m a n u s c r i p t a math. 64, 485 - 502 (1989) manuscripta mathematica �9 Springer-Verlag 1989

Functional equations for zeta functions of

non-Gorenste in orders in global fields

Barry Green 1)

1. I n t r o d u c t i o n

In 1973 Galkin published a paper, [8], which deals with the zeta function of a non-maximal order in an algebraic number field or function field in one variable over a finite field of constants. By using the methods of Haar measure and duality he was able to establish a functional equation for non-maximal orders in a closed form involving only zeta functions together with an elementary factor, provided they were Gorenstein. He also gave examples to show that this functional equation does not hold in general in the absence of the Gorenstein condition. In the proof of the functional equation a key role is played by the dualizing module (also called the canonical module by Kunz in [9]) for the order. For Gorenstein orders the dualizing module may be chosen to coincide with the ring and so no distinction between the two is needed. However for non-Gorenstein orders they do not coincide.

In this paper we show how the zeta function of a non-maximal order may be redefined so that in the absence of the Gorenstein condition one still obtains a functional equation. Recently much work has been done on the zeta and L - functions of ari thmetic orders, particularly for orders in a finite dimensional semi- simple algebra over the rational number field, by Solomon [12], Bushnell and Reiner [5], [6] and [7]. However for non-maximal non-Gorenstein orders in general it appears tha t little if any work has been done.

2. Bas i c de f in i t i ons

Throughout the paper K will be either the field of rational numbers Q, or a field of rational functions of one-variable over a finite field of constants F. Let O0

1) Financial assistance from the University of Stellenbosch, South Africa, is gratefully acknowledged.

485

GREEN

be either the ring of integers 2 or the polynomial ring Fix], respectively. E is a finite separable extension of K and O the integral closure of O0 in E. A ring O intermediate between Oo and O is an order in E if it has field of fractions E. Let M(O) denote the monoid of all fractional O-ideals in E.

A dualizing module J for O is a fractional O-ideal such that ( J : ( J : L)) = L for each L E M ( O ) . For L, M E M ( O ) we let ( L : M ) = { x � 9 I x M C L } , the residual quotient of M by L in E. Note that if J is a dualizing O-module then we have ( J : J ) = O and ( J : O) = J. The existence of dualizing modules for O is established in [9], p. 22. It is also shown there that the dualizing module is unique up to multiplication by a projective (invertibIe) fractional O-ideal. The property of being a dualizing module is preserved under localisation [1], p. 43, and the ring O itself being a dualizing module is equivalent to O being Gorenstein [2].

Given L , M �9 M(O) with L ___ M , [M : L] will denote the index of L in M. The index is extended to all fractional O-ideals by defining [L : M] = [L : W]/[M : W] for any W �9 M(O) and W _ L M M . This definition is independent of the choice of W and one easily sees that [L : M] = [M : L] -1 and [n : M ] [ M : N] = [L: N]. Furthermore if J is a dualizing O-module then [n : M] = [(J : M) : ( J : L)] by [9], p. 19. The O-norm of L �9 M(O) can be expressed in terms of the index as ][LII o = [ O : L ] -1. If x � 9 E • []xll will denote its norm. For any order O , Ilxil = llxOIIo, [3], p. 126.

De f in i t i on . Let M E M(O) and s be a complex variable. The zeta function of O relative to M is defined as

= I I L I I o . LEM(O)

L C M

For some r E E , r M ___ O and it follows that these zeta functions con- verge absolutely in the same region as ~o(S) = to(O,s), that is for Re(s) > 1. Consequently they are analytic in the half-plane Re(8) > 1.

E u l e r P r o d u c t I d e n t i t y . Given M E M(O), for each prime ideal P of O we let ~op(Mp,s) denote the zeta function of the localisation, relative to MR. Then

r s) = 1-I ~op (MF' s) P

under the assumption that at least one side of the equation is absolutely conver- gent. This result is proven by similar method to that of [8], p. 4. See also [5], p.

135 and [12], p. 316.

486

G R E E N

3. M a i n R e s u l t s

L o c a l c a s e . Let O be an order in E and P a pr ime ideal. Then:

T h e o r e m 3.1 . For each dualizing OF-module JR,

~ov(Jp, s)/~-~p(s) �9 ~[u, u -1]

8 where u = IIPl[op and OF denotes the integral closure of OF in E . 2)

R e m a r k . As a consequence of 3.1 and the result t ha t r (s) admi ts an analyt ic

cont inuat ion to a meromorphic function of the complex p~ane this is also t rue of

~o~(Jp, s) .

The following theorem shows the way the zeta functions depend on the choice

of J v .

T h e o r e m 3.2 . Let JR and J1P be dualizing Op-modules in E . Then

~o~(J~ , s ) = II(J~ : J , P ) l l ~ ~ o ~ ( J , ~ , s ) .

T h e o r e m 3.3 . For each dualizing Op-module JR , ~op(Jp,s) satisfies the func- tional equation

r --* ~_~ _ _ IIO~11o~, ~'oeCJP, 1 s)/~'~p(1 s)

where ~ , = ( g . : ~ ) .

