a cuspidal class number formula for the modular curvesx1(n)

Math. Ann. 252, 197-216 (1980)

© by Springer-Verlag 1980

A Cuspidal Class Number Formula for the Modular Curves X,(N) Jing Yu Mathematics Department, Yale University, New Haven, CT 06520, USA

Table of Contents

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 1. Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 2. A Variation of the Stickelberger Ideal . . . . . . . . . . . . . . . . . . . . . . . 202 3. Computation of the Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 4. Cuspidal Divisor Classes on X I ( N ) . . . . . . . . . . . . . . . . . . . . . . . . . 212 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Introduction

Let N > 4 be an integer. A cusp on the modular curve X~(N) is said to be of the first type if it is lying above the cusp 0 on X0(p ) for every prime number p which divides N. We consider:

f f ° (N)= the group of function on X,(N) whose divisors have support within those cusps of the first type,

@°(N)=the divisor group of degree 0 on X,(N) whose support lies among the cusps of the first type,

and

cg°( N) = ~°( N)/diwY;°( N) .

(~°(N) is a subgroup of the full cuspidal divisor class group on the modular curve Xa(N ). We are interested in its order h°(N).

Our main result is the following class number formula:

h°(N) = 1--[ pL(p) ]-] iB2. z ( 1 - p •(p)) , pin z * l

Present address : Institute of Mathematics, Academia Sinica, Taipei, Taiwan

0025-5831/80/0252/0197/$04.00

198 .r. Yu

where

L, . f~o(N/pn~P~).(p n~p)-l- 1 ) -2n (p ) -2 if N is composite. L v ) = ~ f - ~ - 2 n - 2 if N = p ~ > 4 .

ftp) = the highest power of p dividing N Z= primitive characters associated to the group

G = ( z / N z ) * / _ + 1.

Bz,z=the generalized second Bernoulli numbers belonging to Z.

The prime power case was previously obtained by Kubert-Lang [KL 8]

h°(P")=P up) I-I ½B2,z' x¢-I

We note that in this case, all the factors of Euler type reduce to 1. If N is a product of more than one distinct primes, our class number is simply

13 [½B2. 1-Ift-p2z p) l. z*l [ t in J We consider those functions in the group o~-°(N) which can be generated in a

specific manner from a suitable set of Siegel functions. We then analize the group of functions thus obtained, proving that it has not only the maximal possible rank, but also no cotorsion inside the original group ~°(N), therefore it is the whole group of functions ~'°(N). This description of ~-°(N), then enables us to represent our divisor class group as a quotient Ro/J, where R o is the augmentation ideal in the group ring R = Z[G], and J is a variation of the usual Stickelberger ideal. We are therefore concerned with a purely algebraic question of determining the index of our ideal J inside R 0.

Determining indices of Stickelberger ideals is certainly a wellknown move- ment. For Stickelberger ideals of order 1 (formed with the first Bernoulli numbers and relevant for the ideal class groups of cyclotomic fields) this was done by Iwasawa [Iw] in the prime power case, and extended by Sinnott I-Si] to the composite case. In carrying out our computation for a "constrained" Stickelberger ideal of order 2, we combined the approach of Kubert-Lang [KL 8] with that of the Sinnott computation for cyctotomic fields. This will occupy us in the first three chapters.

As a final remark, we note that the disappearance of cotorsions for groups of Siegel functions is a subtle phenomenon. For the group of all Siegel functions, 2-cotorsion indeed exist, once N is composite, cf. [KL 4]. It is fortunate that in our situation here, the 2-cotorsion always disappears, even when N is composite.

1. Algebraic Preliminaries

Let N be a fixed positive integer, G= G(N)=(Z/NZ)*/+ 1. We write the elements of G also in the form %, with b~(Z/NZ)*/+ 1. For prime number p, we let Np denotes the highest power of p dividing N. We

associate to every positive divisor f of N, a subgroup of G:

H r = the set of ab, b = + 1 (modf) and (b, N) = 1.

Cuspidal Class Number Formula 199

Clearly H N = { 1 }, H 1 = G, and HI, D H r 2 if f l If2. Moreover:

Lemma 1.1. Let ft , f2 by any two divisors of N, then

Hf1" H f2 = H(f t , f2)"

Proof Given an integer a - 1 mod (fl,f2), it suffices to find a and fl such that

g - - - l (modf0 , fl-=l(modf2), and gf l -a(modS) .

Let flp be the highest power of the prime p dividing f r Let m 1 be the product of all flp, with flp.p still dividing f r It is then clear that mll(fl,f2 ). We may write (A,f2)=mlm2 •

Consider a, fl to satisfy the following:

~=a(mod I-I N,], ~=a(modp~ Np) \ plmt ]

It is immediate that eft congruent to a mod N = 1-[ Np. On the other hand, e - a ( m o d I - I N p t implies e - a ( m o d m l ) . Since

- / \ plml a - 1 (mod m~), this ensures ~ = 1 (rood m 0, therefore

e = l ( m o d f 0 , because film1 ~I N, . pXml

In a similar way, one also obtains f l - 1 (modf2). QED

Let R = ZIG], R o the augmentation ideal in R, that is

R o = the set of elements ~ u(b)a b with ~ u(b) =0.

