STARK S CONJECTURE IN MULTI-QUADRATIC EXTENSIONS, REVISITED

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1 STARK S CONJECTURE IN MULTI-QUADRATIC EXTENSIONS, REVISITED Davd S. Dummt 1 Jonathan W. Sands 2 Brett Tangedal Unversty of Vermont Unversty of Vermont Unversty of Charleston Abstract. Stark s conjectures connect specal unts n number felds wth specal values of L-functons attached to these felds. We consder the fundamental equalty of Stark s refned conjecture for the case of an abelan Galos group, and prove t when ths group has exponent 2. For bquadratc extensons and most others, we prove more, establshng the conjecture n full Mathematcs Subject Classfcaton. Prmary 11R42; Secondary 11R27, 11R11, 11R29. 1 Research supported by NSF grant DMS and NSA grant MDA Research supported by NSF grant DMS and NSA grant MDA

2 I. THE ELEMENTS OF STARK S REFINED ABELIAN CONJECTURE UNITS. Let: L/F be an abelan extenson of number felds n whch a dstngushed (fnte or nfnte) prme of F denoted by v splts completely. These and all number felds wll be assumed to le n a fxed algebrac closure Q of the feld Q of ratonal numbers. w be the normalzed absolute value at a fxed prme w of L above v. w L be the order of the group µ L of roots of unty n L. U (v) L be the group of elements of L havng absolute value equal to 1 at each (fnte or nfnte) absolute value of L except for those assocated wth prmes above v, n other words, those whch are conjugates of w. We sometmes refer to U (v) L as the v-unts of L. L-FUNCTIONS. Let: G be the abelan Galos group of the extenson L/F. Ĝ be the character group of G. S be a fxed fnte set of prmes of F of cardnalty S 3, and assume that S contans v, all fnte prmes whch ramfy n L/F, and all nfnte prmes. The Stark conjecture we are concerned wth must be formulated dfferently when S = 2, and s known to be true n ths case by [4] and [5]. S 0 = S {v}. S fn = the set of fnte prmes n S. p run through the fnte prmes of F not n S. a run through ntegral deals of F, prme to the elements of S. Na denote the absolute norm of the deal a. σ a G be the well-defned automorphsm attached to a va the Artn map. For each χ Ĝ, we have the Artn L-functon wth Euler factors at the prmes n S removed: L S (s, χ) = a ntegral (a,s)=1 χ(σ a ) Na s = prme p / S ( χ(σ p )) 1. 1 Np s It s known that L S (s, χ) has an analytc contnuaton and a functonal equaton relatng t to L S (1 s, χ). The order of ts zero at s =0s { S 1 r S (χ) = {q S : q splts completely n the feld fxed by the kernel of χ} dependng on whether or not χ s the trval character χ 0. See [5] for further background and references. 2

3 THE CONJECTURE. We frst sngle out the key equalty n Stark s refned abelan conjecture for frst dervatves of L-functons whch posts the exstence of a specal v-unt ɛ servng as an L-functon evaluator. Conjecture St (L/F,S). There exsts an element (often called a Stark unt ) ɛ U (v) L such that L S (0,χ)= 1 χ(σ) log( ε σ w ) for all χ w Ĝ. L σ G Remark 1. The condtons on ɛ specfy all of ts absolute values and thus determne ε up to a root of unty n L. Ths ambguty stll remans when we mpose Stark s addtonal condton below. Nevertheless, we sometmes refer to ε as the Stark unt. The full Stark conjecture n ths settng (cf. [4], [5]) says more. Conjecture St(L/F,S). St (L/F,S) holds, and furthermore L(ɛ 1/w L )/F s an abelan Galos extenson. II. STATEMENTS OF RESULTS We assume from now on that there are at least 2 nfnte prmes 1, 2 n S. Otherwse St(L/F,S) s known to be true by [4] (see also [5,IV.3.9]). We may then assume that 2 v. Also assume from now on that G = Gal(L/F ) s somorphc to (Z/2Z) m for some postve nteger m. We then call L/F a multquadratc extenson of rank m. Our am n