied iite pwerful pr-` extesis f k, Galis ver k 0, with cstat eld extesi. Q t Remark 4.8. Let A be iseus t i=1 Am i i with the A i iseus elemetary abelia varieties, as i the Picare-Weil Therem. (See, fr example, Part I f [32] fr a summary f the classicati f abelia varieties up t isey. We use a umber f facts frm that summary i this Q remark, especially frm Sectis 1 ad 6.) The the characteristic t plymial is P (x) = Q f i=1 i(x) e im i fr sme e i, with f i (x) irreducible. Clearly the miimal plymial is m(x) = f i (x). The cdimesi f the ceter f the edmrphism ri f A m i i is e 2 i m 2 i, s if each A m i i has a edmrphism ri with ceter f cdimesi less tha r equal t 4, the cditi the characteristic plymial P is true. Mre lbally, the cdimesi f the ceter f the edmrphism ri f A is e 2 i m 2 i, s if A has a edmrphism ri with ceter f cdimesi less tha r equal t 4 the the cditi is true. Prf (f Therem). We keep the tati f the previus prfs. As befre, we may assume 0 = 1. As i the prf f Therem 4.2, the acti f ' L has characteristic plymial equal t a factr f P (x) mdul `. The ur cditis the rts f P (x) mdul ` imply that the acti f ' L is as i Crllary 4.1.1, s L = L 0 L 1 fr L 1 a ideal as i the lemma. But all f the eievalues have multiplicity 1 r 2, s L 0, the eiespace f L crrespdi t eievalue 1, is e-dimesial r tw-dimesial, ad L 1 is a ideal f cdimesi 1 r 2. If the cdimesi is 1 the we cclude as i the prf f Therem 4.2.

12 12 JOSHUA BRANDON HOLDEN If the cdimesi is 2, the by Prpsiti 3.12, we have a rmal subrup N = L 1 f G such that G=N is a trsi-free (ad thus uifrm) pwerful rup with 2 eeratrs. The Prpsiti 4.6 shws that G=N has a qutiet rup ismrphic t Z`. But aai, this meas there is a uramied iite abelia extesi f k with cstat eld extesi, which is impssible. Aai, we ca lk at this result i terms f which ` it ca be applied t: Therem 4.9. Let k 0 be a fucti eld ver a ite eld F 0, such that its crrespdi abelia variety is as described i the previus therem. Fr all but itely may primes `, ive ay cstat eld extesi k = k 0 F, k has uramied iite pwerful pr-` extesis, Galis ver k 0, with cstat eld extesi. Prf. This time, the primes ` that eed t be excluded are ` = p (f curse), the primes dividi the discrimiat f the miimal plymial, the primes dividi (als fr the miimal plymial), ad the primes dividi P (?1). Remark I fact, we ca squeeze eve mre ut f Crllary al these lies. Fr istace, ly the multiplicities f the rts f P (x) which are 1 r?1 mdul ` are relevat, s i a specic example we ca lk at ly the irreducible factrs f P (x) ctaii these rts. If the ttal multiplicity f the rts which are 1 mdul ` is mre tha 2, ad the ttal multiplicity f the rts which are?1 mdul ` is mre tha tha 1, the there ca be extesis f the srt described. Als, ly the discrimiat f the miimal plymial fr the acti mdul ` is relevat, ad this may be smaller tha the reducti mdul ` f m(x). (It is the larest square-free plymial ver Z=`Z dividi the reducti f P (x) mdul `.) Determii whether these lser cditis are satised, hwever, ca be de ly e ` at a time, ad ly if the plymial P (x) is kw fairly explicitly. 