For each subset C of G, let S(C)-- ~,, % then S(C)eR. beC

If Z is any character of G, we write e x for the corresponding idempotent associated to X in the group ring C[G] :

1

s(6) Note that when Z is trivial, e x = e~ = [GI

For any prime number p, we also define the following element:

a , = 2 2(P)ex, l

where ~ denotes the complex conjugate of the primitive character associated to Z. Note that when N = p", a prime power, then ~p = 0.

We shall always extend characters to the group ring via linearity. We also fix an integer t(p) prime to N, for each p, satisfying:

t(p)-- 1 (mod Np)

t(p)- p (mod N/Np) .

200 J. Yu

- -I S(HN/s~) for all prime p, pIN. L e m m a 1.2 . o-p = o't~ ) lI-1N:s,t

Proof. I t suffices to show that for all characters X,

1 z(S(Hs/N~)) z(~)= z(tfp))- IHN/~,I

Let X be any character of G with conductor fx" If Plfz, • is not of conductor dividing N/Np, hence ~ is non-trivial on HNm p and

z(s(H~,:,))=O= ~,(p).

If pXfz, then fxl(N/Np), and Z(t(p))- 1 = )~(p), by the definition of t(p).

We also have ~ : ~ 1 = 1, as Z is now trivial on HN/N. This proves the \ IHN/~,I ,/

lemma. Now, let the group G = G(N) act via multiplication on the set

TN = (Q/Z)N/+ ~,

where (Q/Z)N denotes the subgroup of Q / Z consisting of elements of order N. We shall parametrize T N by cosets of the various subgroups H:. For each a s T N, let f(a) be its (exact) period, and let d(a)= N/f(a). Define

C, = the set of a~, with d(a)c--- +_ Na (mod N).

C, is a coset of the subgroup Hf(a) , and every coset of H: in G is equal to C,, for an unique element a in T N having period f(a)= f.

I f f (al) f=f(a2) holds for a 1 and a 2 in TN, then obviously

CaiDC,~ if and only if fa2=a 1.

For each prime pIN, we let 1 Z/Z operates on T N by addition. P

We need to look more closely at the various orbits of 1 Z/Z. P

Suppose a s T N, with p2lf(a), then we find f(a)=f(a'), for all a' in the same orbit

o f l z / z as a. In other words, the period is constant on such kinds of orbits. On the P

other hand, suppose a e TN with (p,f(a))= 1, then f(a')=f(a)p, for all a'4~a in the

same orbit o f l Z / Z as a. Moreover, every a ' s T s with plf(a'), but p2Xf(a), lies in an p

orbit of 1 Z/Z containing some a with f(a') = f(a)p. P

Passing to the various cosets in G, we find the following:

For an orbit K of 1 Z/Z in T N with constant period, P

,UrC.=Cp,,, for any a'sK.


For those orbits K of 1 Z/Z containing some a with (f(a), p)= 1, P

a~r Ca'= ,'~KU Co'=Cva", for any a"~K, a"4:a. plf(a') a" ~=a

Lemma 1.3. Let K be an orbit o f l z / z , a ,a ' eK with f ( a ) = f ( a ' ) = f I f pXf' , then P

co .H e, = Co, .H~,.

Proof We may assume, without loss of generality, that Plf It suffices to prove the equality C a.Hf = Cpa .Hf. Write C a .H f, = tr cH / . H y, = a cH~y,,S ~ by Lemma 1.1. Then C,~ . H I, = a c H f /p • H f = tr cH( f /p , f , ). Now PXf' implies ( f /p , f ' )=(f , f ' ) , thereby gives

Cpa. H f , = C a. H f , .

This proves the lemma.

Lemma 1.4. Let a~K, an orbit of l z / z , ( f(a),p)= 1. Then P

tr t~v) 'Ca= U C,, a' ~K a*a'

=Cva,,, for any a"eK , a"~a .

ct Proof Let a '= a + --, ~ =~ 0, it suffices to show that

P

d(a)c =- +_ Na (mod N)

implies

t(p)d(a)c = +_ Na'p (mod N).

This follows from the definition of t(p) as follows:

t(p)d(a)c- pd(a)c (mod N/Np)

= + Nap (mod N/N, )

=- +_ Na' p (mod N/Nv)

t(p)d( a)c - d( a)c (mod Nv)

= 0 (mod N v )

=- + Na'p (mod Nv)

thereby proving the lemma. Combining Lemmas 1.3 and 1.4, we also obtain:

Corollary 1.5. Let K be an orbit of 1 Z / Z containino a, with ( f (a) ,p)=l . Then P

trt~v). C a = C,,. Hu/~p, for any a' 4= a in K.