ths paper s to prove the followng theorems. Theorem 1. Let S fn S consst of the fnte prmes n S, and let r F (S) denote the 2-rank of the S fn -class group of F.If S >m+1 r F (S), then St(L/F,S) holds for the multquadratc extenson L/F. Theorem 2. St (L/F,S) holds for the multquadratc extenson L/F, hence for an arbtrary multquadratc extenson. Theorem 3. St(L/F, S) holds for the multquadratc extenson L/F f v s a real nfnte prme or a fnte prme, except possbly when L s the maxmal multquadratc extenson of F whch s unramfed outsde of S and n whch v splts completely. Theorem 4. St(L/F, S) holds for the multquadratc extenson L/F when the rank of ths extenson s m =2,.e. L/F s bquadratc. Thus t holds for an arbtrary bquadratc extenson. Remark 2. St(L/F, S) was proved for the multquadratc extenson L/F n [2] and [3] under the assumpton that ether S >m+1, or that no prme above 2 (.e. no dyadc prme) s ramfed n L/F. 3

4 III. THE RELATIVE QUADRATIC CASE Assume K/F s a relatve quadratc extenson. Ths secton summarzes some basc results from [3] and [5] on St(K/F,S). We set: Gal(K/F) = τ of order 2. η K =1fS contans two splt prmes of K/F. η K = generator of the nfnte cyclc group U (v) K /µ K wth η K w < 1, otherwse. Note that snce 2 does not splt n ths case, we may also descrbe U (v) L as the S-unts u of L such that u 1+τ =1. If w s a real nfnte prme, choose η K to be postve n the embeddng nduced by w. Snce η 1+τ K = 1, ths then mples that η K s postve at both of the prmes above v. Cl F (S) =Cl F (S fn )=S fn -deal class group of F, the quotent of the deal class group Cl(F )off by the subgroup generated by the deal classes of the prmes n S fn. S K = the set of prmes of K lyng above those n S. Cl K (S) =S K -deal class group of K. H K = H K (S) =Cl K (S)/ι(Cl F (S)), the cokernel of the map ι nduced by extenson of deals. M K = M K (S) = H K, the order of ths group. Theorem(Stark-Tate, cf. [5, IV.5.4]). St(K/F,S) holds wth Stark unt ε K = η M K 2 S 3 K. Remark 3. The extra factor e + n [5, IV.5.4] equals 1 when η K 1as ths mples that the nfnte prme 2 of F does not splt n K. IV. PASSAGE TO THE MULTIQUADRATIC CASE VIA L-FUNCTION PROPERTIES We have assumed that L/F s multquadratc wth the dstngushed prme v of F splttng completely n L, and that 2 v s an nfnte prme of F. From now on, we also assume that: 2 does not splt completely n L/F. (Otherwse St(L/F ) s trvally true wth ε = 1.) Let: τ = complex conjugaton at 2 n L/F. K for =1, 2,...,2 (m 1) be the relatve quadratc extensons of F n L whch are not fxed by τ. (These generate L.) η = η K. M = M K. w = w K. 4

5 Proposton 1. If ε = 2 m 1 =1 η M 2 S m 2 (w L /w ) les n L, then t s the Stark unt ε L satsfyng St (L/F,S). Proof. (Ths s a straghtforward adaptaton of the proof of Theorem 2.6 of [3].) Clearly ε U (v) L because each η U (v) K U (v) L. In partcular, ε1+τ = 1 because ths represents the absolute value of ε above 2. We now show that ε s an L-functon evaluator. Fx an arbtrary character χ Ĝ. Ifχ(τ) = 1, then r S(χ) > 1 by the formula for ths quantty, and therefore L S (χ, 0) = 0. At the same tme, 1 w L σ G χ(σ) log( ε σ w )= w L σ G χ(σ) log( ε (1+τ)σ w )=0, by the observaton n the last paragraph. So ε s an L-functons evaluator for ths type of χ. Now suppose that χ(τ) = 1. The fxed feld of the kernel of χ must then be one of the K for some = (χ). Lettng G = Gal(L/K ), we observe that { 2 m 1, f = (χ) χ(σ) = 0, otherwse. σ G We use the defnton of ε, the fact that χ(τ) = 1, and the fact that G fxes η, along wth the evaluaton of the last sum and fnally the Stark-Tate theorem for relatve quadratc extensons and the nflaton property of Artn L-functons to see that 1 w L σ G χ(σ) log( ε σ w )= 1 w L = = = 2 m 1 =1 2 m 1 2 m 1 =1 =1 σ G 1 1 w 2 m m 1 2 m 1 χ(σ)(w L /w ) log( η M 2 S 3 σ w ) =1 σ G χ(σ) log( η M 2 S 3 σ w ) 1 1 w 2 m 1 χ(σ) log( η M2 S 3 (1 τ)σ w ) σ G 1 1 χ(σ) log( η M2 S 3 (1 τ) w 2 m 1 w ) σ G = 1 log( (η M S 3 (χ)2 w (χ) ) (1 τ) w )=L S(0,χ). (χ) So ε s an L-functon evaluator for ths type of χ as well, and the proof s complete. 5

