A MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET. March 11, 2018

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1 A MOD- ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET DIPENDRA PRASAD March, 208 Abstract. Followig the atural istict that whe a grou oerates o a umber field the every term i the class umber formula should factorize comatibly accordig to the reresetatio theory (both comlex ad modular) of the grou, we are led i the sirit of Herbrad-Ribet s theorem o the -comoet of the class umber of Q(ζ ) to some atural questios about the -art of the classgrou of ay CM Galois extesio of Q as a module for Gal(K/Q). The comatible factorizatio of the class umber formula is at the basis of Stark s cojecture, where oe is mostly iterested i factorizig the regulator term whereas for us i this aer, we ut ourselves i a situatio where the regulator term ca be igored, ad it is the factorizatio of the classumber that we seek. All this is resumably art of various equivariat cojectures i arithmetic-geometry, such as equivariat Tamagawa umber cojecture, but the literature does ot seem to address this questio i ay recise way. I tryig to formulate these questios, we are aturally led to cosider L(0, ρ), for ρ a Arti reresetatio, i situatios where this is kow to be ozero ad algebraic, ad it is imortat for us to uderstad if this is -itegral for a rime of the rig of algebraic itegers Z i C, that we call mod- Arti-Tate cojecture. As a attetive reader will otice, the most mior term i the class umber formula, the umber of roots of uity, lays a imortat role for us it beig the oly term i the deomiator, is resosible for all the oles! Cotets. Itroductio 2. The Herbrad-Ribet theorem 3 3. Proosed geeralizatio of Herbrad-Ribet for CM umber fields 7 4. Itegrality of Abelia L-values for Q 9 Refereces. Itroductio Let F be a umber field cotaied i C with Q its algebraic closure i C. Let ρ : Gal( Q/F) GL (C) be a irreducible Galois reresetatio with L(s, ρ) its associated Arti L-fuctio. Accordig to a famous cojecture of Arti, L(s, ρ) has a aalytic cotiuatio to a etire fuctio o C uless ρ is the trivial reresetatio, i which case it has a uique ole at s = which is simle. More geerally, let M be a irreducible motive over Q with L(s, M) its associated L- fuctio. Accordig to Tate, L(s, M) has a aalytic cotiuatio to a etire fuctio o C uless M is a twisted Tate motive Q[j] with Q[] the motive associated to G m.

2 2 DIPENDRA PRASAD For the motive Q = Q[0], L(s, Q) = ζ Q (s), the usual Riema zeta fuctio, which has a uique ole at s = which is simle. This aer will deal with certai Arti reresetatios ρ : Gal( Q/F) GL (C) for which we will kow a riori that L(0, ρ) is a ozero algebraic umber (i articular, F will be totally real). It is the a imortat questio to uderstad the ature of the algebraic umber L(0, ρ): to kow if it is a algebraic iteger, but if ot, what are its ossible deomiators. We thik of the ossible deomiators i L(0, ρ), as existece of oles for L(0, ρ), at the corresodig rime ideals of Z. It is thus aalogous to the cojectures of Arti ad Tate, both i its aim ad as we will see i its formulatio. Sice we have chose to uderstad L-values at 0 istead of which is where Arti ad Tate cojectures are formulated, there is a ugly twist by ω the actio of Gal( Q/Q) o the -th roots of uity throughout the aer, givig a atural character ω : Gal( Q/Q) (Z/), also a character of Gal( Q/L) for L ay algebraic extesio of Q, as well as a character of Gal(L/Q) if L is a Galois extesio of Q cotaiig -th roots of uity; if there are o o-trivial -th roots of uity i L, we will defie ω to be the trivial character of Gal(L/Q). We ow fix some otatio. We will fix a isomorhism of Q with C where Q is a fixed algebraic closure of Q, the field of -adic umbers. This allows oe to defie, a rime ideal i Z, the itegral closure of Z i C, over the rime ideal geerated by i Z. The rime will always be a odd rime i this aer. All the fiite dimesioal reresetatios of fiite grous i