202 J. Yu

Finally, one more notational convention. We denote by N o, the product of all distinct primes dividing N. We reserve r to denote the divisors of N o, corresponding to the subsets of the set of primes dividing N.

2. A Variation of the Stickelberger Ideal

Let B2(X ) = X 2 - X +-~, the second Bernoulli polynomial. To each a t Ts, we define a Stickelberger element in Q[G]:

O(a)N = N b~ B2(<ba>)ffb- 1 '

where, as usual, (x> denotes the smallest number > 0 in the residue class modulo Z of a rational number x.

These elements are permuted by G, in fact acO(a)s = O(ca)N. Hence

O(a)N=acO\N ] , for ac~C a and a~T x.

We want to consider the set of all elements of the form

~, m(a)O(a)N, a * Oe TN

where the coefficients re(a) are integers and satisfying:

DQ. The quadratic relation mod N, that is

~. m(a) (Na)2 _ 0 (mod N), for N odd.

~" m(a) (Na) 2 -- 0 (mod 2N), for N even.

DO. For each piN, the following linear condition on orbits of 1 Z/Z: P

m(a)=0 for aU orbits K of 1Z/Z . a~r P

Since G permutes all orbits of 1 Z/Z, these elements from an R-module J in Q[G]. P

We shall see later that J is in fact an ideal in R o. We observe first that:

Lemma 2.1. I f N is composite, DO implies DQ.

Proof.. It suffices to show that the following congruences hold:

~. re(a) (Na) 2 - 0 (mod Np)

~. re(a) (Na) 2 = 0 (mod 2N2)

Let q~-p, qlN be another

for p 4: 2, plN .

for p = 2, if N is even.

prime. Consider the action of 1Z/Z. Claim: q

! ~. m(a)(Na) 2 - 0 (rood 2Np) for all orbits K of L Z/Z. Our lemma then follows by ,~K q


partitioning T N into orbits. Choose an a' in each orbit K. From DO we have:

~, re(a) (Na) z = ~, re(a) [(Na) 2 - (Na')2]. a ~ K a' * a

On the other hand, since a and a' are in the same orbit,

Na = Na' (mod N/q)

Na = Na' (mod Np)

(Na) 2 = (Na') 2 (rood 2Np).

Therefore

~, re(a)(Na) 2 =-0 (mod 2Np). aEK

This proves the lemma,

Lemma 2.2. I f DO holds for {re(a)}, then

m(a)=O for any f, f[N. a~TN

f ( a ) = f

Proof. Suppose first that there is a prime P,P:If, then the set of all a in T~ with

f(a) = f can be partitioned into orbits of 1 Z/Z. Thus our lemma is immediate in P

this case. There remains to consider the case f = r, a product of distinct primes. This is proved by induction on the number of primes dividing f = r. First

~ ' m(a)=O aeTN

f ( a ) = p

because those a, together with 0, form an orbit of ! Z / Z is T N. P

Assume that our lemma is proved for all f = r' with r - 1 distinct prime factors. We then consider the case f = r = pr', a product of n distinct primes. We partition all those a in TN, with period f(a) either equal to r or equal to r', into orbits of 1 -z/z. P

By DO, we then have

~, re(a)+ ~ m(a)=O. a~TN a ~ T ~

f ( a ) = r" f ( a ) = r

But

Z re(a) = 0 , a~TN

f ( a ) = r"

by induction hypothesis. This completes the proof of our lemma.

204 J. Yu

Let J~ be the ideal in R consisting of elements of the form

/de\dc ~, ~ dm{~yr}ac= ~, ~ dm(a)S(C,,)

¢~G No6IN \~" / f a No[fIN f(a) = f

with the families {rn(a)} satisfying the conditions DQ and DO. In view of Lemma 2.2 above, one sees that dl CRo. We are going to prove:

Theorem 1. J =J1 .co. ~I (1 -p2~v), where oo=½ ~. B2,~e z and B2, z is the Bernoulli v[N z * 1

numbers formed with the character Z

ceG(f x)

Lemma 2.3. For f i n and df = N, we have

( 1 - e l ) 0 ( d ) =dco. S(H :). I-[ (1-- p~v). N Pl f

Proof. Let ~( be any character of G, non-trivial, with conductor fz" If fz'l'f, then Z(c)4:1 for some c in He, thus

o[ d'~ o:dc~ o: d'i

=,<c, =o

On the other hand, ~f(S(H:)) also equals 0, hence both sides are equal. Iffz[f, then

b~G(N)

= dJ :1½8:, . I-I (1 vlf

This last step follows the well-known distribution relation for the Bernoulli polynomials, cf. I-KL, Chap. 1, Theorem 2.1].