C o r o l l a r y 3 .4 . There exists a dualizing OF-module Jp such that -O*p = Yp, where Yp is the conductor of Op in Op .

T h e o r e m 8.5 . Let Jp be a dualizing Op-module and k @ 7Z be the least integer such that

r = ~ o ~ ( J , , , ~ ) / ~ - ~ ( s ) �9 ~.[u], ~ = lIPllop-

Then IIJ~llop = II-PIIo~ and degCv -- 26p, where ~ is the singularitr number

of Op in Op ; that is the unique integer such that l O P : Op] = [ [P][o? "

R e m a r k . Let Cp(u) = r an element of Z[u] by [8], p. 9, and

set d e g C p -=- r/p. When OF is Gorens te in lOP : Op] 2 - lOP : 7p] so by 3.5 26p ~- r/p and [Op : 7p] = IIPllo~ v. Following the compu ta t i on of a number

2) For the case of the basefield K = Q, theorem 3.1 is a special case of Bushnell and Reiner's theorem 1 of [5].

487

GREEN

of cases it seems probable that lOP : Fp] = IiPllo~ P even when Op is non-

Gorenstein.

G l o b a l c a s e .

T h e o r e m 3.6 . Let O be an order in E and J a dualizing O-module. Then

to(J , ~) admits an analytic continuation to a meromorphic function of the whole complex plane. It has only one pole, which is a simple pole at s = 1.

The theorem below shows the way the zeta funct ion depends on the choice of dual iz ing module J .

T h e o r e m 3.7 . Let J, J1 be dualizing O-modules in E. Then

~o(J,s) = I[( J : J1)l[o ~ o ( J l , s ) �9

T h e o r e m B.8. Let 0 be an order in E and J a dualizing O-module. Then

r162 = I1 *11 " -1 Co(J , 1 - -

where O* = ( J : O ) .

C o r o l l a r y 3 .9 . For each order 0 there exists a dualizing O-module J1 such that O* = (J1 : -0) = ~r, the conductor o f 0 in O .

4. A u x i l i a r y r e s u l t s

T h r o u g h o u t this sect ion we fix a pa r t i cu la r pr ime P of a given order O . To simplify no t a t i on we wr i te A = O F , Ao = A N K , F for the conductor of

in A and ][ [] in place of ][ IIA when tak ing the A-norm. Two fract ional ideals L , M E M(A) are equivalent , wr i t t en L ~ M, if L = x M for some x E E* . The

class semigroup of A is finite and has only one uni t , namely the class of pr inc ipa l ideals [8], p. 5.

Let ~, $ and s denote the complet ions of E , A and L , an A-ideal , respec- tively, in the F - topo logy on E . The comple t ion of the ideal • coincides wi th

its closure in ~ and we have the following relat ions /~ = L~ , L = s N E and for any o the r A- idea l M , [Z : A{] = [L : M] . Fu r the r thei r exists a one to one

no rm preserv ing correspondence between the non-zero ideals of A and the ideals

of finite index in ~. The funct ion I] ]] extends by cont inui ty to ~ .

The f rac t ional A-ideals in ~ are defined to be ideals of the form x ~ where /2 C ~ is an ideal of finite index and x E ~ an invert ible element. By approxi-

m a t i n g x sufficiently closely by x0 E E one ob ta ins x s = Xos which implies tha t x s is the closure of some f rac t ional A-ideal . The n o r m of a f rac t ional ideal

488

G R E E N

equals the no rm of its intersect ion wi th E. For equivalence of f rac t ional A-ideals

the coefficient of p ropor t iona l i ty must be invert ible in E. Two fract ional ideals

in M(A) are equivalent if and only if thei r intersect ions wi th E are equivalent in

M ( A ) .

L e m m a , t .1. Let J be a dualizing A-module. Then J, the completion of J in ~, is a dualizing A-module.

Proof. Let f EM(~q) and set f * = ( J : s We first show tha t f * = L * ~ , where L* = ( J : L). If x E L*$ then xZ = xLA C JA = J and L*~ _c_ f * .

For the o ther inclusion we approx imate x 6 s by an element x0 E E so tha t

x o L C J and x E x 0 + L * A . Now xoLC_ J A E = J and consequently x 0 E L * .

This implies x E L * A and ~* C L * A . F ina l ly f** = ( L * ~ ) * = L * * A = L A = s so t ha t J is a dual iz ing module .