6 Recall that H K =Cl K (S)/ι(Cl F (S)). V. CLASS FIELD THEORY Proposton 2. Let K be any of the K for whch η 1. Then rank 2 (H K ) rank 2 (Cl F (S)) = r F (S), wth equalty holdng f S =3. Proof. We wll show that the norm map nduces a surjectve homomorphsm H K /HK 2 Cl F (S)/Cl F (S) 2 whch s an somorphsm when S =3. The assumpton that η 1 mples that 2 and the other prmes of S 0 do not splt n K/F. Thus the complex conjugaton τ at 2 restrcts to a generator of Gal(K/F). Let I K denote the group of fractonal deals of K, P K denote the subgroup of prncpal fractonal deals, and I F denote the subgroup of fractonal deals of K whch are extended from fractonal deals of F. Also let S fn (K) denote the set of deals of K whch le above those n S fn. Ths contans all of the deals of K whch are ramfed over F. From the factorzaton of deals nto prmes, we see that I F S fn (K) = I 1+τ K J K S fn (K), where J K denotes the group generated by the prme deals of K whch are nert over F. Then H K /HK 2 = I K /P K IK 2 I F S fn (K) = I K /P K IK 2 I1+τ K J K S fn (K) = I K /P K IK 2 Iτ 1 K S fn(k) J K. Under the Artn map of class feld theory, I K /P K (= Cl K ) corresponds to the maxmal unramfed abelan extenson (Hlbert Class Feld) H of K, n the sense that ths map nduces an somorphsm of I K /P K wth Gal(H/K). From ths t s clear that I K /P K IK 2 corresponds to the maxmal abelan unramfed elementary 2-extenson of K n the same way. Smlarly, I K /P K IK 2 Iτ 1 K corresponds to the maxmal abelan unramfed elementary 2-extenson F of K havng the property that τ acts (by conjugaton) trvally on Gal(F/K). By maxmalty, F/F s Galos. Let G F = Gal(F/F ) and N F = Gal(F/K). Then N F s normal of ndex 2 n G F, and τ acts trvally on N F.SoN F and any lft of τ commute wth N F, whch suffces to show that N F les n the center of G F. Now G F /N F s cyclc of order 2, so that G F modulo ts center s cyclc. Ths mples that G F s abelan. Hence n fact F/F s an abelan extenson. We now know that I K /P K IK 2 Iτ 1 K corresponds to the maxmal unramfed elementary 2-extenson F of K whch s abelan over F. So I K /P K IK 2 Iτ 1 K S fn(k) corresponds to the maxmal such extenson L K of K n whch all prmes of S fn (K) splt completely. For each fnte or nfnte prme p S 0, let D p denote ts decomposton group n the abelan extenson L K /F. Under our assumptons, such a prme p does not splt n the quadratc extenson K/F, whle the prme P above t n K splts completely n L K /K. Thus D p = τ p has order 2. Let D be the subgroup of Gal(L K /F ) generated by all the D p for fnte and nfnte prmes p S 0 except 2. Hence D s an elementary abelan 2-group wth 2-rank rank 2 (D) S 2. Let L F be the fxed feld of D. Snce L F L K, no prmes ramfy n L F /K. Also, only prmes n S0 can ramfy n K/F. So only prmes n S 0 can ramfy n L F /F. The defnton of L F requres that the prmes n S0 other than 2 splt completely n L F /F. Hence L F /F can ramfy only at 2. Thus L F s contaned 6