Hence ~(o(d) ) = (dc°'S(Hy)" l-I (l-P~v)) ' if vl:

For ;f= 1, trivial, X(1-el)=0=X(co). Consequently

X((I\ -et)o(dlv •/N:')=~(dc°'S(HI"~ vI: ['[ (1 - P~v)) ' for all X.

This proves the lemma.

C u s p i d a l Class N u m b e r F o r m u l a 205

Consider

, ,S(C°) ,~{d~ I; Y in J.

f i n f(a~= f

Let r(f), be the product of all distinct primes dividing N, but not dividing f Using the above Lemma and the fact that ~ .e I =deg(~)-e 1, we have

l =o9-~, ~ dm(a)S(C,) . I-I (1-pSv)

f f a pIN No IN f(a)= f

+"Zf <'I S'~ 2,, dm(a)'S(Ca"p~ls(1-PSp) ] •

Lemma 2.4. 1; dmfa)S(C,).~p=O, if p2lf for some p. a

f(a) = f

Proof For any a and a' in the same orbit of 1 Z/Z, we have P

S(Ca).~p=S(C,,).~ p, by Lemmas 1.2 and 1.3.

Hence we can divide our sum into sums over orbits. Our lemma then follows from the orbit condition DO.

Lemma 2.5. Let f iN and No,(f, then

3", dm(a)S(Ca)= ~ d(a')m(a')S(Ca')S(C,')" I] (1-p)p~p. a' pit(f) a

f(a) = f f(a') = f r(f)

Proof Let f be as given above, at T N, f (a)=f Write r ( f ) = p 1 ...p,. It follows from Corollary 1.5 and Lemma 1.2 that

I Hs,/p~ I S(Ca,). S(H~/NP~ ) °"Pl)S(C")= Ins, l IHN/N, i

for any al lying in the same orbit of 1 Z/Z as a. P

Hence S(Ca)= S(Co,). (Pt -1)av, al having period fPl. Similarly S(Ca,)= S(C~). (P2-1)~p, holds for any a 2, lying in the same orbit of

1 Z/Z as a 1, which is of period f(a2)=fplp :. P

Continued in this way, we arrived finally at

S(Ca, - ~)= S(C,).(p t - 1)~p,, at having period fr(f).

This holds for any a t lying in the same orbit of 1 Z/Z as a t_ 1- P

206 J. Yu

Let K, denote the set of all a' in TN, obtained from a in the above manner, as a t. These a' can be characterized as the set of all elements having period fr(f) and

in the same orbit of ~ Z/Z as a. We then obtain 4

lying rtJ )

S(Ca)=S(C.') I-I (p- 1)%, for all a'eK a. pit(f)

On the other hand, the orbit condition DO gives

re(a)=(-1)' ~. m(a'). a" ~Ka

Since any a', having period f(a') = fr(f), lies in K, for precisely one a in T N with period f(a)= f We are done.

It follows from the above two lemmas that:

~. dm(a)S(C.)= Z Z d'm(a)S(C.)'l-[(1-p)PSv"

Hence

~m(a)O(a)N=o,). ~ ~ dm(a}S(C,). 1-[ (1 -P~v) ~f a piN No I N f(a)= f

"~-¢JJ'I~<[No~ ' N f(a;=fZ dl~(a)S(Ca)'l-~(l-p)e~pl'p~N(l-P~p)plr _J pXr

=a). ~ Z ~ dm(a)S(C,)l" I-I ( 1 - p a , - ( p - 1)p~p)

LNo::, N :,o,:: j ,,N Y. Y.

Nor:IN :

This completes the proof of our Theorem 1. In the next section we will determine the index of Jt in Ro, namely

(Ro :J1) = I-I pL(p), pl N

where

I rp ( p ~ ) (p'~P)- t - 1 ) - 2n(p)- 2, if N is composite.

L(p) = [ p,(~)- 1 _ 2n(p) - 2, if N > 4 is a prime power.

p"(P) = Np, the highest power of the prime p dividing N. It implies, in particular, that Ro and Jl have the same rank, namely

~o(N) 1. 2


Consequently, from Theorem 1 we also obtain:

Theorem 2. The ideal J is of rank - ~ - 1, and furthermore

(R0,J)= I I t ' H ½B2,z (I - p2x(p)) PiN z * l

where L(p) is defined as above, X reads as the primitive character associated to the character X.

Proof. We consider the action of co. ~ (1 _pZ~p) on the augmentation ideal of the pIN

group ring Q[G] via multiplication. The determinant of this linear transformation can be computed on the augmentation ideal of C[G], which is a vector space over C and splits into 1-dimensional eigenspaces corresponding to the various characters ~e 1, therefore

det ( co" H (l - p2~p)) = x,I,]l [ ~ p l N 1B 2.xp~l(l--p2z(p))].

