(1 α (1) l s )(1 α (2) a n n m b n n m p <ε

Transcription

1 L-INVARIANT OF p-adic L-FUNCTIONS HARUZO HIDA Let Q C be the fied of a agebraic numbers We fix a prime p>2and a p-adic absoute vaue p on Q Then C p is the competion of Q under p We write W = { x K x p < 1 } for the p-adic integer ring of sufficienty arge extension K/Q p inside C p We write Q p for the fied of a numbers in C p agebraic over Q p We consider a Dirichet series a n n s = ( ) a n ns = ( ) 1 (1 α (1) s )(1 α (2) s ) (1 α (d) s ) n=1 n=0 with an Euer product over primes Heuristicay, if n m n m p <εand a n b n p <ε for a integers n, we woud expect that a n n m b n n m p <ε n=1 n=1 up to some constant (the transcendenta factor of the L-vaues) at good integers s = m Even if s s mod p M (p 1) for g 0( s s p p M ), p s and p s may not be very cose p-adicay, whie s s p p 1 M if p Ifm is negative, we wi have further troube interpoating the L-vaues if we do not remove Euer p-factors Thus mod p cass of L-vaues is better represented by the L-vaue with a certain Euer p-factor removed Here is an exampe Start with a Dirichet character χ :(Z/N Z) Q with χ( 1) = 1 Here the word Dirichet character means that it is mutipicative and χ(n) =0ifn has a nontrivia common factor with N For positive integer m and m, as ong as n m n m p < 1 (that is, m m mod (p 1)), it is known from the time of Euer (1 χ(p)p m 1 )L(1 m, χ) (1 χ(p)p m 1 )L(1 m,χ) p < 1 By a work of Kubota Leopodt and Iwasawa, we have a p-adic anaytic L-function L χ (s) =Φ(γ 1 s 1) for a power series Φ(X) Λ=W [[X]] and γ =1+p such that L χ (m) =Φ(γ 1 m 1) = (1 χ(p)p m 1 )L(1 m, χ) for a positive integer m as ong as n m n p < 1 for a n prime to p The Iwasawa s anayticity L χ (s) =Φ(γ 1 s 1) guarantees that there are ony finitey many zeros (counting with mutipicity) of L χ (s) inw If we suppose χ = ( ) for a square free positive ( integer ) D, the modifying Euer factor vanishes at s = 1 if the Legendre symbo =1 (p) =pp in Z[ ] Date: August 8, 2008 A cooquium tak at TIFR on 8/7/2008 The author is partiay supported by the NSF grant: DMS , DMS and DMS p

2 L-INVARIANT OF p-adic L-FUNCTIONS 2 with p = { x Z[ ] x p < 1 } In other words, L χ (1) = 0 and λ 1 This type of zeros of a p-adic L-function is caed an exceptiona zero We may regard χ as a Gaois character Ga(Q[µ N ]/Q) =(Z/N Z) {±1}, χ and we remark that χ(frob p )=1to have the exceptiona zero For a given p-adic L-function, we write e for the number of (inear) Euer factors producing the exceptiona zero Here is another exampe Start with an eiptic curve E /Q defined by the equation y 2 =4x 3 g 2 x g 3 with g j Z If 4x 3 g 2 x g 3 0 mod has three distinct roots in F, the reduced curve E = E mod over F defined by y 2 4x 3 g 2 x g 3 mod remains to be an eiptic curve Counting the number of points of E (F ), we define a = P 1 (F ) E (F ) =1+ E (F ) Then the Hasse-Wei L-function of E twisted by χ is given by L(s, E, χ) = (1 a χ() s + χ() 2 1 2s ) 1 (Hasse) Take the Gaois representation ρ E of E on im n Ker(p n : E E) = Z 2 p Then L(s, E) = det(1 ρ E (Frob ) VI s ) 1 (Wei) Spit the Euer factor as a product of inear factors (1 a p p s + p 1 2s )=(1 αp s )(1 βp s ), if one of α and β, sayα, isap-adic unit (so, α p = 1), E has either ordinary or mutipicative reduction moduo p We suppose this ordinarity condition Then by the soution of Shimura-Taniyama conjecture by Wies et a, this L-function has p-adic anaogue constructed by Mazur such that we have Φ E (X) Λ with Φ E (ε(γ) 1) = (1 α 1 ε(p)) G(ε 1 )L(1,E,ε) Ω E for a p-power order character ε : Z p W ; in other words, L p (s, E) =Φ E (γ 1 s 1) Here Ω E is the period of the Néron differentia of E The ρ E (Frob p ) has eigenvaue 1 if and ony if E has mutipicative reduction mod p if and ony if E(C p ) = C p /qe Z as Gaois modues 1 L-invariant For a p-adic Gaois representation ρ acting on V = W d, we define L(s, ρ) = det(1 ρ(frob ) VIp p s ) 1, assuming that det(1 ρ(frob ) VI X) T [X] for a number fied T Q independent of We suppose that ρ is p-ordinary in the sense that ρ restricted to Ga(Q p /Q p ) is upper trianguar with diagona characters N a j on the inertia I p for the p-adic cycotomic character N ordered from top to bottom as a 1 a 2 0 a d Thus ( N a 1 ) 0 N ρ Ip = a 2 (Ordinarity) 0 0 N a d Under the existence of a good anaytic p-adic L-function L p (s, ρ) of Iwasawa type for ρ, wemake