5. Examples I Fr varieties ver ite elds ive by equatis f the frm a 1 x m a 2 x m a r x mr r (a i ; c 6= 0), there is a relatively easy way t calculate the zeta fucti assciated with the variety, as explaied i Chapters 8 ad 11 f [20]. I particular, we will use this methd fr curves f the frm ax 2 + by m = c: These crrespd t quadratic extesis k f the fucti eld F(t), ad we will use their zeta fuctis t try t d examples f ur therems. Dee the zeta fucti fr k, k (s), i the usual way, ad let Z k (T ) be the fucti such that k (s) = Z k (q?s ), where q is the umber f elemets f F. Nw if C=F is the curve assciated t k, the we have a alterative expressi Z k (T ) = Z(C=F; T ) = exp 1X =1 = c jc(f q )j T where F q is the ite eld with q elemets, ad pits are cuted a -siular mdel f C=F. Let Z(V=F; T ) be the zeta fucti f a variety V, deed aalusly t that f a curve.! ;

13 ON THE FONTAINE-MAZUR CONJECTURE 13 Prpsiti 5.1. The zeta fucti Z(V=F; T ) is equal t Q Q i(1? it ) j (1? jt ) if ad ly if there exist cmplex umbers f i ad f j such that fr all 1, jv (F q )j = X j j? X i i : This is Prpsiti f [20], ad is prved by larithmic dieretiati. We specialize t the curve C=F q, q dd, ive by ax 2 + by m = c; where m > 2 divides q? 1. (Sice q is dd, 2 als divides q? 1.) The it turs ut that C(Fq d) = q d m?1 X i6=m=2; i=1 (?1) d+1? i (c)(a?1 ) i (b?1 )J(; i ) d where is the quadratic character F q, is a eeratr f the cyclic rup f characters f rder dividi m F q, ad J(; i ) is their Jacbi sum. (See Chapter 8 f [20], fr hw t cut the ae pits, ad Example II ad Exercise 2.14 f [30] fr the prjective pits.) Applyi Prpsiti 5.1, we btai a frmula fr the zeta fucti, amely: m?1 Y i6=m=2; i=1? 1? i (c)(a?1 ) i (b?1 )J(; i )T (1? T )(1? qt ) Fially, this ives us the plymial P (T ) which we have bee usi i ur therems, amely P (T ) = m?1 Y i6=m=2; i=1? T? i (c)(a?1 ) i (b?1 )J(; i ) : The fllwi examples were calculated usi the prram PARI. (See [2].) Example 5.2. C = fx 2 + y 5 = 1; F = F 11. The i terms f ur therems, k 0 = F 11 (t; p 1? t 5 ): P (T ) = T 4? 4T 3 + 6T 2? 44T + 121; P (1) = 2 4 5, = , ad = Fr every ` 6= p except 2 ad 5, we ca apply Therem 2.4. Fr every ` 6= p except 2, 5, ad 211, we ca als apply Therem 4.2. Example 5.3. C = fx 2 + y 7 = 1; F = F 29. The k 0 = F 29 (t; p 1? t 7 ): P (T ) = T 6? 6T 5? 13T T 3? 377T 2? 5046T ; P (1) = , =? , ad = Fr every ` 6= p except 2, 7, ad 43, we ca apply Therem 2.4. Hwever, fr ` = 43, we ca istead apply Therem 4.2. :

14 14 JOSHUA BRANDON HOLDEN Example 5.4. C = fx 2 + y 9 = 1; F = F 19. The k 0 = F 19 (t; p 1? t 9 ): P (T ) = (T 6 + 9T T T )(T 2? 8T + 19); P (1) = , = , ad = Fr every ` 6= p except 2, 3, ad 37, we ca apply Therem 2.4. Hwever, fr ` = 37, we ca istead apply Therem 4.2. There are may similar examples where Therem 4.2 prves useful. Here is e where istead we ca apply Therem 4.7. Example 5.5. C = fx 2 + ( + 2)y 6 = 1, where = 0; F = F 25. The k 0 = F 25 (t; p 1? ( + 2)t 6 ): P (T ) = (T 2? 