Knowing the faet that B2, x =~ 0 from the study of L-series, cf. [L 2, Chap. 3] we see that our transformation is non-singular, therefore J, as image of Jx, is also of

rank ~0_~v p_ _ 1, and its index (R o :J) is precisely the one described in our theorem.

3. Computation of the Indices

Let J~, be the ideal in R o consisting of elements of the form

2 ~, dm ac= E 2 dm(a)S(C,), "° ,,°bin

where the families {re(a)} satisfies the orbit condition DO only. If N is composite, J~, = J1, by Lemma 2.1. Otherwise:

Lemma 3.1. I f N = f , a prime power >4, then

(J'~ : J1 )=p .

Proof. Write ~ ~ dm ac = 2 ~ ffm ~= ~ u(e)~, . c~G NolflN \ ' N / c~G t=O

Let M = N if N is odd and M = 2N if N is even. Consider the map tp:J~ ~ Z/MZ defined by

~, u(c)a,~--> ~ u(c)c ~ mod M.

Since ~m(a)(Na)2=~ ~ p-lm we that J1 is the t=0

kernel of this map tp. Now we contend that @J~) is a group of order p. Let K be an

208 J. Yu

1 Z orbit of ff /Z, for a'eK, then

Na' = + Na(mod p"- t), for any a6 K.

Hence as a consequence of the condition DO, we find

~. m(a) (Na) z = ~. re(a) [(Na) z - (Na') z] -- 0 (rood Mp- 1). a~K a e~ a" ~K

It follows that the image of ~0 is contained in Mp-~Z/MZ. Let K be an orbit consisting of elements of period N. Since N > 4, such an orbit contains more than one element. We then define a function m(a) which is zero outside K and

m(a) = O, acK

~,, re(a) (Na) 2 --- Mp- 1 (mod M). aeK

Thus the image of ~ is precisely Mp- tZ/MZ.

We shall characterize J~, by a congruence condition, which allows us to perform an inductive procedure, ultimately leading to the computation of the index (R 0 :Yt).

Let F be the set of all elements ~ u(c)a, in Ro, satisfying for every f, with NolflN, and every coset C of H:, the following congruence:

~, u(c)- 0 (mod de), where d = N/f . ¢~C

Lemma 3.2. J] =F .

Proof We first prove J'~ CF. This is equivalent to the congruences

~ ~, d'm =O(modd 2) for any d, d f=N, and NolflN. c¢C No(fiN

(d'~k If did', then f ' l f , and m {~-) is constant on the cosets of us. Knowing that

ICt = In:l =d in the case NolflN, we find

~ d m -~- - 0 (rood d2), for C any coset of H:. eeC \ /

If d~/d ', it suffices to show that for every coset C of H:,

., /d'c\ Xdmt -)=0. ¢¢C

This is equivalent to the following equality in the group ring:

d'm(a)S(Co)S(n ~)=O. f(a)=f'


Summing over orbits of _1 Z/Z, with pLd, pXd', we see that the equality is an P

immediate consequence of Lemma 1.3 and DO. Conversely, given ~ u(c)ac in F, we have to find {re(a)}, such that it satisfies the

condition DO on all relevant orbits in TN, and

d' [a'c\ Z } =u¢c) No(f 'IN

This equality certainly holds if the following is true:

~. ~ d'm(a)S(Ca)= ~,u(c}a c S(HuYl), for any f, NolflN. No( ',i s,Z-s' c / l Z f l

o r

, d'c ~ S{Hy) for any f, Not f iN. ~ dm(---~-)trc= u(c)a~lHst, ceG No(f "If

As can be seen directly by letting f = N and comparing coefficients. We therefore define re(a) to make the above equality true. To start with, we define m(a) for those a in T N with Nolf(a), which can be done

recursively by the following defining equation:

E dm(a)S(C,) + Z E do~m(a')S(C_ ) = 3' u(c~a S(H[) . ,~:,. 1 . , " ' ' 7 " " c I H s l "

f(a) = f =Nolf S(a') = $/a

Comparing coefficients of C, on both sides, this equation reduced to

d(a)m(a)+ ~, d(a)ctm(cta)= ~ u(c)/d(a). ~t > 1 cECa

aNolf(a)

In other words, we define m(a), in terms of those re(b), where b has smaller period. Next, we define re(a), for those a=~0 in T N with No/~f(a ), by

re(a)= ~ (--1)k("~m(a'), a' ~Ka

where k(a) is the number of distinct primes dividing r(f(a)). There remains to check the orbit condition DO for our {m(a)}.

Let K be an orbit of 1 Z/Z in T~. There are two cases, depending on whether P

Nolf(a ) for all a in K, or NoXf(a) for some a in K. We will prove DO inductively in the first case. Suppose DO is already proved for all orbits K having constant period strictly

dividing f. We proceed to show that DO is then also satisfied for those orbits K having period exactly f.