3 L-INVARIANT OF p-adic L-FUNCTIONS 3 Conjecture 11 If the eigenvaue of Frob p contains 1 with mutipicity e, then L p (s, ρ) has zero of order e + ord s=1 L(s, ρ) and for a nonzero constant L(ρ) C p, L p (s, ρ) im s 1 (s 1) = e L(ρ)E+ (ρ) L(1,ρ) c + (ρ(1)), where c + (ρ(1)) is the transcendenta factor of the critica compex L-vaue L(1,ρ) and E + (ρ) is the product of nonvanishing modifying p-factors The probem of L-invariant is to compute expicity the L-invariant L(ρ) The L-invariant in the cases where ρ = χ = ( ) as above and ρ = ρe for E with spit mutipicative reduction is computed in the 1970s to 90s, and the resuts are Theorem 12 Let the notation and the assumption be as above (1) L(χ) = og p(q) = og p(q) ord p(q) for q C h p given by q = ϖ/ϖ, where h is the cass number h of Q[ ] and p h =(ϖ) (Gross Kobitz and Ferrero Greenberg); (2) For E spit mutipicative at p, writing E(C p )=C p /q Z for the Tate period q Q p, we have L(ρ E ) = og p(q) ord p(q) This was conjectured by Mazur Tate Teitebaum and ater proven by Greenberg Stevens Here og p is the Iwasawa ogarithm and x p = p ordp(x) Starting with a 2-dim p-adic Gaois representation for a number fied F, there is a systematic way to create many Gaois representations whose eigenvaues of Frob p contain 1 Take a symmetric n-th tensor of ρ E twisted by m times det(ρ) 1 Then det(ρ E) m has exceptiona zero at s =1ifn =2m Ifm is odd and F is totay rea, ρ 2m,m is critica at s = 1, and e = {p p} There is an arithmetic way of constructing p-adic L-function due to Iwasawa and others We can define Gaois cohomoogicay the Semer group ρ n,m = ρ n E Se M (ρ) H 1 (Ga(Q/M ),ρ Q p /Z p )(M = F, F ) for the Z p -extension F /F inside F (µ p ) The Gaois group Γ = Ga(F /F ) acts on H 1 (Ga(Q/F ),ρ Q p /Z p ) and hence on Se F (ρ), making it as a discrete modue over the group agebra W [[Γ]] = im n W [Γ/Γ pn ] Identifying Γ with (a subgroup of) 1+pZ p by the cycotomic character, we may regard γ Γ Then W [[Γ]] = Λbyγ 1+X By the cassification theory of compact Λ-modues, the Pontryagin dua Se (ρ) is pseudo-isomorphic to f Ω Λ/fΛ for a finite set Ω Λ The power series Φ ρ = f Ω f(x) is uniquey determined up to unit mutipe We then define Larith p (s, ρ) = Φ ρ (γ 1 s 1) Greenberg verified in 1994 the conjecture for this L p (s, ρ) except for the nonvanishing of L(ρ) (under some restrictive conditions) I aso did verify in my book from Oxford university press his conjecture under mider assumptions by an automorphic way If there exists a good anaytic way of making the p-adic L-function L an p (s, ρ) = Φ an ρ (γ 1 s 1) interpoating compex L-vaues, the main conjecture of Iwasawa s theory confirms Φ ρ =Φ an ρ up to unit mutipe An important point is to describe L without recourse to the above formua invoving L-functions Greenberg s computation of the L-invariant is via Gaois cohomoogy groups; for exampe, for Ad(ρ) = ρ 2,1 : He found a unique subspace Se cyc F (ρ) H 1 (F, Ad(ρ)) of dimension e = {p p} responsibe to the order e zero at s = 1 This space is represented by cocyces c p : G F = Ga(Q/F ) Ad(ρ) such that