5T + 25) 2 ; P (1) = 441 = , P (?1) = 961 = 31 2, =?75 =? , ad = 576 = ( ad are take fr the miimal plymial.) Thus fr ` = 7, Therem 4.7 may be applied, eve thuh Therem 2.4 cat. Remark 5.6. Nte that fr m = 3 ad m = 4, C is a elliptic curve, i.e. has eus 1. The by Abhyakar's Cjecture, prved by Harbater, Gal(K ur =K) is a free pr-` rup with tw eeratrs. (See [15]). I this case, e f Therem 2.4, Therem 4.2, r Therem 4.7 ca always be applied, fr all ` 6= p. 6. Iwasawa thery ad cycltmic extesis We w tur t umber elds. Let ` be a dd prime, ad let k be a extesi f sme ttally real umber eld k 0, ive by k = k 0 (`) fr > 1, where ` will dete a primitive `-th S rt f uity. Let k 1 = k 0 (`) be the subextesi f deree prime t `. Let K = k 0 (`1) = k d 0(`d) ad let K urab be the maximal abelia uramied `-extesi f K. The Gal(K=k 0 ) =?, where = Gal(k 1 =k 0 ) is a ite abelia rup f rder prime t `, ad? = Gal(K=k 1 ) is ismrphic t Z` ad is eerated by sme. Let X = Gal(K urab =K), as i Chapter 13 f [31]. The? acts X by cjuati, s des. Sice ` des t divide the rder f (which divides `?1), ad the values f the (e-dimesial) characters,! (Z=`Z) lie i Z` (ad t a extesi), the as explaied i Secti 13.4 f [31], we ca decmpse X as M X = X accrdi t the rthal idemptets f the rup ri Z`[], where We ca als decmpse = 1 jj X 2 ()?1 : X = X + X? = + X? X usi the elemets = (1 J)=2, where J 2 is cmplex cjuati. The, we have 1 + J X 1? J X = ad = 2 2 dd eve

15 s ON THE FONTAINE-MAZUR CONJECTURE 15 X + = M eve X ad X? = M dd X: (See Secti 6.3 f [31] fr mre details.) Nw, sice k 1 is a CM eld, by Prpsiti f [31], X? ctais ite - submdules, where = Z`[[T ]] ad T crrespds t? 1 2 Z`[?]. Thus we have X?,! M i =(`mi ) M j =( j (T )) with ite ckerel, ad likewise X,! M i =(`m i ) M j =( j (T )) with ite ckerel fr all dd characters. (The plymials are distiuished plymials i Z`[T ], meai that j (T ) = T + a?1 T?1 + + a 0 with ` dividi a i fr 0 i? 1. As we have P writte the P decmpsiti, they are t ecessarily irreducible.) Fr all such, let = m, i = de j, ad let Y (T ) = ` j (T ): Q (This is t ecessarily a distiuished plymial, but X X j (T ) is.) Let? = ;? = : dd Barsky [1], Cassu-Nues [5], ad Delie ad Ribet [9] have each shw that e ca cstruct a `-adic L-fucti L`(s; ) fr eve characters ver a ttally real eld k 0. Furthermre, L`(s; ) is Iwasawa aalytic if is a -trivial character, i.e. L`(s; ) ca be expressed as a pwer series i u s? 1, where u is a elemet f the pricipal uits f Z`, deed by ` = ù fr 1. (Fr mre ifrmati Iwasawa aalytic fuctis, see Secti 4 f [27].) Let dd! : = Gal(k 1 =k 0 )! h`?1 i Z ` be the `-adic Teichmuller character (which is dd), ad let be a dd character. The, mre explicitly, it ca be prved that there exists a pwer series f 2 such that L`(1? s;!?1 ) = f (u s? 1)=h (u s? 1) fr s 2 Z`, where h (T ) = T if =! ad h (T ) = 1 therwise. (This simple characterizati f h cmes abut because all f the characters we csider are e-dimesial ad f what Greeber calls type S, i.e. k \ k 0 B 1 = k 0, where k is this extesi f k 0 attached t ad B 1 is the cycltmic Z`-extesi f Q. Fr mre eeral `-valued Arti characters, we have h (T ) =!?1 ()(1 + T )? 1 if!?1 is f what Greeber calls type W, i.e. k k 0 B 1, ad h (T ) = 1 therwise.) I rder t relate these pwer series t X?, we use the Mai Cjecture f Iwasawa thery, which was prved by Mazur ad Wiles fr abelia extesis f Q i [24] ad by Wiles fr arbitrary ttally real elds i [33]. Usi tati frm [33] ad [31], the therem says:

16 16 JOSHUA BRANDON HOLDEN Therem 6.1 (The Mai Cjecture). Fr a dd character, with U (T ) a uit i. f (u(1 + T )?1? 1) = (T )U (T ); (This is a cmbiati f Therems 1.2 ad 1.4 f [33]. See als Secti 13.6 f [31], r Secti 5 f [6] fr mre the Mai Cjecture.) Let = [k 1 : k 0 ] = jj, ad let `e be the larest pwer f ` such that `e 2 k 0 (`). Deiti 6.2. Let L(s; ) be the L-fucti fr characters assciated t k 0. We say that ` is k 0 -reular if ` is relatively prime t L(1? m; 1) = k0 (1? m) fr all eve m such that 2 m? 2 ad als ` is relatively prime t `el(1? ; 1) = `e k0 (1? ). Remark 6.3. This ew deiti is aalus t ` bei reular, sice if k 0 = Q we have = `? 1, L(1? m; 1) =?B m =m fr B m the m-th Berulli umber, 2 m `? 3, ad ` ever divides `el(2? `; 1) =?`B`?1 =(`? 1) by V Staudt-Clausse's Therem (Therem 5.10 f [31]) r by the arumet ive i Secti 2 f [26]. Als, k 0 -reularity seems t share at least sme f the prperties f reularity. Fr example, by a prf aalus t e fr the case ver Q, I have shw that there are iitely may k 0 -irreular primes fr ay ive k 0. (See, e.., Therem 5.17 ad Exercise 4.3.(a) f [31].) Remark 6.4. We will shw belw that the umbers i these cditis are `-iteral. (It is well-kw that they are ratial umbers; see, e.., [25].) Therem 6.5. Suppse ` is k 0 -reular. The there are uramied iite pwerful pr-` extesis M f k, Galis ver k 0, such that K \M = k ad Gal(M el =k ) = Gal(M el =k )? accrdi t the acti f, where M el is the maximal elemetary abelia subextesi f M=k. Remark 6.6. The cditi K \ M = k is equivalet t sayi that k ctais all the `-th pwer rts f uity i M. Nte that fr sucietly lare, K=k is ttally ramied (see Lemma 13.3 f [31], e..). Hwever, M=k is uramied by hypthesis, s fr these, K \ M = k always. Remark 6.7. The acti f Gal(M el =k ) cmes frm the acti f Gal(k =k 0 ), which is by cjuati i Gal(M el =k 0 ) by sme represetative. (Nte that M el =k 0 is Galis, because Gal(M=M el ) is a characteristic subrup f a rmal subrup f Gal(M=k 0 ), ad the last extesi is Galis by hypthesis. Furthermre, Gal(M el =k ) is a rmal subrup f Gal(M el =k 0 ), s the acti abve is deed.) The acti is idepedet f the chice f represetative, sice Gal(M el =k ) is abelia. Prf (f Therem). Let M be such a extesi f k. As i previus therems, we may assume by Lemma 4.3 that M is a uifrm extesi f k. Let G = Gal(M=k ), ad let d be the dimesi f G. Sice M is Galis ver k 0, acts G by cjuati i Gal(M=k 0 ) (equivaletly i Gal(M=k 1 )) by sme represetative. Let K urel be the maximal elemetary abelia subextesi f K urab =K ad let M el be the maximal elemetary abelia subextesi f M=k. Sice acts X, ad Gal(M el =k ) = Gal(M el =k )?, we ca csider M el as a subextesi f (K urab )? ad (K urel )?. (See Fiure 2.)