this aer will take values i GL ( Q ), ad therefore i GL (C), as well as GL ( Z ). It thus makes sese to talk of reductio modulo of (comlex) reresetatios of fiite grous. These reduced reresetatios are well defied u to semi-simlificatio o vector saces over F (theorem of Brauer-Nesbitt); we deote the reductio modulo of reresetatios as ρ ρ. If F is a fiite Galois extesio of Q with Galois grou G, the it is well-kow that the zeta fuctio ζ F (s) ca be factorized as ζ F (s) = ρ L(s, ρ) dim ρ, where ρ rages over all the irreducible comlex reresetatios of G, ad L(s, ρ) deotes the Arti L-fuctio associated to ρ. Accordig to the class umber formula, we have, ζ F (s) = hr w sr +r 2 + higher order terms, where r, r 2, h, R, w are the stadard ivariats associated to F: r, the umber of real embeddigs; r 2, umber of airs of comlex cojugate embeddigs which are ot real; h, the class umber of F; R, the regulator, ad w the umber of roots of uity i F. This aer cosiders ζ E /ζ F where E is a CM field with F its totally real subfield, i which case r + r 2 is the same for E as for F, ad the regulators of E ad F too are the same excet for a ossible ower of 2. Therefore for τ the comlex cojugatio o C, ζ E /ζ F (0) = L(0, ρ) dim ρ = h E/h F, w ρ(τ)= E /w F

3 A MOD- ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET 3 where each of the L-values L(0, ρ) i the above exressio are ozero algebraic umbers by a theorem of Siegel. I this idetity, observe that L-fuctios are associated to C-reresetatios of Gal(E/Q), whereas the classgrous of E ad F are fiite Galois modules. Modulo some details, we basically assert that for each odd rime, each irreducible C-reresetatio ρ of Gal(E/Q) cotributes a certai umber of coies (deedig o -adic valuatio of L(0, ρ)) of ρ to the classgrou of E tesored with F modulo the classgrou of F tesored with F (u to semi-simlificatio). This is exactly what haes for E = Q(ζ ) by the theorems of Herbrad ad Ribet which is oe of the mai motivatig examle for all that we do here, ad this is what we will review ext. 2. The Herbrad-Ribet theorem I this sectio we recall the Herbrad-Ribet theorem from the oit of view of this aer. We refer to [Ri] for the origial work of Ribet, ad [Was] for a exositio o the theorem together with a roof of Herbrad s theorem. There are actually two a riori imortat asects of the Herbrad-Ribet theorem dealig with the -comoet of the classgrou for Q(ζ ). First, the Galois grou Gal(Q(ζ )/Q) = (Z/), beig a cyclic grou of order ( ), its actio o the -comoet of the classgrou is semi-simle, ad the -comoet of the classgrou ca be writte as a direct sum of eigesaces for (Z/). We do ot cosider this asect of Herbrad-Ribet theorem to be imortat, ad simly cosider semisimlificatio of reresetatios of Galois grou o classgrous to be a good eough substitute. The secod ad more serious asect of Herbrad-Ribet theorem is that amog the characters of Gal(Q(ζ )/Q) = (Z/), oly the odd characters, i.e., characters χ : (Z/) Q with χ( ) = reset themselves as it is oly for these that there is ay result about the χ-eigecomoet i the classgrou, ad eve amog these, the Teichmüller character ω : (Z/) Q lays a role differet from other characters of (Z/). (Note that earlier we have used ω for the actio of Gal( Q/Q) o the -th roots of uity, givig a atural character ω : Gal( Q/Q) (Z/), as well as to its restrictio to Gal( Q/L) for L ay algebraic extesio of Q. Sice Gal(Q(ζ )/Q) is caoically isomorhic to (Z/), the two roles that ω will lay throughout the aer are actually the same.) To elaborate o this asect (the role of odd characters i Herbrad-Ribet theorem), observe that the class umber formula ζ F (s) = hr w sr +r 2 + higher order terms, ca be cosidered both for F = Q(ζ ) as well as its maximal real subfield F + = Q(ζ ) +. It is kow that, cf. Pro 4.6 i [Was], R/R + = 2 3 2, where R is the regulator for Q(ζ ) ad R + is the regulator for Q(ζ ) +. We will similarly deote h ad h + to be the order of the two class grous, with h = h/h +, a iteger.

