L-fucios ad Class Numbers Sude Number Theory Semiar S. M.-C. 4 Sepember 05 We follow Romyar Sharifi s Noes o Iwasawa Theory, wih some help from Neukirch s Algebraic Number Theory. L-fucios of Dirichle Characers. Dirichle Characers A Dirichle characer is a compleely muliplicaive fucio χ : Z C which is periodic wih some period ad saisfies χa) = 0 precisely whe a, ) =. A Dirichle characer ca also be regarded a characer χ : Z/Z) C. I is o erribly hard o see hese wo oios are equivale, ad we will use his equivalece wihou warig hough we will use he secod oio wheever possible). The coducor of a Dirichle characer χ : Z/Z) C is he smalles posiive ieger f such ha χ facors hrough Z/ f Z). Z/Z) χ C mod f Z/ f Z) χ We say ha a Dirichle characer is primiive if is coducor is equal o is period i.e. i does o facor as above). From ay Dirichle characer χ : Z/Z) C of coducor f we ca produce a associaed primiive Dirichle characer of period ad coducor f by akig he characer χ : Z/ f Z) C i he above diagram. If we regard Dirichle characers as maps Z C, he χ is he Dirichle characer wih leas period such ha χa) = χa) wheever a, ) =. Iuiively, we re fillig i as may zeros as possible i he o-primiive Dirichle characer). We say a Dirichle characer χ is eve or odd accordig as χ ) = or χ ) =, respecively.
Table : A Dirichle characer of period 4 ad is associaed primiive characer of period. Here ζ = e πi/6 is a 6 h roo of uiy. 0 3 4 5 6 8 9 0 3 period coducor pariy χ 0 0 ζ 0 ζ 5 0 0 0 ζ 0 ζ 4 0 4 odd χ 0 ζ ζ ζ 4 ζ 5 0 ζ ζ ζ 4 ζ 5 odd. L-fucios To every Dirichle characer χ we associae a L-series Lχ, s), defied by χ) Lχ, s) = s. This series coverges absoluely for s C wih res) >, ad coverges uiformly o res) > + ε for ay ε > 0). The complee muliplicaiviy of Dirichle characers imply ha he L-series has a Euler produc: for res) >. Lχ, s) = p prime χp)p s Theorem.. The L-series Lχ, s) has a meromorphic coiuaio o he whole complex plae which we also deoe Lχ, s)). I is o he rivial characer he Lχ, s) is i fac holomorphic, while i is rivial he ζs) = Lχ, s) has a simple pole wih residue a s =. This meromorphic coiuaio we call he L-fucio o. To express he special values of L-fucios ad heir fucioal equaios, we ll eed Gauss sums. The Gauss sum associaed o a Dirichle characer χ of period is τχ) = χa)e πia/. The ex lemma records some of heir basic properies. Lemma.. I is a primiive Dirichle characer he ad for all b Z, χb)τχ) = τχ) = Here χa) = χa) is he cojugae Dirichle characer)., χa)e πiab/. I order o give a fucioal equaio for our L-fucios, we make he followig defiios. Le δ χ = χ ) { 0 i is eve = i is odd,
a idicaor of wheher χ is eve or odd; le ε χ = τχ) i δ χ, some algebraic umber of absolue value ); ad le Λχ, s) = fχ π ) s/ ) s+δχ Lχ, s). Recall ha Lχ, s) is a produc over primes; we should hik of he -fucio i he above expressio as addig i he ifiie prime. We have he followig fucioal equaio for Λ. Theorem.3. For χ a primiive Dirichle characer,.3 Special Values Λχ, s) = ε χ Λχ, s). We are ieresed i hese L-fucios for heir special values a iegers). I order o compue hese special values we iroduce Beroulli umbers, alog wih a sligh geeralizaio. Defie he Beroulli umbers B for 0 by The iverse of his power series is e = B!. 0 e = + )!, 0 ad we ca use his fac o iducively compue he B. Table : The firs few Beroulli umbers. 0 3 4 5 6 8 9 0 B 6 0 30 0 4 0 30 5 0 66 Noe i paricular ha B = 0 for all odd > which ca be see by showig ha e + is a eve fucio). For χ a primiive Dirichle characer, defie geeralized Beroulli umbers B,χ for 0 by χa) e a e fχ = B,χ!. 0 I fac i his defiiio we ca replace by ay muliple of i, usig he ideiy r k=0 x k x r = x. Noe ha χ is odd, ad B, χ = 0 for eve. I geeral, B,χ = 0 for δ χ mod, wih he sigle excepio of B, = where deoes he rivial characer). Now we ca give he followig special values for our L-fucios. 3
Table 3: The firs few Beroulli umbers associaed o he primiive characer χ of Table.. 0 3 4 5 6 8 9 0 B, χ 0 4 3 i 0 3 + 3 3i 0 445 565 3 i 0 49 + 30049 3 i 0 83 38555 3i 0 Proposiio.4. Le χ be a primiive Dirichle characer. The for all iegers, we have Proof. Complex aalysis. Lχ, ) = B,χ. Theorem.5. Le χ be a o-rivial primiive Dirichle characer. The πiτχ) B f,χ i is odd, χ Lχ, ) = τχ) χa) log e πia/ i is eve. Proof. Odd case: fucioal equaio ad /) = π give fχ π Λχ, ) = ε χ Λχ, 0) ) / ) + Lχ, ) = τχ) fχ i π Lχ, ) = τχ)iπ Lχ, 0) Lχ, ) = τχ)iπ B,χ ) 0/ 0+ ) Lχ, 0) Eve case: usig he Gauss sum formula, he chagig he order of summaio, he recogizig he power series of log z), χ) Lχ, ) = = = τχ) τχ) e χa) πia/ χa)e πia/ = τχ) χa) log e πia/ ) Sice χ is eve we have τχ) = τχ), so = τχ)τχ). Also sice χ is eve ad we are summig over all a mod, we ca replace he log wih ) log e πia/ ) + log e πi a)/ ) = log e πia/. This chages he above equaio o as desired. Lχ, ) = τχ) χa) log e πia/ 4