17 ON THE FONTAINE-MAZUR CONJECTURE 17 Fiure 2. The relatiships betwee the subelds f M ad K i the umber eld case. (K urab )? K (Z=`) d? (Z`) () s?? (Z=`) (=`) s? M el K (K urel )? M K hi=? k k 1 k 0 h`?1 i (Z=`) d M el G G` M By hypthesis, M \ K = k. Let Frat(G) = [G; G]G` be the Frattii subrup f G, which is equal t G` sice G is pwerful ad itely eerated. The G= Frat(G) = G=G` = Gal(M el =k ) = Gal(M el K=K); which is a qutiet f Gal((K urel )? =K). Sice X? = M dd X; we ca write M s t M! M X?,! =(`m i ) =( j (T )) dd i=1 j=1 with ite ckerel. We wat t kw whether has ay xed pits as a autmrphism f Gal((K urel )? =K) = X? =`X?. S far we ca deduce that X? =`X?? = (Z=`) (=`) s? P, where s? = s. We will ext shw that fr all dd, s = 0 i the decmpsiti f X?, s that i fact ly the secd rup f terms exists, ad X? =`X?? = (Z=`). Claim Uder the hyptheses f the therem, ` is relatively prime t f (u? 1) fr all dd. (We will defer the prf f this util later.) By the Mai Cjecture, f (u? 1) = f (u(1 + 0)?1? 1) = (0)U (0). U (T ) is a uit pwer series, s it is well-kw that U (0) must be a uit i Z`. Thus if ` is relatively Q prime t f (u? 1) the it is als relatively prime t (0) i Z`. But (0) = ` j (0); s we

18 18 JOSHUA BRANDON HOLDEN must have 0 = = P m i ; s s = 0. Thus X?,! M dd t M j=1 =( j (T )) with ite ckerel. Nw let R be the? by? matrix, with Z`-ceciets, represeti the acti f X?, which is ismrphic t Z? `. The acts X? =`X? accrdi t the reducti f R mdul `. If 1 is t a eievalue f R mdul `, the this acti has -trivial xed pits, i.e. is reular. Sice X? L Q` = L dd j =( j (T )) Q`, as Q`[[T ]]- mdules, their characteristic plymials fr are the Q same. Sice T crrespds Q Q t? 1, the characteristic plymial f the riht-had side is dd (x?1) = dd j (x?1), j s this is the characteristic plymial f R als. Nw the eievalues f R mdul ` are the rts f the varius (x? 1) mdul `, s 1 is such a eievalue if ad ly if (0) 0 mdul ` fr sme. Aai, the Mai Cjecture says this is ly if f (u? 1) 0 mdul `, ad we kw frm the claim that this des t happe. S 1 is t a eievalue fr fr ay. Thus, ive the claim, the acti f ' Gal((K urel )? =K) = X? =`X? is reular. We will w prve the claim. Prf (f Claim). By the deiti f f, we have f (u? 1) = f (u 1? 1) = L`(0;!?1 )h (u? 1): (1) Further, as i [33], we have by deiti, fr 1: where L`(1? ; ) = L S (1? ;!? ); (2) Y L S (1? ;!? ) = L(1? ;!? ) (1?!? ((l; k 1 =k 0 ))(N l)?1 ); (3) S is the set f primes f k 0 lyi ver `, L is the cmplex Dirichlet L-fucti, N is the rm, ad (l; k 1 =k 0 ) is the Arti reciprcity symbl. (We use the cveti that the prduct f characters is always primitive. This frmula is a eeralizati f Therem 5.11 f [31]; see als Secti 4.1 f [6].) Furthermre, L`(1? s; )h (u s? 1) is a pwer series i u s? 1 (i.e. Iwasawa aalytic) s sice u s? 1 