Let K be such an orbit of 1 Z/Z in TN, containing a, f = f(a). P

210 J. Yu

By our recursive definition, it suffices to show that:

E X dotm(~a)= X Z .(~)td. a~K a > 1 aeK ceCa ~olf

If (g,p)=l, 0~K is also an orbit of 1Z/Z with P

~ d~m(ota)=O by inductive hypothesis. <'°'< < '>

Thus we have only to show the following:

period f/~, hence

Z ~, d~m(o~a)= ~, u(c)/d acK a ~ f c~Cp,,

but this is just the recursive equation defining pa, after multiplied by p:

~,, dpflm(pfla)= ~ u(c)/pd, ~=o~/p. #N~olf ceCp~

Now suppose that K contains some a with NoYf(a ).

If this orbit K of 1 Z/Z is still of constant period f, then p)~r(f) and P

~Ka=K,e+~Z/Z , for all a' in K

therefore, by reducing to the first case, we obtain

~. m(a)=(- 1) kta') ~ m(b) =0.

Finally, if K is an orbit of 1 Z/Z, with changing period, then P

K~ = .~,. Ka,, where a in K with (f(a), p) = 1. a' ~a

If a4:0, from our definition of m(a), we have:

m(a) + .'.xX m(a')=(--1)~" (t.x.,X re(b)- ,~'..X ~,, . re(b)) =0. a' @a a" # a

If a=O, Ka=K o is the set of all elements in T s with periodf(b)=N o, and k = k(O) is the number of distinct prime factors of N. Thus:

Z m(a')=(-t) '-1 E Z re(b)=(-1) ' - j Z re(b) a'¢K a" cK lmKa, b~T~ a' :0:0 a':~a f(b)= NO

'-1 E E ~T~ ~C~

f (b)=No


Thus the condition DO is indeed satisfied by our {re(a)}. This proves Lemma 3.2.

The congruences needed to characterize F can be reduced. Given any p,p[N, let Np=p "~'~ and O<_l<n(p). We define F~ p~ to be the ideal of all elements ~u(c)a c in R o, satisfying, for

every f , with Nff/NplflN, and every coset C of H r that

~. u(c) = 0 (mod d2), d = N / f . ceC

When/=n(p) , we have rtv~ - o for every pIN.

Lemma 3.3. ~t~ FT) = e .

Proof. Clearly we have F C c~F(t p) by definition. Conversely, assuming that ~ u(c)ac~F~ p) for all pIN. We contend that for any f, NolflN and every coset C of Hy that

E u(c)- 0 (mod d2).

Let (d, Np) = pl~a) and f~P) = N/p tea), all we have to prove is

u(c)=O (mod p2na~) , for all piN. ceC

Since f i r ~x'), this follows from the observation that we may write C as a disjoint union of cosets of Hf(p) for each p.

Now we consider the following filtration for each p, pIN.

c- b"(P) - - F~ p) C F (p) C . . . . . n(p) - - " '0"

Lemma3.4. (F~ p):F~I)=p 21GIIp'''' . . . . - 2 , for 2<l<n(p).

Proof. Let ~u(c)o'c~FiP) and consider the following map

~p~: F~ )~(Z/p 2~tp)- ~ + ,Z)~cz)

given by

~_,u(c)ac~( .... c~ u(c), . . .) mod p2"'P'-'+ " ,

where the components are indexed by the cosets C of Hy, f = NP t(p)- 1/N,, and • (/) = the number of cosets of H e = IGI/p "~t')-z+ ~.

The kernel of our map ~Pz is precisely FI~ 1. As to the image, we contend that:

Ipt( Fl p)) ~ ( Z/p2 Z) ':tO- J. .

Let C be a fixed coset of Hf. For each coset C + C' of Hf, we may define a function u c on G such that u c vanishes outside CuC' and

E~c(~)~o~ ~,

c~C'

212 J. Yu

This ensures that the image of ~ is the hyperplane consisting of

( . . . . X c . . . . ),Y~Xc=0 inside

( p 2(n(p) -- l l Z / p 2 ( n ( p ) - l + t)Z)t(i)"

Corollary 3.5.

~pLtV) if N is composite (Ro:F~P)) = [pZt,)-,, if N = p " > 4 ,

where

] q)( ~)(pn'p ' - ' - l ) -2n(p)-2 ' if N is /

L(P) = [ p"~P~- 1 - 2n(p)- 2, /f N = p n > 4 .

Finally, we obtain our index:

Theorem 3. (R o :Jl)= I~ pL(p). pIN

Proof. Since (R o :FI (p~) is always a power of p, we have

If N is composite, we are done. If N = p"> 4, by Lemma 3.1 we have to multiply both sides by p to get the desired index. QED

4. Cuspidal Divisor Classes on X~(N)

We recall first that the cusps (primes at infinity) on the modular curve X(N) can be represented as column vectors

where x, ye Z/NZ and (x, y, N) = 1. The cusps on XI(N) are orbit classes under the action of

I) that is, under the following equivalence relation

::t:[;],,,+__[X; ry] with r ~ Z .