4 4 DIPENDRA PRASAD Dividig the class umber formula of Q(ζ ) by that of Q(ζ ) +, we fid, L(0, χ) = h h χ a odd character of (Z/) , ( ) the factor / arisig because there are 2 roots of uity i Q(ζ ) ad oly 2 i Q(ζ ) +. It is kow that for χ a odd character of (Z/), L(0, χ) is a algebraic umber which is give i terms of the geeralized Beroulli umber B,χ as follows: It is easy to see that B,ω 2 L(0, χ) = B,χ = a= aχ(a). a= ( ) mod sice aω 2 (a) is the trivial character of (Z/) whereas for all the other characters of (Z/), L(0, χ) is ot oly a algebraic umber but is -adic itegral (Schur orthogoality!); all this is clear by lookig at the exressio: L(0, χ) = B,χ = Rewrite the equatio ( ) u to -adic uits as, χ a odd character of (Z/) χ = ω 2 = ω a= aχ(a). a= L(0, χ) = h h +, where we ote that both sides of the equality are -adic itegral elemets; i fact, sice all characters χ : (Z/) Q take values i Z, for χ = ω, L(0, χ) Z. This whe iterreted just a iterretatio i the otic of this aer without ay suggestios for roof i either directio! for each χ comoet o the two sides of this equality amouts to the theorem of Herbrad ad Ribet which asserts that divides L(0, χ) = B,χ for χ a odd character of (Z/), which is ot ω 2, if ad oly if the corresodig χ -eigecomoet of the classgrou of Q(ζ ) is otrivial. (Note the χ, ad ot χ!) Furthermore, the character ω does ot aear i the -classgrou of Q(ζ ). It ca hae that L(0, χ) is divisible by higher owers of tha, ad oe exects this is ot rove yet! that i such cases, the corresodig χ -eigecomoet of the classgrou of Q(ζ ) is Z/ (vall(0,χ)), i articular, it still has -rak. (By Mazur-Wiles [Ma-Wi], χ -eigecomoet of the classgrou of Q(ζ ) is of order (vall(0,χ)).) The work of Ribet was to rove that if B,χ, χ -eigecomoet of the classgrou of Q(ζ ) is otrivial by costructig a uramified extesio of Q(ζ ) by usig a cogruece betwee a holomorhic cus form ad a Eisestei series o GL 2 (A Q ). To be able to use the class umber formula i other situatios, we will eed to have the itegrality of L(0, χ) for χ a character associated to the Galois grou of a umber field, or eve of L(0, ρ) for geeral irreducible reresetatios ρ of the Galois grou

5 A MOD- ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET 5 of a umber field, i more situatios that we call mod Arti-Tate cojecture. We begi with the followig lemma. Lemma. Let l, be ay two odd rimes (l = is allowed). Let F be a totally real umber field, ad E = F(ζ ) be a quadratic CM extesio of F. Let h l (E) be the order of the l-rimary art of the classgrou of E o which comlex cojugatio acts by ; defie h l (Q(ζ )) similarly. The h l (Q(ζ )) h l (E). Proof. By classfield theory, it suffices to rove that a cyclic degree l uramified extesio, say L, of Q(ζ ) o which comlex cojugatio acts by o Gal(L/Q(ζ )) = Z/l whe iflated to E remais cyclic of degree l, i.e., the degree of LE/E is l. Assumig the cotrary, we have LE = E, ad sice degree of E/F is 2, ad is odd, we must have LF = F, i.e., L F, therefore comlex cojugatio must act trivially o L. O the other had, we kow that comlex cojugatio does ot act trivially o L. Remark. It may be oted that we are ot assertig that if Q(ζ ) E, h(q(ζ )) h(e). Let E be a CM umber field which we assume is Galois over Q. Assume that E cotais -th roots of uity but o + -st root of uity. Let F be the totally real subfield of E with [E : F] = 2. Let G = Gal(E/Q) with G, the comlex cojugatio i G. We have, ζ E (s) = L Q (s, ρ) dim ρ, ρ ζ F (s) = ζ E /ζ F (s) = L Q (s, ρ) dim ρ, ρ( )= L Q (s, ρ) dim ρ, ρ( )= where all the roducts above are over irreducible reresetatios ρ of G = Gal(E/Q). By the class umber formula, Similarly, h (E)/ = h (Q(ζ ))/ = L Q (0, ρ) dim ρ ρ( )= Dividig the equatio ( ) by ( 2), we have, h (E)/h (Q(ζ )) = L Q (0, χ) χ( )= ρ( ) =, ρ = χ ( ). ( 2). L Q (0, ρ) dim ρ ( 3), where the roduct o the right is take over irreducible reresetatios ρ of G = Gal(E/Q) for which ρ( ) =, ad which are ot cyclotomic characters of the form χ : Gal(Q(ζ )/Q) = (Z/ ) C. It is kow that L(0, ρ) Q for ρ( ) =. This is a simle cosequece of Siegel s theorem that artial zeta fuctios of a totally real umber field take ratioal