7 n the ray class feld modulo 2 for F. But the ray classes modulo 2 are the same as the ray classes modulo 1, due to the presence of the unt 1. Thus L F /F s unramfed at 2 as well, and s therefore everywhere unramfed, wth all prmes n S splttng completely. The fact that 2 splts n L F /F but not n the quadratc extenson K/F mples that L F K = F. Thus the elementary abelan 2-group N L = Gal(L K /K) and the elementary abelan 2-group D = Gal(L K /L F ) generate the abelan group Gal(L K /F ), whch s therefore also an elementary abelan 2-group. Hence f q s any prme of F whch s nert n K, we may consder the Frobenus of the extended prme Q of K n the extenson L K /K and use the propertes of the Frobenus n relatve extensons (see [1, III.2.4]): σ(q, L K /K) =σ(q, L K /F ) 2 = 1. Ths shows that J K has trval mage n Gal(L K /K) under the Artn map. We return now to the somorphsm from I K /P K IK 2 Iτ 1 K S fn(k) to N L = Gal(L K /K) whch s nduced by the Artn map as descrbed above. Snce the mage of J K les n the kernel of ths somorphsm, we conclude that (1) H K /HK 2 = I K /P K IKI 2 τ 1 K S fn(k) J K = Gal(LK /K) =N L Now we observe that L F has an ntrnsc defnton n terms of F. Snce L F /F s an unramfed elementary abelan 2-extenson n whch all prmes of S splt completely, t s contaned n the maxmal such extenson, whch we denote by L F. Then L F K s an unramfed elementary abelan 2-extenson of K n whch all prmes of S fn (K) (ndeed S(K), as unramfed s the same as splt for the nfnte prmes) splt completely, and s abelan over F. But L K was defned to be the maxmal such extenson, so L K L F.As all prmes n S splt completely n L F /F, L F must be fxed by the decomposton groups generatng D. Ths means that L F L F. We conclude that L F = L F. Fnally defne D 0 = Gal(L K /(K L F )), whch has ndex 2 n D = Gal(L K /L F ). Thus D 0 s an elementary abelan 2-group wth rank 2 (D 0 ) S 3. Then we have an exact sequence: (2) 1 D 0 N L Gal(L F /F ) 1 Ths smply comes from the natural restrcton map dentfyng N L /D 0 = Gal((K L F )/K) wth Gal(L F /F ). Interpretng Gal(L F /F ) va the class feld theory of F,wehave (3) Gal(L F /F ) = I F /P F I 2 F S fn = Cl F (S fn )/Cl F (S fn ) 2 Thus n terms of class groups (usng (1) and (3)), the exact sequence(2) becomes (4) 1 C 0 H K /H 2 K Cl F (S fn )/Cl F (S fn ) 2 1, where the kernel C 0 s an elementary abelan 2-group of rank S 3 and the map on the rght s nduced by the norm map on deals. The concluson of the theorem follows. 7

8 Corollary 1. The nteger 2 r F (S) dvdes M when η 1. Proof. Ths s clear snce r F (S) = rank 2 (Cl F (S fn )) rank 2 (H K ) for K = K. VI. PROOF OF THEOREM 1 The assumpton s that S m +2 r F (S). In vew of Proposton 1, we consder ε = η M 2 S m 2 (w L /w ), where the product may clearly be taken over for whch η 1. For such, the expresson e = M 2 S m 2 s an nteger multple of 2 r F (S) 2 S m 2, by Corollary 1, and ths n turn s ntegral by assumpton. Thus e s ntegral and so ε = η e (w L /w ) does n fact le n L, snce each η les n K L. Then St (L/F,S) holds by Proposton 1. Furthermore ε 1/w L = η e /w, and each η 1/w les n an abelan extenson of F, by the Stark-Tate theorem. As the composte of abelan extensons s abelan, we conclude that ε 1/w L les n an abelan extenson of F. Ths completes the proof of Theorem 1. Let: m S = p S fn p VII. KUMMER THEORY L = L S be the composte of all quadratc extensons of F n Q wth relatve dscrmnant dvdng 4m S. O F be the rng of ntegers of F. O F (S fn ) be the rng of S fn -ntegers of O F. Lemma 1. Suppose [K : F ]=2. Then K = F ( γ) for some γ F whch generates a fractonal deal of F of the form (γ) =a 2 b wth b supported n S fn f and only f the relatve dscrmnant δ(k/f) of K over F dvdes 4m S. In partcular, f K/F s unramfed outsde S, then δ(k/f) 4m S. Proof. Frst suppose that K = F ( γ) wth (γ) =a 2 b and b supported n S fn. The relatve dscrmnant may be computed locally, so we reduce to the case of an extenson of local felds K P /F p by passng to the completons at a fxed arbtrary prme p of F and a prme P over p n K. That s, the p-part ( δ(k/f) ) of the relatve dscrmnant of K/F equals p the relatve dscrmnant of K P /F p, and t suffces to show that ths dvdes 4m S for each 8