We can take a normalized representative in such a class with

ON_y<N and 0_Nx<(N,y).


The cusps of the first type are those cusps which tie above the cusps 0 on the modular curve Xo(p), for every pIN. These cusps can be represented by column vectors

+[0y 1, with ( N , y ) = l .

Let

~ ° ( N ) = t h e group of functions on X I ( N ) whose divisors have support within those cusps of the first type,

~° (N)=the divisor group of degree 0 on XI(N ) whose support lies among the cusps of the first type,

and

cg°(N) = ~°(N)/div ~°(N) .

cg°(N) is a subgroup of the full cuspidal divisor class group on the modular curve X I ( N ) . We are interested in its order h°(N).

The cusps of the first type obviously form a principal homogeneous set over G = ( Z / N Z ) * / + _ 1, so that the divisor group ~°(N) can be identified with R 0, the augmentation ideal in the group ring Z[G]. We are thus concerned with determining the subgroup div ~ ° ( N ) .

We will characterize .~°(N) in terms of Siegel functions. Let a = ( a p a2) be rationals with denominator dividing N. The Siegel function g, is defined by

where A(v) 1/12 is the square of the Dedekind eta function

a(Z) is the Klein form associated to a:

r,,('r) = e -½("''7' + a2"2)Za(z, [~, 1]).

Here ql, ~/2 are the quasi periods of the Weierstrass zeta function associated with the period lattice [r, 1], a is the corresponding Weierstrass sigma function, Z=al ' r + a 2.

We know that the Siegel functions has the following q-product

g.(z) = - O~B:(at)e 2'a"2~"' - a)/2(1 - qz) f i (1 - qn, q~) (1 - q,/q,)," n = l

where q, = e2~i~, q~ = e 2~ and Bz(X) =X 2 - X + ~, the second Bernoulli polynomial. From a well-known transformation law for the sigma function we observe that

if we change a by an integral vector in Z 2, then g, changes by a root of unity. Furthermore, g, and g_. also differs only by a root of unity. Since we are interested in modular functions only up to a constant factor, we shall abuse our language, speak of go, ae(Q2/Z2) + 1, not only as a function, but also as the class of functions modulo constant factors.

From now on, we let a = (0, a) with ae T N =(Q/Z)N/-t-i.

214 J. Yu

We abbreviate our notation and write for these a, a # 0

g= = g(o, a)"

Let F'~(N) be the group of products of Seigel functions

g = l-I

such that the exponents re(a) satisfy the conditions DQ and DO, that is the quadratic relation mod N and the orbit condition.

By [Ku 1], we see that these functions g are modular functions on the congruence subgroup FI(N). Moreover, these are functions with neither zeros nor poles on the upper half plane.

From the q-product for Siegel functions, we can analyze the orders at infinities for these functions g in ~-~(N).

We observe that

Ord ±[~]ga = ½B2((ay)).

l_emma 4.1. ~-~(N) C ~°(N).

Proof. This is a consequence of the orbit condition DO. Let g=I-Ig~*=~-~(N), it suffices to show that

Ord±l~ lg=0 , if ( N , y ) > l .

That is ½ Z m(a)B2((aY)) = 0 if (N, y) > 1. a

4

Let p[(N,y) be a prime number. We consider orbits of X-Z/Z in T N. Since P

B2((ay)) is clearly constant on such orbits, we can rewrite our sum as sums over orbits which is therefore 0. This proves the lemma.

We can thus determine the whole divisor of any function in ~,~(N)

(g)=(l-Ig~(=)) = ~ Z ~.m(a)B2((ab))a; 1 =~m(a)O(a)N 4" b ~ G a a

thereby we conclude that our ~rl(N ) is just our ideal J. Since J consists of divisors of functions we see that J C R o. Our main result is the following characterization of ~r°(N).

Theorem 4. ~';(N) = ~'°(N).

Proof. By Theorem 2 and the above remark, we know that ~ ( N ) , considered as a

group of functions modulo constants, is of rank ~ - 1, which is the maximal

possible rank for ~q~-~(N). Therefore, in view of Lemma 4.1 above, these two groups of functions must have the same rank. All we have to do is to show that ~'~(N) has no cotorsion in ~,a~°(N).

Lemm_n 4.2. Let I be a prime number, and g be a modular function of level N. Suppose

o' = I- [


then

On(a) for all a,t= 0 in T N.

Consequently

g = 1-[ g~(~)/l is itself in ~ ( N ) .

Our proof of this Lemma is based on the method of Kubert-Lang [KL 4]. We recall briefly the arguments which we need from [KL 4].