6 6 DIPENDRA PRASAD values at all o-ositive itegers, cf. Tate s book [Tate]. (Note that to rove L(0, ρ) Q for ρ( ) =, it suffices by Brauer to rove it for abelia CM extesios by a Lemma of Serre cf. Lemma.3 of Chater III of Tate s book [Tate].) By Lemma, the left had side of the equatio ( 3) is itegral (excet for owers of 2), ad we would like to suggest the same for each term o the right had side of the equatio ( 3). The followig cojecture about L(0, ρ) exteds the kow itegrality roerties of L(0, χ) = B,χ = a= a= aχ(a), ecoutered ad used earlier. The formulatio of the cojecture also assumes kow itegrality roerties about L(0, χ) for χ : Gal(Q(ζ )/Q) = (Z/) C discussed i the last sectio of this aer. Cojecture. (mod aalogue of the Arti-Tate cojecture) Let ρ be a irreducible reresetatio of Gal( Q/Q) cuttig out a CM extesio E of Q with ρ( ) = where is the comlex cojugatio i Gal(E/Q). The uless ρ is oe dimesioal reresetatio factorig through Gal(Q(ζ )/Q) (for some rime ) with ρ the reductio of ρ modulo beig ρ = ω, L(0, ρ) Q is itegral outside 2, i.e., L(0, ρ) Z[ 2 ]. We ext recall the followig theorem of Delige-Ribet, cf. [DR], which could be cosidered as a weaker versio of Cojecture. Theorem. Let F be a totally real umber field, ad let χ : Gal( Q/F) Q be a character of fiite order cuttig out a CM extesio K of F (which is ot totally real). Let w be the order of the grou of roots of uity i K. The, wl(0, χ) Z. I fact Cojecture ca be used to make recise the above theorem of Delige-Ribet as follows; the simle argumet usig the fact that the Arti L-fuctio is ivariat uder iductio from Gal( Q/F) to Gal( Q/Q) will be left to the reader. Cojecture 2. Let F be a totally real umber field, ad χ : A F /F Z character, cuttig out a o-real but CM extesio. The if L F (s, χ) Z, a fiite order () χ mod is ω. (2) χ is a character of A F /F associated to a character of the Galois grou Gal(F(ζ q )/F) for some q which is a ower of. Remark 2. I the examles that I kow, which are for characters χ : Gal( Q/Q) Q with χ = ω (mod ), if L(0, χ) has a (mod ) ole, the ole is of order ; more recisely, if L = Q [χ(gal( Q /Q ))] is the subfield of Q geerated by the image uder χ of the decomositio grou at, the L(0, χ) is the iverse of a uiformizer of this field L. It would be ice to kow if this is the case for characters χ of Gal( Q/F) for F arbitrary. This would be i the sirit of classical Arti s cojecture where the oly ossible oles of L(, ρ), for ρ a irreducible reresetatio of Gal( Q/F), are simle. Cojecture 3. Let F be a totally real umber field, ad let ρ : Gal( Q/F) GL ( F ) be a semi-simle modular reresetatio of the Galois grou Gal( Q/F) cuttig out a fiite CM extesio E of F with E ot totally real. Assume that ω ρ does ot cotai the trivial

7 A MOD- ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET 7 reresetatio of Gal( Q/F) where ω is the actio of Gal( Q/F) o the -th roots of uity. The it is ossible to defie L(0, ρ) F with L(0, ρ + ρ 2 ) = L(0, ρ ) L(0, ρ 2 ), for ay two such reresetatios ρ ad ρ 2, ad such that, if ρ arises as the semi-simlificatio of reductio mod- of a reresetatio ρ : Gal( Q/F) GL ( Q ) cuttig out a fiite CM extesio E of F with E ot totally real, the L(0, ρ) which belogs to Z by Cojecture has its reductio mod- to be L(0, ρ). The cojecture above requires that if two reresetatios ρ, ρ 2 : Gal( Q/F) GL ( Q ) have the same semi-simlificatio mod- ad do ot cotai the character ω, the L(0, ρ ) ad L(0, ρ 2 ) (which are i Z by Cojecture ) have the same reductio mod-. By a well-kow theorem of Brauer, a modular reresetatio ρ ca be lifted to a virtual reresetatio i ρ i i characteristic 0. However, sice L(0, ρ i ) may be zero mod-, for some i (for which i < 0), the theorem of Brauer does ot guaratee that L(0, ρ) ca be defied. 