For the convenience of the reader we now review the integrals we shall need

to consider . To this end we first define a Haar measure on the addi t ive group

of ~ . Recal l t ha t the complet ion ~ of A decomposes into a direct sum of the

comple t ions of A in each of the B- topolgies where B runs th rough the max ima l

ideals of A ; A = ~ B A B , where ~B is the r ing of B-ad ic integers having max ima l ideal B-~B , which we will denote by B. Similar ly ~ has decomposi t ion

= ~-~B ~B, where ~B is the B-ad ic field. Note t ha t ~ (and also ~q ) is compact and ~ is local ly compact . For each max imal ideal B, let #B be a Haar measure

defined on the addi t ive group of EB and normal ised so tha t #B(~B) = 1. Let

# = 1-[8 ttB, a Haa r measure defined on the addi t ive group of 8 as the p roduc t of of the measures on each of the summands . Then tt(A) = 1 and for ~ E M ( A ) ,

# ( f ) = IiflI~/I]~I]~. Note tha t by our earl ier remarks # (~ ) = IILI[/IIAII where L = f M E .

Let r be a bounded complex valued funct ion of ~ and for each n E Z let

Un = { x e ~ : HxH-= ]]PiIn}. We define:

/ 11=11"-'r a= = d=, n6 ~..

whenever this is convergent. We shall be concerned with integrals for which r is

e i ther the character is t ic funct ion of some fract ional A-ideal or a cer ta in continuous charac te r of the addi t ive group of ~ , i.e. a continuous homomorph i sm from the

addi t ive group of ~ to the mult ipl icat ive group of complex numbers . The proofs

of the following results due to Galkin are included here since the arguments used

are needed in the proof of 4.8.3)

3) The author is grateful to the referee for pointing out that a result of Galkin, namely [8] corollary to lemma 3.3, is false. Originally this result had been used in the proof of the main theorem. In order to avoid the corollary in question, theorem 4.8 of this paper was needed. Consequently its proof has been given with full detail.

489

GREEN

L e m m a 4.2 [8]. Let X be a character o f the additive group o f ~B . Then for Re(s) > 0 we have:

L IIzll~-~x(=) d= =

1 - II.Bil ~ for Z = o 1 - I IBI I~

1 -I IBII-}-" for l > 1, IIBII~ 1-IIBII~-

where l is defined as the least integer such that X is trivial on B t .

Proof. We divide the domain of in tegrat ion into domains of the form B'~\ B n+ 1

for rt > O, so that :

d= = ~ /,.\,.+ ll=ll~-~x(=) d= n>_o

= ~ IIBII~ -( '-~) s ,,>o o\s~ x(=) d=.

Now for each n ~ 0 by shifting the variable of integrat ion by an element t in

B'~+I\B'~+2 the integral becomes

s176176 x(=) ~= = x(t) s d~.

If n + 1 < l - 1, t can be chosen so tha t x ( t ) # 1 and consequently the integral

vanishes when n < l - 2. For n = I - 1, X(X) will be constant on the cosets of

B '~ modulo B '~+1 and so induces a non-triviM character of the group Bn/B '~+1 .

Hence,

/~%~.+I X(X ) d x = X(5)#B(B"+I) = --,~B(B'~+~) = -]IBI]~"

Finally, for n _> l we have:

= ~B(8") - ~B(~ '~+~)

= IIBII~(1 - ] IBI Ix) .

We obtain:

E IIBII~( 1 - I IBI Ix) nko

dx = -]IBII~-(*-I)(I-*)+ E IIBII~( 1 - IIBIIx)

for l = 0 ,

for / > O.

490

G R E E N

By summing the geometr ic progressions the resul t follows.

L e m m a 4 .3 [8]. Let X be a non-trivial character o f the additive group o f SB.

Then for Re(s) > 0 :

e 1 -IIBII . -s d= = IIBII - 1 - IIBII

where l is defined as the least integer such that X is trivial on B l .

Proof. This integral is computed by identical me thod to tha t of l > 0 in 4.2.

L e m m a 4.4 [8]. The integral = o rot negative sumciently

large in absolute value. Further this integral has order O(nk-1)[IP[I '~R+(') as

n -+ oo, where k is the number of max ima l ideals o f A .

Proof. Firs t we note tha t by performing the subs t i tu t ion x ~ ax for sui table

a E ~, the proof is reduced to cases when ei ther r = 0 for x E ~ \ ~ , or r is a

charac te r for which there are uni ts Us E •B, for all /3, such t ha t ~(UB) • 1. Let B i , 1 < i < k, be the max imal ideals of A and suppose [[Bill X -- ][P[[~' " By the p roduc t formula for the norm, the domain of integrat ion Urn, can be represented as

k n the disjoint union of the domains U_~ = I I~=I (B '~ ' \B '~ '+ I ) , where n = ~ i = 1 ei i and n = (ni) , the k- tup le de te rmining this representa t ion of n . Note tha t if

k x @ U~_ then [[x][ = I-[i=1 [[B,[[~ = ]IP[[ n . We observe tha t the integral over U,~

is zero if one of the nr are negat ive. This is obvious if r is of the first type; t ha t is r ---- 0 for x E 8 \ ~ , and here Un c ~ \ ~ . For r of the second type we

have: k

~-- i = 1 ~ '

and if ni < 0 for some i then the corresponding factor is zero by the same

a rgument as in 4.2. The first assert ion now follows for n < 0.