We let M be maximal such that there exists a' with m(a')~ 0 and a' has period M. It suffices to show that lid for all a' in T N with f (a ' )=M. Having done that we can absorb in g those factors of the product with a' of maximal period M, and continue by induction to get the desired/-divisibility for all re(a).

Let a' be a given element in T N with maximal period M. We rely on a theorem of Shimura about the integrality of the coefficients in

q-expansions of modular cusp forms, which allows us to conclude that g must have a q-expansion at each cusp whose coefficients are algebraic and have bounded denominators. What we have to do here is to check the q-expansion at the cusps a - 1p~ (which is not necessary of the first type), where or, depending on a', is the following element in the Cartan group C(N):

(0 1 ~ = with ceG(N) and c a ' = - - .

c M

In other words, we consider the q-expansion at P~ of the function crg which satisfies

re(a) 0o) '= H g°~ •

We will study the leading coefficient after the constant term. From the q-product of Siegel functions we see that the reduced power series g*,

(that is, the q-expansion of go~ at P~o divided by its lowest term), have coefficients in the ring of cyclotomic integers. It follows that (o'g) t also has a reduced power series with algebraic integral coefficients. Furthermore, by the Gauss 1emma for power series with bounded denominators, we see that in fact (t~g)* itself already has algebraic integral coefficients.

As we can take roots on those reduced power series formally, we have

(~g)* = I~ g.~C.~/~. We observe that if (al, a2) has period M, the coefficient ~q of qt/M in the

reduced power series gt=l.,2)* is given by

t 0 if at#: ~

0~1 = 1 --e z"i"~ if a 1 = ~ .

Hence the coefficients of ql/M in the power series (ag)* is

216 J. Yu

where

c= -~ ,a , O < u < M

and

~c = e2~iulM ;

this is because that M was chosen to be the largest period. Now, since our family ofexponents {m(a)} is defined only on TN, m(c)= 0 except

for c = (0, a') the above coefficient is therefore equal to m(a')/l. The integrality then implies llm(a'), as desired.

This also completes the proof of Theorem 4. Our class number formula is obtained by combining Theorems 2 and 4.

Theorem 5.

h°(N) = 1-I pL(~,) I-I [½B2,x I-I (1 -p2x(p))]. pIN x * 1 L plJv 1

T h i s f o r m u l a c a n b e u s e d to g ive expl ic i t b o u n d s fo r t h e c u s p i d a l c lass n u m b e r s .

Acknowledgements. I am deeply indebted to my advisor Professor Serge Lang, for proposing the problem and for his advice and constant encouragement during my stay at Yale University. I would also like to express my gratitude to the Mathematics Department at Yale University for their support during the years 1976-1980. Last, but not least, I want to thank my parents and my wife back home in Taiwan, for their love, as well as their sacrifices.

References

[Iw] [K 1] [Ku 1]

[Ku 2] [KL] [KL 1] [KL 2]

[KL3]

[KL4]

[KL 5] [KL 6]

[KL 7] [KLS]

[L 1] EL 2] [Sil

lwasawa, K. : A class number formula for cyclotomic fields. Ann. Math. 76, 171-179 (1962) Klimek, P. : Thesis, Berkeley 1975 Kubert, D. : Quadratic relations for generators of units in the modular function field. Math. Ann. 225, 1-20 (1977) Kubert, D. : The universal ordinary distribution. Bull. Soc. Math. France 107, 179-202 (1979) Kubert, D., Lang, S. : Modular units (in preparation) Kubert, D., Lang, S. : Units in the modular function field. I. Math. Ann. 218, 67-96 (1975) Kubert, D., Lang, S. : Units in the modular function field. II. A full set of units. Math. Ann. 218, 175-189 (1975) Kubert, D., Lang, S. : Units in the modular function field. III. Distribution relations. Math. Ann. 218, 273-285 (1975) Kubert, D., Lang, S. : Units in the modular function field. IV. The Siegel functions are generators. Math. Ann. 227, 223-242 (1977) Kubert, D., Lang, S. : Distributions on toroidal groups. Math. Z. 148, 33-51 (1976) Kubert, D., Lang, S. : The p-primary component of the cuspidal divisor class group on the modular curve X(p). Math. Ann. 234, 25-44 (1978) Kubert, D., Lang, S. : Stickelberger ideals. Math. Ann. 237, 203-212 (1978) Kubert, D., Lang, S. : The index of Stickelberger ideals of order 2 and cuspidal class numbers. Math. Ann. 237, 213-232 (1978) Lang, S. : Introduction to modular forms. Berlin, Heidelberg, New York : Springer 1976 Lang, S.: Cyelotomie fields. Berlin, Heidelberg, New York : Springer 1978 Sinnott, W. : On the Stickelberger ideal and the circular units of a cyclotomic field. Ann. Math. 108, 107-134 (1978)

Received April 9, 1980

a cuspidal class number formula for the modular curvesx1(n)

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