3. Proosed geeralizatio of Herbrad-Ribet for CM umber fields The Herbrad-Ribet theorem is about the relatioshi of L values L(0, χ) with the χ -eigecomoet of the classgrou of Q(ζ ). I the last sectio, we have roosed a recise cojecture about itegrality roerties for the L values L(0, ρ). I this sectio, we ow roose their relatioshi to classgrous. Let E be a Galois CM extesio of Q with F the totally real subfield of E with [E : F] = 2. Let G deote the Galois grou of E over Q. Let τ deote the elemet of order 2 i the Galois grou of E over F. As i the last sectio, ζ E /ζ F (0) = L(0, ρ) dim ρ = h E, h ρ(τ)= F w E where the roduct is take over all irreducible reresetatios ρ of G = Gal(E/Q) with ρ(τ) =, ad w E deotes the grou of roots of uity of E cosidered as a module for G. Let H E (res. H F ) deote the class grou of E (res. F). Observe that the kerel of the atural ma from H F to H E is a 2-grou. (This follows from usig the orm maig from H E to H F.) Therefore sice we are iterested i -rimary comoets for oly odd rimes, H F ca be cosidered to be a subgrou of H E, ad the quotiet H E /H F becomes a G-module of order h E /h F. Cojecture 4. Let E be a CM, Galois extesio of Q with F its totally real subfield, ad τ Gal(E/F), the otrivial elemet of the Galois grou. Let ρ : Gal(E/Q) GL ( Q ) be a irreducible, odd (i.e., ρ(τ) = ) reresetatio of Gal(E/Q) with ρ its reductio mod for a odd rime. Let ω : Gal(E/Q) (Z/) be the actio of Gal(E/Q) o the -th roots of uity i E (so ω = if ζ E). Write ρ = i ρ i, ad let ρ j be a comoet i this sum. The if ρ j = ω, ad L(0, ρ j ) = 0 F (cf. Cojecture 2) [H E /H F ] F cotais j dim(ρ) ρ j i its semi-simlificatio, with differet ρ s cotributig ideedetly to the semi-simlificatio of [H E /H F ] F, fillig it u excet for the ω -comoet. If ω =,

8 8 DIPENDRA PRASAD we make o assertio o the ω -comoet i [H E /H F ] F, but if ω =, there is o ω -comoet iside [H E /H F ] F (see remark 3 below). Remark 3. I the cotext of this cojecture, there is a questio i reresetatio theory which lays a role too: Give a fiite grou G, what are the fiite dimesioal irreducible reresetatios ρ : G GL ( Q ) such that the mod- reresetatio ρ cotais the trivial reresetatio of G i its semi-simlificatio? See the lemma below. I our cotext with CM fields etc., there is oe case which does ot ivolve dealig with this subtle questio. To elaborate o this, let G be a fiite grou together with a character ω : G (Z/), ad with a cetral elemet τ of order 2, i.e., τ 2 = ; the questio is to uderstad the fiite dimesioal irreducible reresetatios ρ : G GL ( Q ) such that the mod- reresetatio ρ cotais the character ω of G i its semi-simlificatio with ρ(τ) =? If the character ω =, the we further are give that ω (τ) =. However, ω : G (Z/) might be the trivial character corresodig to the CM field E havig o -th roots of uity. I this case, there are o such irreducible reresetatios ρ : G GL ( Q ) such that the mod- reresetatio ρ cotais the character ω =, because ρ(τ) = cotiues to hold mod-! Lemma 2. Let G be a fiite grou, ad a rime umber dividig the order of G. The the umber of trivial reresetatios of G aearig i the semi-simlificatio of F [G] equals P a (G), where P is a -Sylow subgrou of G, ad a (G) deotes the umber of trivial reresetatios of G aearig i the semi-simlificatio of F [G/P]. I articular, there are always o-trivial irreducible C-reresetatios of G whose reductio mod- cotais the trivial reresetatio of G. Proof. Observe that F [G] = Id G P IdP {e} F = Id G P (F [P]). Sice ay irreducible reresetatio of P (i characteristic ) is the trivial reresetatio, the semi-simlificatio of F [P] is the same as P coies of the trivial reresetatio. Coclusio of the lemma follows. Remark 4. The iteger a (G) itroduced i this lemma seems of iterest. For G = PGL 2 (F ), a (G) =. For ay grou G i which a -Sylow subgrou is ormal, or more geerally i a grou G with a -Sylow subgrou P, ad aother subgrou H of G of order corime to with G = HP, we have a (G) =. The author is grateful to Bhama Sriivasa