To evaluate the in tegral over Un it suffices to consider t e rms corresponding to Un. The number of such terms is equal to the number of solut ions of the

k Diophant ine equat ion n = ~ i = 1 elni. We recall tha t if (mi) E ~k and m = k ~ = l e i m l < n , then the number of solut ions of the Diophant ine equat ion n = k ~-~i=lelnr with m i <_ nl is bounded above by D ( n - m) k-1 for some constant

D E ]R depending only on k and the integers e i , 1 < i < k , and not on m

or n , by [11], p. 4, p roblem 27. In the s i tua t ion above the bound is D n k-1 as

n i ~ 0 for each i , l < i < k .

The measure of each domain U,~ can be expressed in te rms of the norms as:

/z(U,~) = [[P[[~ 1-I/k=1 (1 - ][B/[[ ~ ) .

Note tha t the cons tant y I ~ = l ( 1 - I I B , II x) is not dependent on n , bu t on the order A . We shall denote it by C . It follows tha t the integral over U~_ has absolute

491

G R E E N

value bounded by,

Hence the est imate is established.

< IIPII"R (').

As an immediate consequence we obtain:

T h e o r e m 4.5. Let r be either the characteristic function for some ~ E M(~), or a continuous character of the additive group of s which is non-trivial on s for each maximal ideal B in A. Then: (i) there exists ~ E M(~) such tha t for all, ~ E M(A), r _D )r implies,

/ell=lls- C(=) = / ll=ll=-x c(=) d= ;

(ii) for each -~ e M ( ~ ) ,

d=

is absolutely convergent for Re(s) > 0.

In order to simplify the notat ion in the proposit ion below we shall use ~-~'~n_ to represent the series obtained by summing over all representations n of n for all n E 7Z.. It will also be convenient to order the tuples n partially by m > n if mi ~ ni for each i , 1 < i < k .

P r o p o s i t i o n 4.6. Let r be the characteristic function for some A4 C M(~q) . Then ill 0 < Re(s) < 1, for each h E Rk the series

m+~>h/v~_ /v~_[ []x[[8-1r dt d=

is absolutely convergent.

Proof. The result is established directly using the integral estimate over U s

in 4.4 and making a comparison. Suppose At~ = 1-Ii=lk 3/n,o and let n o = (nio) be the tuple determined by AI in this way. Then:

+~n> /U /U I]'x'[S-l~)(t)ldtdJ:: = E ]'Ip]]rn(a-1)'fu /u ~(t) dtdx m h m ~_ r n + n > h m ~_

= E I lPllm(Re(s)- l ) ' (Um)#(Un-)--C2 E IlPlinlIPllmRe(a) rn+n>h rn+n>h

n>r% n > n o

= c= Z : IIPII n>n.o rn+n>h

<_ CUD = ~ (n - n o ) k - ' ] ] e l ] " ( l - a ~ ( ' ) ) ( ~ (m + n - h)k-'ilPII (''§ n > n o m + n > _ h

= C2D2 E ( n - n o ) k - l l l P l l n ( i - R e ( s ) ) ( E ( l - h)k-illPl[lRe(s)) �9 n>_no l > h

492

GREEN

Since the first series is convergent for Re(s) < 1 and the second series for Re(s) > 0 the result follows.

L e m m a 4.7. Let X be a continuous character of the additive group of ~, which is non-trivial on ~B for each maximal ideal B in A . Suppose ~ , = I-[i=13i,k z is the largest module on which X is trivial and let h_ = (hl) = (l{ - 2). Then if m + n ~ h , f o r e a c h t �9 Urn and x �9 Un

Proof. The proof follows directly from lemma 4.3. Indeed if m + _n ~ h , then for at least one i , 1 < i < k , m i + n i _ < l l - 2 = hi and consequently as

k

I s " x(tx) dt = O. Bm'+l i~ \ i I From the product decomposition for the integral

k

-- i=l ?'keg ̀ +` x(tx) dt,

we deduce the result.

T h e o r e m 4.8. Let r be the characteristic function for some .M E M(A) and X a continuous character of the additive group of ~, which is non-trivial on s for each maximal ideal B in A. Then if 0 < Re(s) < 1,

/~ /~ ]'X['a-lx(tx)(D(t ) d tdx: /~ /c ]"T"[a--xx(tx)~(t)dxdt.

Proof. We first observe that since these integrals are not absolutely convergent it is not possible to simply apply Fubini's theorem to interchange the order of integration. The theorem is proved here using the preceeding results and Fubini's theorem internally. We simply outline the main steps: By the definition of these integrals we have

/ c L l 'xll~-lx(tx)r dt dx = ~ / c r m ( ~ / u [lXllS-lx(tx)r dt) dx

: E f u ( ~ f u . ]]~lls-lX(tx)(/)(t) dr)dx. m___ m -- --

We next observe that by ]emma 4.7 when ra + _n ~ h_ (with _h defined as in the lemma) then while x E U,~

/ )l = l l S - X x ( t x ) r dt = o.