9 p. Let π be a unformzng parameter for the rng of ntegers O p of F p. Then γ = uπ 2e+a where u s a unt of O p and a equals 0 or 1. So K P = F p ( γ)=f p ( uπ a ). We treat the two possbltes for a ndvdually. When a =0wehaveK P = F p ( u). Then the relatve dscrmnant δ(k P /F p ) dvdes the dscrmnant of the polynomal x 2 u whch s (4u) = (4), and ths clearly dvdes 4m S. When a = 1, t evdently must be the case that p dvdes b, and therefore p dvdes m S. We have K P = F p ( uπ) =F p ( π ), where π s another unformzng parameter. Thus K P /F p s an Esensten extenson for whch t s known that O P = O p ( π ). Therefore δ(k P /F p ) equals the dscrmnant of x 2 π, namely (4π )=4p. Agan ths dvdes 4m S, as p dvdes m S. Ths completes the frst half of the proof. Next assume that the relatve dscrmnant δ(k/f) ofk over F dvdes 4m S. Snce K/F s a relatve quadratc extenson, we know that K = F ( γ) for some γ F. Wrte (γ) =a 2 b, and b a square free fractonal deal. If a prme p appears n the factorzaton of b, let P be a prme above p n K. Then we are n the stuaton appearng n the frst half of the proof where a = 1 and K P /F p s an Esensten extenson. In ths case we saw that δ(k P /F p )=4p. We are assumng that ths dvdes 4m S, so may clearly conclude that p dvdes m S and thus p s n S fn. Ths shows that b s supported n S fn, and concludes the proof. Proposton 3. The feld L = L S contans L({ η : =1dots, 2 m 1 }). Proof. We show that L contans L({ η : =1...2 m 1 }) by showng that L contans L and each η. Frst, each K /F s a quadratc extenson, so K = F ( γ ). We may wrte (γ )=a 2 b wth b square free. Then K s ramfed at the dvsors of b, by Kummer theory. Snce K /F s unramfed outsde S, we conclude that b s supported n S fn. It now follows from the Lemma that K s a quadratc extenson of F wth relatve dscrmnant dvdng 4m S. But L was defned to be the composte of all such extensons. Thus L contans the composte of all the K, whch s L, as we observed n the begnnng of secton III. Havng shown that L contans L, we proceed to show that L contans each η. Ths s trval f η = 1, so we may assume that we are not n ths stuaton. Then the mage of η generates the nfnte cyclc group U (v) K /µ K. Thus η s not a square n K, and η does not le n F. So F ( η ) s an extenson of degree 4 over F. We know that η τ =1/η,so the conjugates of η over F are ± η and ±1/ η. Thus F ( η )/F s a Galos extenson of degree 4. It s n fact the composte of the relatve quadratc extenson K = F (η )n whch 2 ramfes, and the relatve quadratc extenson K = F ( η +1/ η ) n whch 2 splts. We have already seen that K les n L, so we now show that K les n L. Ths wll mply that the composte F ( η ) les n L, as desred. Above we saw that δ(k /F ) 4m S. In fact, δ(k /F ) 4m S 0, snce ths extenson s unramfed at v. Snce η les n U (v) K, the Lemma yelds δ(k ( η )/K ) 4v. Hence δ(k ( η )/F )=(N K /F δ(k ( η )/K ))δ(k /F ) 2, whch dvdes 16v 2 δ(k /F ) 2. Smlarly, δ(k /F )2 dvdes δ(k ( η )/F ), and thus dvdes 16v 2 δ(k /F ) 2. We conclude that δ(k /F ) 4vδ(K /F ). 9

10 We examne ths dvsblty statement one prme at a tme and show that t mples δ(k /F ) p 4m S for each p. Observe that K /F s unramfed outsde of S, soδ(k /F ) p = (1) for p not dvdng m S. Consequently δ(k /F ) p 4v whch dvdes 4m S n ths case. Now for p dvdng m S, the Lemma appled to K /F mples that δ(k /F ) p dvdes 4p whch n turn dvdes 4m S. Ths shows that δ(k /F ) dvdes 4m S. Hence K les n L, by ts very defnton. Proposton 4. [L S : F ]=2 r F (S)+ S Proof. Let {[a ]: =1,...,t} be a mnmal set of generators for the 2-torson subgroup Cl F (S fn )[2] of the S fn -class group Cl F (S fn ). So t = rank 2 (Cl F (S fn )) = r F (S). We vew Cl F (S