0 mdul `Z` fr all s 2 Z`, we have that the expressi l2s L`(1? s; )h (u s? 1) (4) has the same value mdul `Z` fr ay s 2 Z`. We are lki at f (u? 1) = L`(0;!?1 )h (u? 1), where is a dd character the Galis rup, ad thus equal t! m fr sme dd m, 1 m? 1. Sice f 2 ad u 2 Z`, we have f (u? 1) 2 Z`. Nte that + 1? m 1? m mdul, s! 1?m =! +1?m ad 2 + 1? m. The we have f (u? 1) = L`(0;!?1 )h (u? 1) frm (1) = L`(0;! +1?m )h (u? 1) L`(1? ( + 1? m);! +1?m )h (u +1?m? 1) (md `) frm (4) = L S (1? ( + 1? m);! +1?m )h (u +1?m? 1) frm (2)

19 ON THE FONTAINE-MAZUR CONJECTURE 19 = L(1? ( + 1? m);! +1?m!?(+1?m) ) Y l2s frm (3) (1?! +1?m!?(+1?m) ((l; k 1 =k 0 ))(N l)?m ) = L(1? ( + 1? m); 1) Y l2s (1? (N l)?m ) Nw (N l)?m is a pwer f `, sice m <. S this is cruet t!! h (u +1?m? 1) L(1? ( + 1? m); 1)h (u +1?m? 1) mdul `. h (u +1?m? 1) Fr 6=!, we have h (T ) = 1, ad s this equals L(1? ( + 1? m); 1), which is thus `- iteral sice f (u?1) is. Als, 6=! implies m > 1, s we actually have 2 +1?m?2. If ` is k 0 -reular the ` is relatively prime t L(1? ( + 1? m); 1) fr all f these + 1? m, s f (u? 1) is relatively prime t ` fr ay 6=!. If =! the h! (T ) = T s we have f! (u? 1) L(1? ( + 1? m); 1)(u +1?m? 1) (md `) = L(1? ; 1)(u? 1); sice m must equal 1. Writi u = 1 + t`e fr t a uit i Z` ad usi the bimial therem, we ca see that (u? 1)=`e is a uit i Z`, i fact cruet t t = (u? 1)=`e mdul `. I particular, sice f! (u? 1) is i Z`, we must have `el(1? ; 1) i Z`, sice they dier by a uit i Z`. Further, ` divides `el(1? ; 1) if ad ly if it divides f! (u? 1) L(1? ; 1)(u u? 1? 1) = `el(1? ; 1) : If ` is k 0 -reular, the ` is relatively prime t `el(1? ; 1), ad thus t f! (u? 1): Nw we kw that the acti f ' Gal((K urel )? =K), which ca be rearded as a vectr space ver Z=`Z, is reular. We als kw that ' acts G, which iduces a acti G=G` = Gal(M el K=K), sice G` is characteristic. We ca als reard G=G` as a vectr space ver Z=`Z, ad the the acti f ' is a qutiet f the acti Gal((K urel )? =K). Thus this acti is reular, s the acti G ca have xed pits utside G`. The rest f the prf fllws as i [4]. `e 7. Greeber's criteri The ccept f k 0 -reularity, thuh t the deiti itself, has appeared earlier i wrk f Greeber a eeralizati f Kummer's criteri fr the class umber f k 1 t be divisible by `. Greeber's wrk ties it my w i a umber f ways. The rst is thruh k 0 -reular primes. Ather is that it allws us, i sme cases, t restate Therem 6.5 i a frm rather mre parallel t Therem 2.4. This statemet appears as Crllary belw. Furthermre, the techiques used i the previus secti allw us t simplify the prfs ad weake the hyptheses i Greeber's therems, ad we will explre sme ways that this ca be de. Fially, i sme limited situatis a prf f Crllary ca be ive that parallels that f Remark 2.5, usi the ideas f this secti.