493

GREEN

Consequently taking this into account and then appealing to the absolute conver- gence condition of proposition 4.6, which entitles shifting the second integral sign to the right past the summation sign, the integral above can be expressed as:

/e fe Ilx"'-ix(tx)r dtdx = ~m__ /~__ (m___+~n_>_h_/tr~_ IxI]'-IX(tx)c(t) dt) dx

=m+~n>h/rY_~ /~_ ''x''8-1X(tx)r dt dx '

provided 0 < Re(s) < 1. By an identical argument:

[IXJ[S-lx(tt)~(t)d:~dt = ~ i (Z ~ [[XHS-Ix(tX)~)(t)dx)dt _ u n m um

App ly ing Fubini 's we have

rum__ fU. ,Ix'js-lX(tx)(fi(t) dtdx: /U. rum HxH'-IX(tx)(fi(t) dxdt. Hence the two integral expressions are identical and the theorem is established.

In section 5 we shall also need the next theorem. It 's proof follows directly from 4.3 and the Eu]er product formula.

T h e o r e m 4.9 [8]. Let X be a continuous character of s which is non-trivial on s for each maximal ideal B in -A. Then for some m E ~. :

e l l = l i b - I x ( = ) d x = IIPII m= - f X ( ' ) / ~ ( 1

P a r t i a l z e t a f u n c t i o n s . Let L, M E M(A). The M partial zeta function of A relative to L is defined as

~A ( L ' M ' s ) = Z IINll ~" N_L N~M

Clearly r s) = E(M) s L, M, s) where (M) runs through the representatives of the class semigroup of A.

The reader is advised that the proof of the following theorem is a straightfor- ward adaptat ion of theorem 4.1 of [8], p. 9 and so will not be given here.

T h e o r e m 4.10. Let J be a dualizing A-module. Then

fA(J, M, s) = c (M) - i I1~11 I1~'11-~ IIMII ~ f .ll=ll dx

494

GREEN

where c(M) is a constant depending only on the class of M .

For later use we remark that c(M) is in fact the order of the multiplicative subgroup of r ( M ) / Y where r ( M ) = ( M : M ) .

D u a l i t y . We recall briefly the duality in s relative to the trace. Let Tr denote the trace map from s to K, K is the completion of K in the p-adic topology where p -- P M Ao. For an additive group L in s we set

Z' : {x �9 s 1 6 3 _ Ao}

where -40 is the ring of p-adic integers of K. If ~ �9 M(A) one has /~' �9 M(A), and / ~ = /Z. For the s tandard theory of additive duality over local fields the reader is referred to [14], chapter II.

L e m m a 4.11. Let J be a dualizing A-module and L �9 M(A). Then F~ = s where f* = ( J : L ) .

Proof. As T r ( L * J ' f ) _ T r ( J J ' ) _ Ao, s C ~ . For the converse (L*J ' ) ' = {~ e e [ Tr(xZ*J ' ) _ Ao} _c {x e e I xZ* c j,, = j} _- z** -- z. Thus (s C_ L giving ffi _C_ s

C o r o l l a r y 4 .12. I f J is a dualizing A-module then J ~ ~ ' .

Proof. Taking L -- ~ql in the lemma gives A = (Ar) . j i . By the remarks at the beginning of this section applied to ~q, J~ must be principal. Hence J~ ~ ~q and taking the dual J N A~.

We conclude this section with some results we shall need concerning characters of the additive group of ~. It will be convenient first to fix a part icular character Xo of the additive group of K such that , for x C K, Xo(xAo) -- {1) implies x E ~qo. The details of how Xo is constructed may be found in [8], p.12. Using Xo and composing with the trace we obtain continuous characters of the additive group of ~ , X(X) = xo(Tr(ax)) , a, x E $, a invertible.

T h e o r e m 4.1~. The following are equivalent for J E M($) . (i) J is a dualizing A-module. (ii) There exists a continuous character X of the additive group of 8 such that X is trivial on J and i f x is trivial on ~ E M ( A ) then LC_ J .

Proof, (i) ~ (ii) Suppose J is a dualizing ~q-module. Then ~q '= aJ for some invertible a C 8. Set X(X) = xo(Tr(ax)). Then X(J) = xo(Tr(aJ)) = x(Tr(~ ' ) ) C_ Xo(~o) = {1} and X is trivial on J. Now suppose X is trivial on s Then Xo is trivial on Tr(as Hence Tr(a~) _ ~o and a s _ A' = aJ. Thus

c J as required.

495

GREEN

(ii) =~ (i) Let X be a character satisfying the conditions of (ii), s e M(~) and suppose r is the characteristic function of s We first observe that the Fourier transform of r realized by X satisfies

r -- ./~ X(tx)r dt

=/ . x(tx) dt

=/Lx(Ct+a)4et, aeL = xCax) /2. x(tx) dr.