fn ) as the group of nvertble deals modulo prncpal fractonal deals of O F (S fn ), the rng of elements of F whch are ntegral at all fnte prmes not n S fn. Usng Chebatorev s densty theorem, we choose the representatves a to be prme deals of O F (S fn ). The unts of ths rng are denoted U F (S fn ) and called the S fn -unts. Now a 2 = α O F (S fn ) for some α. Let A = {α : =1,...,t} U F (S fn ). We begn by notng that A = {α } U F (S fn ). For a non-trval element of {α } generates an deal whch s a non-trval product of the prme deals a, whle each element of U F (S fn ) generates the unt deal. Thus by the Drchlet-Chevalley-Hasse unt theorem, rank 2 (A) = rank 2 ( {α } ) + rank 2 (U F (S fn )) = t + S = r F (S)+ S. We wll establsh a one-to-one correspondence between the non-trval elements of A/A 2 and the relatve quadratc extensons K/F contaned n L. Ths mples that rank 2 (Gal(L/F )) = rank 2 (A/A 2 ), whch combned wth the dsplayed equalty yelds the statement of the proposton. Now observe that A (F ) 2 = A 2 as follows. If γ 2 (F ) 2 les n A, then γ 2 = αc u, for some u U F (S fn ). Hence γ 2 O F (S fn )= a2c and therefore γo F (S fn )= ac. The fact that ths s a prncpal deal generated by the a mples by ther defnton that all of the exponents are even, c =2b. We now have γ 2 = α2b u, and ths shows that u = v 2 s a square. Clearly v U F (S fn ), so γ = αb v, after choosng the correct sgn for v. From ths we see that γ 2 A 2, whch was to be proved. Gven a γ representng a non-trval class n A/A 2, ths wll correspond to the feld K = F ( γ). Accordng to the last paragraph, K wll n fact be a relatve quadratc extenson of F. We check that K les n L by showng that the relatve dscrmnant δ(k/f) dvdes 4m S. The fact that γ A means that γ = u α e for some ntegers e and some u U F (S fn ). Then the prncpal O F -deal generated by γ s (γ) = 2e ã (v) =a 2 b, where b = (v), and ã s the (prme) deal of O F supported outsde of S fn such that ã O F (S fn )=a. Snce b =(v) s supported n S fn, Lemma 1 allows us to conclude that δ(k/f) dvdes 4m S, as desred. Conversely, gven a relatve quadratc extenson K/F contaned n L, we wll produce the correspondng γ A. Frst we note that the relatve dscrmnant of a relatve quadratc extenson s equal to the fnte part of ts conductor, by the conductordscrmnant theorem. Thus every relatve quadratc extenson of F wth dscrmnant 10

11 dvdng 4m S s contaned n the ray class feld of F wth conductor equal to 4m S multpled by all of the nfnte prmes. Hence the feld L generated by all of these relatve quadratc extensons s also contaned n ths ray class feld. Then any quadratc extenson of F contaned n L wll have conductor dvdng the product of 4m S wth all of the nfnte prmes, so that ts dscrmnant also dvdes 4m S. We can conclude that the dscrmnant of our gven K dvdes 4m S. Lemma 1 now mples that K = F ( γ ), where (γ )=a 2 b and b s supported n S fn. Hence γ O F (S fn )=(ao F (S fn )) 2, so that ao F (S fn ) represents an element of Cl F (S fn )[2]. But ths group s generated by the mages of the a. Thus ao F (S fn ) = a c βo F (S fn ) for some β F. Then γ O F (S fn )= a 2c β 2 O F (S fn )= α c β2 O F (S fn ). Let γ = γ /β 2. We clearly have K = F ( γ )=F( γ), whle γo F (S fn )=( a c )2 O F (S fn )=( α c )O F (S fn ). Thus γ = u α c for some u U F (S fn ) and therefore γ A. Corollary We have [L : L] =2 r F (S)+ S m. 2. Let ζ be a generator of µ K. When η 1, the exponent M 2 S m 2 (w L /w ) s n 1 2 Z. If t s not n Z, then ether L = L or [L : L] =2and ζ / L. Proof. 1. From the fact that [L : F ]=2 m and Propostons 3 and 4, we conclude that [L : L] = 2 r F (S)+ S m, and thus ths ratonal number s n fact an nteger. 