20 20 JOSHUA BRANDON HOLDEN We keep the tati f the previus secti, ad als let k + 1 dete the maximal real subeld f k 1, which is equal t k 0 (` +?1 ` ). Let h(k 1 ) dete the class umber f k 1 ad h + (k 1 ) dete the class umber f k + 1. By part (a) f Therem 7.8 belw, h + (k 1 ) j h(k 1 ); we let the relative class umber h? (k 1 ) be the qutiet. I [13], Greeber ave the fllwi eeralizatis f Kummer's criteri: Therem 7.1 (Greeber, 1973). Assume that ` - [k 0 : Q] ad that prime f the eld k + 1 lyi ver ` splits i k 1. The ` divides the class umber f k 1 if ad ly if ` divides the umeratr f ` Y i eve;i=2 k (1? i): Therem 7.2 (Greeber, 1973). Assume that ` - [k 0 : Q] ad that prime f the eld k + 1 lyi ver ` splits i k 1. Further, assume that k 0 is abelia ver Q. The ` divides the class umber f k 1 if ad ly if ` is t k 0 -reular. Remark 7.3. Nte that Greeber uses ` i his deitis ad therems where we use `e; his hypthesis that ` - [k 0 : Q] implies that e must be equal t 1. k + 1 Greeber als made a strer cjecture. Let R ad R + dete the reulatrs f k 1 ad respectively, let w dete the umber f rts f uity i k 1, ad let The Greeber cjectured that Ah? (k 1 ) A = lim Q Q l2s s!0 w 2(R=R + ) (1? (N l)?s ) l2s (1? (N l)?s ) : + Y i eve;i=1 k0 (1? i) (md `); which implies Therem 7.1. Kud, i [21], prved this cjecture i the case where k 0 is abelia ver a real quadratic eld with restricti the deree f k 0, usi the `-adic L-fucti cstructed fr these elds by Cates ad Sitt i [7]. Usi the same cstructi, Kud als prved Therem 7.2 i the case where k 0 is abelia ver a real quadratic eld ad ` - [k 0 : Q]. The eeral cstructi f the `-adic L-fucti ver ay ttally real umber eld ad the crueces derived i the prf f the Claim frm Therem 6.5 allw us t prve this statemet fr ay ttally real k 0 whatsever. This ives the fllwi therem: Therem 7.4. Assume that prime f the eld k + 1 h? (k 1 ) if ad ly if ` is t k 0 -reular. lyi ver ` splits i k 1. The ` divides Remark 7.5. Nte that we ler require a cditi the deree f k 0 ; this wuld ly be ecessary i rder t replace the cditi h? (k 1 ) by e h(k 1 ) such as Greeber used. (See Therem 7.9 belw.) Furthermre, as we bserved earlier, the cstructi f the `-adic L-fucti tells us that the umbers i the deiti f k 0 -reular are all `-iteral, allwi us t remve the disticti betwee Therems 7.1 ad 7.2. As a crllary f Therems 6.5 ad 7.4, we et:

21 ON THE FONTAINE-MAZUR CONJECTURE 21 Crllary Assume that prime f the eld k + 1 lyi ver ` splits i k 1, ad that ` des t divide h? (k 1 ). The there are uramied iite pwerful pr-` extesis M f k, Galis ver k 0, such that K \ M = k ad Gal(M el =k ) = Gal(M el =k )?. Remark 7.6. Nte that if there are primes f k + 1 which split i k 1, the ` divides A ad s ` is ever k 0 -reular. Thus Therem 6.5 ca ever be applied if this is true. Furthermre, Therem 7.4 cat be used t shw ` divides h? (k 1 ) i this case. (See Greeber's paper fr mre this.) Remark 7.7. If ` des t divide h? (k), the there are uramied abelia `-extesis M f k such that Gal(M el =k) = Gal(M el =k)? ad thus trivially pwerful extesis. Remark 2.5 uses the crrespdi idea t ive a quick prf f Therem 2.4; i certai circumstaces we ca d the same here. Csider the fllwi therems class