It follows that r is non-zero if and only if X is trivial on x/~, i.e. for x• _ J and x E/~*. Hence r differs from the characteristic function of s by the factor

, ( z ) = IILII /11 11

Finally from the theory of Fourier transforms r differs only in a constant factor from fe x(xt)r (t) dt = r (x) which in turn differs only in a constant fac- tor from the characteristic function of /~**. Hence L** = L and J is a dualizing A-module.

:Remark . Notice that theorem 4.13 implies the existence of a dualizing .4-module independently of the foregoing results as #ql satisfies condition (ii) by definition. In view of 4.12 the dualizing ~q-modules are exactly those elements of M(~) in the same semigroup class as ~q~.

5. P r o o f s o f the t h e o r e m s

L o c a l case . The notation here will be as in 4 with A = Op and A = O p .

Proolfof theorem 3.1. Let J be a dualizing A-module in E . Then from the definition of the partial zeta functions of A relative to J and theorem 4.10 we

have:

~A(J, S) ---- ~ ~A(J, M, 8) (1) (M)

= ~ e(M)- ' I IAII ll~ll-lLIMIr f ~ I{xlr - l d x , (M)

where (M) runs through the finitely many representatives of the class semigroup

of A.

We next consider the the integral f~.llxH ~-ldx, the representative for (M) being chosen so that At* C ~ . Suppose r is the characteristic function for A4* in

and let G denote the finite subgroup of the character group of #q, consisting of

496

G R E E N

those charac ters of ~ t r ivia l on 34". Then r can be expressed as the a r i thmet ic 1 mean of the characters X in G ; t ha t is r - carmG'~.t J ~ X. We obta in:

xEG

I1~:11 ~-' d ~ - card(G~ II~lF-"x(~) d~ xc

= / ~ C~) r (4 ,

where ~ ---- IIPII ~ and fM(u) e Q[u,u -1] by lemma 4.2. Substituting in (1) we obtain fA(J, s) = f (u )~ (s) , with f(u) E Q[u, u - l ] . However, from the definition of ~t(J,~) and the Euler product formula for r we have r162 e Z((u)) . Therefore f(u) �9 ~.[u,u -t ] completing the proof of the theorem.

Proof of theorem 3.2. Let J, J1 be dualizing A-modules in E . F rom the resul ts of 4 each dualizing A-module is in the same semigroup class. Therefore

J1 = x J for some x E E • and we obtain:

fA(J, s) = ~ IILII" = I1~11-" ~ ]IL[F = II~II-'~A(JI,~) �9 LEM(A) LeM(A)

LC_J LC_J 1

The resul t follows as ( J : J1) = x - l A .

Proof of theorem 3.3. Let J be a dualizing A-modu le and suppose X is the charac ter associa ted to J as in 4.13. The reader should note tha t wi th this choice

of character , for each max imal ideal B in A , X is non- t r iv ia l on ~B and so 4.9

applies.

For the proof we shall first work with the pa r t i a l ze ta functions and ob ta in

a funct ional re la t ionship between ~A(J, M, s) and ~A(fl, M*. 1 - s) given M C

MCA).

Let r be the character is t ic function of 34 = M ~ . By making a change of

var iables we have the ident i ty

/~ , ,x , ,8-1x( tx ) dx = ,,t,,-s /E,[xi,8-1X(X) dx.

Subs t i tu t ing for lira - " in the integral below and apply ing 4.8 we obta in ,

f lltll-'dt f~ fe IlzllO-Xx(t~)r d~dt f~ II~ll'-~x(x) d~

= f~ fc IlxllS-1x(tx)r dtdz (1) fcIl~lP-'X(X) dx

_ fc Ilxll '-1 fc x(tx) r dt dx - fc[lx[l~-Xx(x) dx

f~ ilxll ,-1 r dx fcIl~ll~-lx(~) d~

497

GREEN

Interchanging the order of integration at (1) above is valid for 0 < Re(s) < 1 by 4.8.

Applying 4.9 to the integral in the denominator we obtain

_ IIPIImSI[AI1-1 IIMII f~t* [IxJl 8-1dx ~x(1 - , ) ~ ( , ) '

for some m E 2g. Using 4.10 we obtain the following equation relating the partial zeta functions:

~A(J,M*, 1 - s) c(M*)IIM*[I "-1 s c(M)IlMl[1-'trP[Imsil~l[ -1 ~x(1 - ~) r

We have c(M) = c(M*) since by the properties of the residual quotient,

r ( M ) = ( M : M) = ( ( J : M * ) : M) = (g : M ' M ) and

r (M*) = (M*: M*) = ( ( J : M ) : M*) = ( J : MM*).

To establish the relationship between the A-norm of M* and M we recall that the J residual quotient gives a duality theory for M(A). Therefore

HM[[ = [A: M ] - 1 = [M*: A*] -1

= ( [M*: A][A: j ] ) - l , (A* = J )

= [A: J ] - I [ A : M*] = [[g[[ [[M*H -1.