2. Now we can see that 2 r F (S)+ S m 2 =[L : L]/4 les n 1 Z. By Corollary 1, t follows 4 that M 2 S m 2 s n 1 4 Z, and that f t does not le n 1 Z, we must have L = L. In 2 ths case, note that the ambguty up to a root of unty n the choce of η allows us to conclude from Proposton 3 that L = L contans both η and ζ η and therefore ζ L. Thus w L /w s even and M 2 S m 2 (w L /w ) les n 1 Z. Fnally, snce 2 [L : L] s a power of 2, the only other stuaton n whch 2 r F (S)+ S m 2 =[L : L]/4 s not ntegral clearly occurs when [L : L] = 2 and t s half-ntegral. Then M 2 S m 2 s n 1 2 Z,soM 2 S m 2 (w L /w ) s ntegral unless w L /w s odd,.e. ζ / L. VIII. PROOFS OF THEOREMS 2 AND 3 Under our standng assumptons that L/F s multquadratc, and that n order to avod specal cases of the conjecture whch have already been proved, S contans at least two nfnte prmes and one other fnte or nfnte prme, we now have: ε = η M 2 S m 2 (w L /w ) by Proposton 1. The exponent M 2 S m 2 (w L /w ) s ether ntegral or half-ntegral when η 1,by Corollary 2. If t s half-ntegral for some, then ether L = L or we have both [L : L] =2and ζ / L, also by Corollary 2. If L = L, then η L for all, snce η L, by Proposton 3. If [L : L] = 2 and ζ / L for some, notce that both η and ζ η le n L by Proposton 3 agan and the ambguty n η. If nether of them le n L, then L( η )=L = L( ζ η ). Ths mples 11

12 that ζ L, whch s not the case. Hence ether η L or ζ η L. By renamng η, we may agan assume η L. Thus n all cases, Theorem 2 follows from Proposton 1. Turnng to the proof of Theorem 3, we now assume that v s ether real or fnte. When L s not the maxmal multquadratc extenson M of F whch s unramfed outsde of S and n whch v splts completely, we clam that 4 [L : L]. Then by Corollary 2, r F (S)+ S m 2 0, and the result wll follow from Theorem 1. To establsh the clam, we frst show that v s not splt completely n L S /F. When v s real, ths s clear snce the defnton of L S mples that 1 L S and thus L S s totally magnary. When v s fnte, we proceed by contradcton. Suppose v splts completely n L S /F. Then v s unramfed n L S, so clearly L S = L S 0. From Corollary 2, we then get 2 r F (S)+ S = [L S : F ] = [L S 0 : F ] = 2 r F (S 0 )+ S 0 = 2 r F (S 0 )+ S 1, so that r F (S) =r F (S 0 ) 1. Now 2 r F (S 0) = Cl F (S 0 )/Cl F (S 0 ) 2, so the class [v] ofv must be non-trval n ths group. By class feld theory, v s then not splt completely n the maxmal unramfed multquadratc extenson of F n whch every fnte prme of S 0 splts completely. However, ths extenson s contaned n L S, by Lemma 1, and we have assumed that v splts completely n L S, a contradcton. Let L v S denote the splttng feld of v n L S. By the clam we have just establshed, 2 [L S : L v S ]. Snce v splts completely n L L S, we also have L v S L and 2 [Lv S : L] unless L = L v S. Thus 4 [L S : L] unless L = L v S. From the defntons and Lemma 1 agan, t follows that L M L v S. Thus n the exceptonal case of L = Lv S, we have L = M, the maxmal multquadratc extenson of F whch s unramfed outsde of S and n whch v splts completely. IX. THE BIQUADRATIC CASE: PROOF OF THEOREM 4 We now assume that m = 2 and turn to the proof of Theorem 4. Snce S 3, Theorem 1 reduces us to the case where S = 3 and r F (S) = 0. By Remark 2, we may assume that some prme p 2 over 2 ramfes n L/F, so that S = { 1, 2, p 2 }, and we must have v = 1 splttng n L/F. (Ths s the only tme we wll make use of [3].) Then by Proposton 2, we have that M s odd for η 1. Thus ε = η 1 M 1 (w L /w 1 ) η2 M 2 (w L /w 2 ), wth both M odd. By Theorem 2, ε L satsfes St (L/F ) and ndeed the proof shows that we may take (w L /w ) η L for =1, 2. Temporarly fx = 1 or 2. Notce that 2 ramfes n K, and only one other prme (namely p 2 ) s allowed to ramfy over F. But some other prme must ramfy, for otherwse K s contaned n the ray class feld for F modulo 2. But the ray class group modulo 2 s the same as the ray class group modulo 1, due to the presence of the unt 1. Ths would mply that K s unramfed at 2, a contradcton. Thus η 1. It remans to check that L(ε 1/w L ) s abelan over F. For ths we use a standard lemma (see [5, p. 83, Prop. 1.2]). 12