rups (see, e.., Chapter 10 f [31]): Therem 7.8. (a) Suppse the extesi f umber elds F=F 1 ctais trivial uramied abelia subextesis. (This is true, e.., if sme prime is ttally ramied.) The h(f 1 ) divides h(f ). (b) Suppse the extesi f umber elds F=F 1 is a ` extesi with at mst e ramied prime. If ` divides h(f ) the ` divides h(f 1 ). Ad ally, Greeber i [13] ives the therem (attributed t Lepldt): Therem 7.9 (Lepldt). Let k 0 be a ttally real ite extesi f Q such that ` - [k 0 : Q]. If `jh + (k 1 ) the `jh? (k 1 ). Nw let k 0 be such a extesi, ad suppse as abve that ` des t divide h? (k 1 ). Thus ` - h + (k 1 ) als, s ` - h(k 1 ). Nw if at mst e prime f k 1 ramies i k the ` cat divide h(k), by part (b) f Therem 7.8. Thus ` certaily des't divide h? (k), s as we said there are trivially pwerful extesis. This ives us a trivial prf f Crllary uder the cditis suppsed, but there is reas t thik that a similar prf will wrk utside these cditis. It is iteresti t cmpare this t the cycltmic case described i Remark 2.5. Remark Kud als discusses sme crueces mdul hiher pwers f ` which are similar t the e cjectured by Greeber. It seems likely that uder the prper cditis, these crueces wuld allw us t prve therems i the umber eld case aalus t thse prved fr the fucti eld case i Secti Examples II Havi deed k 0 -reular primes, it seems i rder t ive sme aalysis f hw likely the situati is t ccur fr ttally real elds k 0. I have therefre cmputed the umbers k0 (?1); k0 (?3); : : : ; k0 (3? ) ad ` k0 (1? ) fr varius real quadratic elds k 0 = Q( p D). (Fr mre abut zeta fuctis fr real quadratic elds, see [16].) I the fllwi tables, the discrimiat D es dw the side ad the prime ` es acrss the tp. The values shw are \umbers f irreularity" m such that ` divides k0 (1? m) r, if m =, ` k0 (1? ). The tables were calculated usi the prram PARI (see [2]) usi a frmula f Sieel (see [29]). The alrithm is discussed further by the authr i [17], which als icludes sme aalysis f this raw data.

22 22 JOSHUA BRANDON HOLDEN Table I cvers 2 D 100 ad 3 ` 50, ad Table II cvers 2 D 100 ad 51 ` 100. Table I. Numbers f irreularity fr 2 D 100 ad 3 ` Ackwledemets The authr wuld like t thak Michael Rse fr his ivaluable advice while this research was bei de ad fr his uwaveri supprt duri the writi ad publicati f this paper. Hearty thaks are als due t Gerhard Frey, Erst Kai, ad Yasutaka Ihara fr piti ut cases where Questi 1 has a eative aswer. Refereces [1] D. Barsky, Fctis z^eta p-adiques d'ue classes de ray des crps de mbres ttalemet reels, i Grupe d'etude d'aalyse Ultrametrique, 1977/1978, 5e aee, Exp.. 16, 23 pp. [2] C. Batut, D. Berardi, H. Che, ad M. Olivier, User's Guide t PARI-GP, Labratire A2X, Uiversite Brdeaux I, versi 1.39 ed., Jauary 14, [3] N. Bst, Sme cases f the Ftaie-Mazur cjecture, J. Number Thery 42 (1992) 285{291. [4] N. Bst, Sme cases f the Ftaie-Mazur cjecture, II, J. Number Thery (Submitted Prelimiary versi). [5] P. Cassu-Nues, Valeurs aux etiers eatifs des fctis z^eta et fctis z^eta p-adiques, Ivet. Math. 51 (1979) 29{59. [6] J. Cates, p-adic L-fuctis ad Iwasawa's thery, i A. Frhlich, editr, Alebraic Number Fields (L-fuctis ad Galis Prperties), paes 269{353, Academic Press, 1977.