Substituting in the equation for the zeta functions we obtain

~A(J,M*,I-s) I IPI Im~llJI l~-~l l~l l - ' ~A(J,M,s) ~(1-~) ~(~) (2)

We shall determine IlPllm'llJll~-'ll~ll -~, which is independent of the class of M, by computing gA(J,A*,s) directly. In order to describe those L E M(A)

with L ~ A* we note that A, being a Dedekind domain with only finitely many maximal ideals is a principal ideal domain and also that A* = ( J : A) E M(A). The A-ideals equivalent to A* will therefore all be of the form A*N for some N E M(A). Further the condition A*N _ J is equivalent to asserting that N _ A, that is, is an integral A-ideal. As the norm is mnltiplicative in A,

IIWNII = [~: A]/[A : A*N]

= [~ : A] / ( [~ : ~* ] [~ : N])

~- [ A : "~ ' * ] -1 [ '~ : Y ] - ' = IIWII Ilgllx.

The partial zeta function can therefore be expressed as:

~ ' A ( J , A * , 1 - s ) = IIWII~-" ~x(1 - ~ ) .

4 9 8

G R E E N

Since A A* we also have ~A(J, A, s) = fIX* I] 8 ~ (s) By compar ing with (2)

above, we obtain [IP[[msIIJHI-'HA[1-1 = NA*I]I-2", so tha t

fA(g, M, s ) / r E (s) = I1 * II 2,-I;A(j ' M*, 1 - s)/~X (1 - s),

for each M E M(A). When summing over all the par t ia l zeta functions the relation M** = M ensures tha t each part ial zeta function is counted exactly once. The functional equat ion has been derived provided 0 < Re(s) < 1. By the principal of analytic cont inuat ion it holds everywhere.

Proof of corollary 3.4. As A is a principal ideal domain, and for any dualizing A-module J, A* = ( J : A) is an A-module, it follows tha t A* ~ jr. the dualizing modules being uniquely determined up to multiplication by an invertible element, it is possible to choose J so tha t A* = jr .

Proof of theorem 3.5. Suppose

~A(J,s) /~x(S ) -~ a_k u - k + a_k+lU -k+l + . . . + am urn,

with u = IIPl[ s and a -k , a--k+1, . . . , am C Z. The t e rm of lowest u-degree comes f rom the t e rm I[J[l" in the zeta function fA(J,s) . Hence [IJ]l" = u - k and [[Jll = [IPll - k proving the first pa r t of the theorem.

Suppose [[A*][ = ][P]Ie so tha t [ [~*[[~,-1= [[p[[-eu2e" Now

~A(J, 1 -- S ) / fx (1 -- s) = b_mu - m + b_m+l u-m+1 + . . . + bkU k,

where bl = a-i]]P]] - i for - r n < i < k, so f rom the functional equation we obtain

+ . . . + a inu = lIpll- u ~ - m + . . . +

and deg (UkfA(J, s ) / f x ( s ) ) = 2(e + k). It remains to show tha t e + k = 6p where

I l p ] l = A]. Since

[]Pl[ -(~+k) = [A: A*]/[A : J] = [ ] : J ] [ J : A] -- [A: A],

the p roo f is complete.

G l o b a l c a se .

Proof of theorem 3.6. Suppose first tha t Re(s) > 1. For almost all pr imes P c O, Jp = OF = OF, so tha t foe(JR,s) = f ~ ( s ) for these primes. By the Euler product formula and 3.1 it follows tha t f ( s ) = fo(J , s ) /g~(s) is an analytic function with analytic continuation an entire function of the whole complex plane. Therefore as f~ (s) admits analytic continuation to a meromorphic function of the whole of e so does fo (J , s ) .

499

G R E E N

Now ~'~(s) has only one pole, the s imple pole at s -- 1, so in order to es tabl ish the cor responding fact for ~o(J,s) we need only show tha t f ( s ) is a p roduc t of the polynomia ls

o (Jp, e 8 with u = liP]lop for each of the finitely many pr imes P c O such tha t OF, Jp

OR. Observe t ha t as a formal Laurent series toe(JR, s) E ]N((u)) by definit ion and

= H (1 - e l < i < n

where ei is the unique integer such t ha t ]]Bi]l-6e = ]]P]]~e for each max ima l O p -

ideal Bi, 1 < i < n. Since ~'z (s) -1 does not v a n i s h a t s = 1 and ~oe(Jp,s) has - - O p

a posi t ive lower bound for s in any real interval 1 < s < b, f (1) ~ 0, comple t ing the proof of the theorem.

The proofs of theorems 3.7 and 3.8 follow direct ly from the corresponding resul ts in the local case and the Euler p roduc t identity. (We r emark t ha t if L E

M(O) then ]]L[] o = l~p I]LP[Iop �9 )

Proof of corollary3.9. Let O be an order in E and J a dualizing O-module . We show there exists a dual iz ing module J1 such t ha t O* -- (J1 : O) = 7, the

conduc tor of O in O. F rom the proof of 3.4, for each pr ime P we can find x p E E

so t ha t for the dual izing O p - m o d u l e x p J p , ( x p J p : Op) = 7p. To complete the proof it therefore suffices to find a dual iz ing O-module J1, wi th J1P = x p J p . Observe t ha t as O is an order and J E M(O) , for a lmost all P one has JR = OF = OR, so t ha t in these cases xp -- 1 will do. For each pr ime P form the

ideal x p O p . Then the xpOp form the fibres of a one dimensional locally free sheaf over Spec( O ) and so by [4], ch. II, w and 5.4, cor respond to an invert ible

(O-pro jec t ive ) module , N E M(O) , which is unique up to isomorphism. Final ly

let J1 = N J, then by [9], p. 20, J1 is a dual iz ing O-modu le and J1P = x p J p .