13 Lemma 2. Suppose L/F s a fnte abelan extenson of number felds wth Galos group G. Let A be the annhlator deal of the group of roots of unty µ L consdered as a module over the group rng Z[G]. Let T be a set of Z[G]-generators for A. Then an element u n the multplcatve group L has the property that L(u 1/w L )/F s abelan, f and only f there exsts a collecton of a α L, ndexed by α T such that both of the followng condtons hold: a. a w L α = u α α T b. a β α = aα β α, β T. Recall that τ G s the complex conjugaton n L over 2. Also let τ 1 be the element of order 2 n G whch fxes K 1. Thus τ and τ 1 generate G. Frst consder the number of roots of unty w L n L. Suppose w L > 2. Then L has no real embeddngs, and the splt prme 1 of F must be complex, whle 2 s real. Hence F s a non-galos cubc extenson of Q and [L : Q] =12. If L contans a pth root of unty ζ p for some odd prme p, then L/F must ramfy at some prme over p, because F (ζ p )/F does. But the only fnte prme whch can ramfy n L/F s p 2 S. If L contans a 16th root of unty, then [L : Q] = 12 must be dvsble by φ(16) = 8, a contradcton. Thus the number of roots of unty n L s w L =2, 4, or 8. CASE I: w L =2. When w L = 2, the above arguments show that we may take K( η 1 )=L = K( η 2 ). Snce τ s a complex conjugaton and η 1+τ = 1 for =1, 2; we conclude that η 1+τ =1 for =1, 2. Usng ths, one can verfy that the condtons of Lemma 2 hold for ε = M 1 M 2 η1 η2 L upon settng a wl = ε, a 1+τ = 1, and a 1+τ1 = M η 1 1. Thus St(L/F, S) holds n ths case. Now f 4 w L, then L contans F ( 1), whch must be ether K 1 or K 2. By renumberng, we may assume that t s K 2. Thus w 1 =2. CASE II: w L =4. M Now w 1 =2,w 2 = 4, and ε = η 1 M 2 1 η2 L. We have notced above that L contans the square roots of all the roots of unty n the K. Thus L contans an 8th root of unty ζ 8. Put a wl = ε, a τ+1 = 1, and a τ1 3 = ζ 8 ( η 1 ) M 1 ( η 2 ) M 2. The argument above shows that η 1 and ζ 8 le n L, but not L, although ther squares le n L. Thus M ζ 1 8 η1 L, snce M 1 s odd. Also η 2 L, so we have confrmed that a τ1 3 L. Agan the condtons of Lemma 2 hold and St(L/F,S) s proved n ths case. CASE III: w L =8. 2M Now w 1 = 2, w 2 = 4, and ε = η 1 M 1 η 2 2 L. Put a wl = ε, a τ+1 = 1, and a τ1 3 =( η 1 ) M 1 ( η 2 ) M 2. We know that η 1 and η 2 le n L, but not n L, although ther squares le n L. Thus a τ1 3 L, snce M 1 and M 2 are odd. Agan the condtons of Lemma 2 hold and St(L/F,S) s proved n ths case. 13

14 Acknowledgements The second author thanks the Mathematcs Department of the Unversty of Texas at Austn for ts hosptalty whle much of the work of ths paper was beng carred out. Thanks also go to Crstan Popescu for several dscussons related to ths work. References 1. G. Janusz, Algebrac Number Felds, Academc Press, New York, J. W. Sands, Galos groups of exponent two and the Brumer-Stark conjecture, J. Rene Angew. Math. 349 (1984), J. W. Sands, Two cases of Stark s conjecture, Math. Ann. 272 (1985), H. M. Stark, L-functons at s = 1IV. Frst dervatves at s = 0, Advances n Math. 35 (1980), J. T. Tate, Les conjectures de Stark sur les fonctons L d Artn en s = 0, Brkhäuser, Boston, Dept. of Math. and Stat., Unversty of Vermont, Burlngton, VT USA Dept. of Math. and Stat., Unversty of Vermont, Burlngton, VT USA Dept. of Math., Unversty of Charleston, Charleston, SC USA 14