ζ-fucios of Number Fields. ζ-fucios ad L-fucios Recall he orm of a ideal a O F o be Na) = [O F : a]. We defie a ζ-series for a umber field F by ζ F s) = Na) a O s, F where he sum is over o-zero ideals of O F. Noe ha ζ Q s) = ζs) is he classical Riema ζ-fucio. As i he case of L-series above, his ζ-series has a Euler produc, ζ F s) = Np) p O s, where he produc is over o-zero prime ideals of O K. These ζ-series also have a meromorphic coiuaio ad fucioal equaio. Now suppose F is a abelia exesio of Q. The Q F Qµ ) for some, ad GalF/Q) = GalQµ )/Q) / GalQµ )/F) = Z/Z) / GalQµ )/F) realizes GalF/Q) as a quoie of Z/Z). Give a characer of GalF/Q), we ca lif i o a characer of Z/Z), which has a associaed primiive Dirichle characer. Defie XF) o be he se of Dirichle characers produced i his way, i.e. he se of primiive Dirichle characers associaed o characers of Z/Z) ha facor hrough GalF/Q). We have he followig relaioship bewee he ζ-fucio of F ad he L-fucios of hese Dirichle characers. Proposiio.. For F a abelia field, ζ F s) = Lχ, s). χ XF) Noe by examiig boh sides poles a s =, we ca see ha Lχ, ) = 0 for a o-rivial characer χ, ad his ca be used o prove Dirichle s heorem o primes i arihmeic progressios.) Proof. Le p be a prime which decomposes i F as p = p p r ) e, wih Np i ) = f. The p coribues p f s ) r o he Euler produc of ζ F s), ad coribues χ XF) χp)p s ) o he produc of L-fucios. We wa o show ha hese are he same. Seems like his is doe esseially by carefully examiig how primes spli, bu he res of he proof is clear o me. For example, cosider F = Qi). The GalQi)/Q) = Z/4Z), which has wo associaed primiive Dirichle characers: he rivial characer, ad he characer χ defied by χ) = { if mod 4, if 3 mod 4. I his field ramifies, primes p mod 4 spli compleely, ad primes p 3 mod 4 are ier. 5
Thus. Regulaors ζ Qi) s) = s ) p mod 4 p s ) p 3 mod 4 p s ) = s ) p mod 4 p s ) p 3 mod 4 p s ) + p s ) = L, s) p s ) p mod 4 = L, s)lχ, s). p 3 mod 4 + p s ) Le F/Q be a umber field. Le r = r F) be he umber of real embeddigs of F, ad r = r F) he umber of cojugae pairs of complex embeddigs. Deoe is discrimia by d F. Say ha a se of uis i O F is idepede if he subgroup of OF i geeraes is free abelia, wih he chose uis as geeraors. Le r = rak Z OF = r + r, ad choose embeddigs σ,..., σ r+ : F C correspodig o he archimedea places of F icludig oe of each cojugae pair of complex embeddigs). Defie he regulaor of a se α,..., α r of uis o be where c log σ α ) c log σ α r ) R F α i ) = de.. c r log σ r α ) c r log σ r α r ) c i = { if σi is real, if σ i is complex. Noe ha we omi σ r+ from he defiiio of regulaor. The choice of embeddig o omi does o affec he resul, because for ay α OF we have r+ i= r+ c i log σ i α) = log i= σ i α) c i = 0, so he omied row wih eries c r+ log σ r+ α j ) is mius) he sum of he rows of he marix. The regulaors of differe ses of uis have he followig relaio. Lemma.. Suppose B = µ β,..., β r A = µf) α,..., α r for β i ad α i idepede ses of uis, ad r = rak Z O F. The R F β i ) = [A : B]. R F α i ) Thus idepede ses of uis geeraig he same subgroup of OF /µf) have he same regulaor. We defie he regulaor R F of F o be he regulaor R F α i ) of a se of uis α,..., α r wih O F = µf) α,..., α r. 6
.3 Class Number Formula Theorem.3. The ζ-series of a umber field F has a meromorphic coiuaio which we also deoe ζ F ) o he whole complex plae, wih he oly pole a simple pole a s =. Seig Λ F s) = r π [F:Q] d F /) s s ) r s) r ζ F s), we have he fucioal equaio Λ F s) = Λ F s). The ζ-fucio of F ecodes much of he arihmeic daa of F i is pole a s =. Le w F = #µf) be he umber of roos of uiy i F. Theorem.4. For a umber field F, he ζ-fucio ζ F has a simple pole a s = wih residue res s= ζ F s) = r π) r h F R F w F d F /. Sice he Riema ζ-fucio i.e. he Dirichle L-fucio of he rivial characer) has a simple pole a s = wih residue, we ca combie Theorem.4 wih Proposiio. o obai he aalyic class umber formula. Theorem.5 Aalyic class umber formula). Le F be a umber field. The χ XF) χ = Lχ, ) = r π) r h F R F w F d F /.