for coveyig a examle of Paul Fog that for = 5, ad G = A 5, the umber a 5 (A 5 ) = 2. Remark 5. I should add that Ribet s theorem is secific to Q(ζ ) ad although this sectio is very geeral, it could also be secialized to a CM abelia extesio E of Q, ad the actio of the Galois grou Gal(E/Q) o the full class grou of E. Sice class grou of a abelia extesio is ot totally obvious from the classgrou of the corresodig cyclotomic field Q(ζ ), eve if we kew everythig i the style of Ribet for Q(ζ ), resumably there is still some work left to be doe, ad ot just book keeig (for which is a comosite umber)!

9 A MOD- ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET 9 4. Itegrality of Abelia L-values for Q The aim of this sectio is to rove certai results o itegrality of L(0, χ) for χ a odd Dirichlet character of Q which go as first examles of all the itegrality cojectures made i this aer. Although these are all well-kow results, we have decided to give our roofs. Lemma 3. For itegers m >, >, with (m, ) =, let χ = χ χ 2 be a rimitive Dirichlet character o (Z/mZ) = (Z/mZ) (Z/Z) with χ( ) =. The, is a algebraic iteger, i.e., belogs to Z Q. L(0, χ) = B,χ = m m aχ(a), a= Proof. Observe that B,χ = m m a= aχ(a), has a ossible fractio by m, ad that i this sum over a {, 2,, m}, if we istead sum over a arbitrary set A of itegers which have these residues mod m, the m a A aχ(a), will differ from B,χ by a itegral elemet (i Z). Sice our aim is to rove that B,χ is itegral, it suffices to rove that m a A aχ(a) is itegral for some set of reresetatives A Z of residues mod m. For a iteger a {, 2,, m}, let ā be a arbitrary iteger whose reductio mod m is a, ad whose reductio mod is. Similarly, for a iteger b {, 2,, }, let b be a arbitrary iteger whose reductio mod is b ad whose reductio mod m is. Clearly, the set of itegers ā b reresets exactly oce each residue class mod m, ad that ā b as a elemet i Z goes to the air (a, b) Z/m Z/. (It is imortat to ote that ā b as a elemet i Z is ot cogruet to ab mod m, ad therei lies a subtlety i the Chiese remaider theorem: there is o simle iverse to the atural isomorhism: Z/m Z/m Z/.) By defiitio of the character χ, χ(ā b) = χ (a)χ 2 (b). It follows that, [ ] [ ] m ā bχ(ā b) m m aχ (a) bχ 2 (b) Z. ( ) a= b= Sice the character χ is odd, oe of the characters, say χ 2 is eve (ad χ is odd). Observe that, B,χ2 = bχ 2 (b) = ( b)χ 2 (b). b= b= It follows that, 2 bχ 2 (b) = χ 2 (b) = 0, b= b= where the last sum is zero because the character χ 2 is assumed to be o-trivial. Sice m ā bχ(ā b) m m cχ(c) Z, c= by the equatio ( ), it follows that: m m cχ(c) Z, c=

10 0 DIPENDRA PRASAD as desired. Lemma 4. For a rime, let χ be a rimitive Dirichlet character o (Z/ Z) with χ( ) =. Write (Z/ Z) = (Z/Z) ( + Z/ + Z), ad the character χ as χ χ 2 with resect to this decomositio. The, L(0, χ) = B,χ = aχ(a), a= is a algebraic iteger, i.e., belogs to Z Q if ad oly if χ = ω. Proof. Assumig that χ = ω, we rove that B,χ belogs to Z Q. By a argumet similar to the oe used i the revious lemma, it ca be checked that, [ aχ(a) a= ] aχ (a) a= b= ( + b)χ 2 ( + b) Z. If χ = ω, a= aχ (a) is easily see to be itegral. To rove the lemma, it the [ ] suffices to rove that, b= ( + b)χ 2( + b) is itegral. Note the isomorhism of the additive grou Z with the multilicative grou + Z by the ma ( + ) + Z. Let χ 2 ( + ) = α with α =. The (the first ad third equality below is u to Z), ( ). b= ( + b)χ 2 ( + b) = = = = c= c= ( + ) c α c [α( + )] c [α( + )] α( + ) [ ( + ) ]. [ α( + )] Note that sice α = either α =, or α is a uiformizer i Q (ζ d) for some d. Therefore either = [ α( + )] if α =, or [ α( + )] is a uiformizer i Q (ζ d). Fially, it suffices to observe that, ( + ) mod, hece, b= ( + b)χ 2( + b) is itegral. If χ = ω, the same argumet gives o-itegrality; we omit the details. The followig roositio follows by uttig the revious two lemmas together, ad makig a argumet similar to what wet ito the roof of these two lemmas. We omit the details.