4 L-INVARIANT OF p-adic L-FUNCTIONS 4 (1) c Dp ( 0 ) for the decomposition group D p at each p p; (2) c is unramified outside p and c moduo nipotent matrices is unramified over F p [µ p ] at a p p (automatic if F p = Q p ) Take a basis {c p } p p of L over K Write ( ) ap (σ) c p (σ) for σ D 0 a p (σ) p Then a p : D p K is a homomorphism His L-invariant is defined by ( ( ) 1 L 2,1 = det (a p ([p, F p ]) p,p ogp (γ)a p ([γ,f p ])) p,p ) The goa of this tak is to reate Greenberg s L-invariant with Gaois deformation theory, and give a coupe of conjectures on L n,m and the deformation ring 2 A conjecture Take an eiptic curve E /F for a totay rea fied F with spit mutipicative reduction moduo at every prime p p; so, E(F p ) = F p /qz p for q p F p Let Q p = N Fp/Q p (q p ) Conjecture 21 (L-invariant) Suppose that the motive Sym n (H 1 (E))(m) for m Z with 0 m<nis critica at 1 ( either n is odd or n =2m with odd m) Then if the arithmetic or anaytic L p (s, ρ n,m ) has an exceptiona zero at s =1, we have L n,m = p p og p (Q p ) ord p (Q p ) 3 Gaois deformation The Greenberg s Semer group Se cyc F (ρ) can be identified with the tangent space at the origin of the universa deformation space of ρ n = ρ n,0 Consider J 1 =( )and put J n = J1 n We have t ρ n (σ)j n ρ n (σ) =N n (σ)j n Define an agebraic group G n over Z p by G n (A) = { α GL n+1 (A) } t αj n α = ν(α)j n for the simiitude homomorphism ν : G n G m (note that G 1 = GL(2)) The Gaois representation ρ n has vaues in G n (Z p ) Consider the p-adic Lie agebra Ad(ρ n ) of the derived group of G n Then σ G F acts on Ad(ρ n )byx ρ n (σ)xρ n (σ) 1 Then (31) Ad(ρ n ) = ρ 2j,j j:odd, 1 j n Start with ρ n, and consider the deformation ring (R n, ρ n ) which is universa among Gaois representations: ρ A : G F G n (A) ρ n mod m A for oca artinian Q p - agebras A with residue fied Q p such that (Q n 1) unramified outside bad primes for E, and p; (Q n 2) ρ A Dp = α 0,A,p 0 α 1,A,p 0 0 α n,a,p with α i,a,p N n i p mod m A with α i,a,p Ip (i =0, 1,,n) factoring through the cycotomic inertia group Ga(F p [µ p ]/F p ) for a p p; (Q n 3) ν ρ A = N n for the goba p-adic cycotomic character N

5 L-INVARIANT OF p-adic L-FUNCTIONS 5 The universa coupe (R n, ρ n ) exists under (Q n 1 3) Each diagona character, say, j- th δ j from the top, of ρ Dp induces a Z p -agebra homomorphism of Z p [[X j,p ]] sending 1+X j,p to the vaue of δ j at the generator of the p-primary part of Ga(F p [µ p ]/F p ) Conjecture 31 We have R n = Qp [[X j,p ]] p p, j:odd, 1 j n for variabes X j,p induced by j-th diagona character of ρ n Dp ; in particuar, dim R n = e rank G n = e n 2 When F = Q and n = 1, the conjecture hods, and for genera totay rea F,it hods if n = 1 and ρ E mod p has nonsoube image (Wies/Tayor, Skinner/Wies, Fujiwara, Kisin, Khare/Wintenberger, Lin Chen) Since G 3 = GSp(4) is the spin cover of G 4 = GO(2, 3) Some progress has been made by A Genestier and J Tiouine towards the identification of Gaois deformation rings and GSp(4)-Hecke agebras (for F = Q), there is a good prospect to get a proof of Conjecture 31 when n = 3 and 4 Further, when F = Q, in view of the recent resuts of Coze Harris Tayor and Tayor (in the paper proving the Sato Tate conjecture for Tate curves), one woud be abe to treat genera n in future not so far away Conjecture 31 impies the L-invariant conjecture for Greenberg s L-invariant: Theorem 32 Suppose Conjecture 31 and that n is odd Then we have L(ρ 2j,j )=L(Ad(ρ n )) = ( ) ogp (Q p ) (n+1)/2 ord p (Q p ) j:odd,0<j n p p ( ) This foows from the fact that ρn X j,p ρ 1 n gives a canonica basis of Se X=0 cyc (Ad(ρ n )); so, we can compute Greenberg s ( L-invariant ) expicity For the abeian case, if χ = M/F for a CM fied in which a p p in F spits into PP with P h =(ϖ(p)), we aso get Coroary 33 Up to a simpe constant, for a haf subset Σ Σ c = {P p}, we have L(χ) = det ( og p (N P (ϖ(p) (1 c) )) ) P,P Σ P Σ ord, p(n p (ϖ(p) (1 c) ) where N P is the oca norm N Mp/Q p and c is a compex conjugation The above two resuts are obtained by expicity computing the universa representation ρ As for Coroary 33, we take a CM Hecke eigenform so that ρ = Ind F M ψ for a CM Hecke character ψ of Ga(Q/M ) Writing κ for the universa character deforming ψ whose restriction to Ga(Q p /F p ) factors through Ga(F [µ p ]/F p ), we have κ([u, M P ]) = (1 + X p ) og p (Np(u)/ og p (γ) ψ([u, M P ]) for P-adic unit u, and we get ρ = Ind F M κ By this fact, we can compute L(Ind Q F χ)=l(indq F Ad(ρ)) The genera case for a n>0 is treated in my paper appeared in IMRN 2007 Vo 2007, Artice ID rnm102, 49 pages doi:101093/imrn/rnm102, and the proof of the case: n = 1 is given in my book Hibert moduar forms and Iwasawa theory from Oxford University Press pubished in 2006