11 A MOD- ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET Proositio. Primitive Dirichlet characters χ : (Z/) Z ot belog to Z are exactly those for which: () = d. (2) χ = ω mod. for which L(0, χ) does The followig cosequece of the roositio suggests that rudece is to be exercised whe discussig cogrueces of L-values for Arti reresetatios which are cogruet. (Our formulatio of Cojecture 3 excludes the characters which are ivolved i the corollary below.) Corollary. Let, q be odd rimes with (q ). For ay character χ 2 of (Z/qZ) of order, defie the character χ = ω χ 2 of (Z/qZ). The although the characters ω ad χ have the same reductio modulo, L(0, ω ) is -adically o-itegral whereas L(0, χ) is itegral. Questio. Let χ : (Z/ d m) Z with (, m) =, m >, be a rimitive Dirichlet character for which χ = ω mod so that by Proositio, L(0, χ) is -itegral. Is it ossible to have L(0, χ) = 0 modulo, the maximal ideal of Z? Our roofs i this sectio are u to Z, so good to detect itegrality, but ot good for questios modulo. The questio is relevat to cojecture 4 to see if the character ω aears i the classgrou H/H + for E = Q(ζ d m ); such a character is kow ot to aear i the classgrou of H/H + for E = Q(ζ d). Ackowledgemet: The author thaks P. Colmez for suggestig that the questios osed here are ot as outrageous as oe might thik, ad for eve suggestig that some of the cojectures above should have a affirmative aswer as a cosequece of the Mai cojecture of Iwasawa theory for totally real umber fields roved by A. Wiles [Wiles] if oe kew the vaishig of the µ-ivariat (which is a cojecture of Iwasawa roved for abelia extesios of Q by Ferrero-Washigto). The author also thaks U.K. Aadavardhaa, C. Dalawat, C. Khare for their commets ad their ecouragemet. Refereces [DR] P. Delige, K. Ribet, Values of abelia L-fuctios at egative itegers over totally real fields, Ivet. Math. 59 (980), o. 3, [Lag] S. Lag, Cyclotomic fields I ad II. Combied secod editio. With a aedix by Karl Rubi. Graduate Texts i Mathematics, 2. Sriger-Verlag, New York, 990. [Ma-Wi] B. Mazur, A. Wiles, Class fields of abelia extesios of Q. Ivet. Math. 76 (984), o. 2, [Ri] K. Ribet, A modular costructio of uramified -extesios of Q(µ ), Ivet. Math., 34, 5-62 (976). [Tate] J. Tate, Les Cojectures de Stark sur les Foctios L d Arti e s = 0, Birkhauser, Progress i mathematics, vol. 47 (984). [Was] L. C. Washigto, Itroductio to Cyclotomic Fields, GTM, Sriger-Verlag, vol. 83, (982). [Wiles] A. Wiles, The Iwasawa cojecture for totally real fields. A. of Math. (2) 3 (990), o. 3, Diedra Prasad

12 2 DIPENDRA PRASAD Tata Isititute of Fudametal Research, Colaba, Mumbai , INDIA.