A n e x a m p l e

Let F be a finite field of order q, K = F ( X ) and E = F ( X 1/3) so t ha t O -- FIX1/3]. Take O = F[X, X4/3, X 5/3] and set t 3 -- Z so t ha t O -- F [ t3 , t4 , th ] ,

and 0 = F[t]. We localize at the pr ime P = t 3 0 + t 4 0 + t S O and work with the P - c o m p l e t i o n th roughout . Then ~q = O ~ = F[[t3,t4,th]], -~ = - O p = F[[t]],

= F(( t ) ) and Jr = P = tSF[[t]]. As [~ : 4] ---- q2 and [4 : 7] = q the

Gorens te in condi t ion fails. One easily checks tha t for J = ~q + t~q is a dual iz ing

module wi th ~* = ( J : ~) = Jr and ]lJ]]~ = q.

We next compute the ideals of given norm. Let /~ C M ( N ) . Then L ~ = t ' ~

for some n and:

500

GREEN

Hence t3~ C t-'~/2 ___ ~ and t-r~/2 corresponds to an A/taA-subspace of ~/t3~. As -]/ t3~ ~ F + tF + te F and A/tS~ ~ F the F-subspaces will be of dimension 1, 2 or 3. There are q2 + q + 1 different subspaces for each of dimension 1 and 2. Thus amongst the fractional A-ideals /2 such that t'~+sA C /? _ t'~A there is one ideal of norm IIt"~llA = a s - " and q2 + q + 1 ideals of norm ql-,~ and q - ~ . By checking cases for different values of n we obtain the following table giving the number of ideals of various norm.

A-Norm Number of Number of ideals C A ideals C J

q 0 1 1 1 q + l

q-1 1 q + l q-2 q2 + q + 1 q2 + q + 1

q-n , n > 3 2q 2 q - q + l 2q 2 - b q q - 1

and

Let u -- IIPlI = q-S , then the zeta functions are:

1 ~'~(s) = l + ( q 2 + q ) u 2 + q u 3 ~ ( s ) - - 1 - - u ' 1 - - u

u -1 + q + q2u2 A- q2uS cA (J, ~) = 1 - - u

Although ~'~ (s) does not satisfy the functional equation, ~,~ (Y, s) does, as can be verified directly.

Re fe r ences

[1] M.F .ATIYAH and I. G. MACDONALD, Introduction to commutative alge- bra, Addison Wesley, London, 1969

[2] H. BASS, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28 [3] Z . J . BOREVICH and I. R. SHAFAREVICH, Number theory, Academic

Press, New York, 1966 [4] N. BOURBAKI, Elements of Math., Commutative Algebra, Addison Wesley,

Massachussets, 1972 [5] C . J . BUSHNELL and I. REINER, Zeta functions of arithmetic orders and

Solomon's conjectures, Math. Z. 173 (1980), 135-161 [6] C . J . BUSHNELL and I. REINER, Analytic continuation of partial zeta func-

tions of arithmetic orders, J. reine angew. Math. 349 (1984), 160-178 [7] C . J . BUSHNELL and I. REINER, Functional equations for Hurwitz series

and partial zeta functions of orders, J. reine angew. Math. 364 (1986), 130-148

501

GREEN

[8] V.M. GALKIN, g-functions of some one-dimensional rings, Izv. Akad. Nauk. SSSR. Ser. Mat. 37 (1973), 3-19

[9] J. HERZOG and E. KUNZ, Der kanonisches Modul eines Cohen-Macaulay Rings, Springer Lecture Notes in Math. 238, Berlin-Heidelberg-New York, 1971

[10] W.E. JENNER, On zeta functions of number fields, Duke Math. J. 36 (1969), 669-671

[11] G. POLYA and G. SZEGO, Problems and theorems in analysis 1, Die Grund- lehren der Math. Wissenschaften in Einzeldarstellungen 144, Berlin-Heidel- berg-New York

[12] L. SOLOMON, Zeta functions and integral representation theory, Advances in Math. 26 (1977), 306-326

[13] J. TATE, Fourier analysis in local fields and Hecke's zeta functions, Thesis, Princeton University, 1950

[14] A. WEIL, Basic Number Theory, Die Grundlehren der Math. Wissenschaften in Einzeldarstellungen 144, Berlin-Heidelberg-New York, 1974

Barry W. Green Mathematisches Institut Universits Heidelberg 6900 Heidelberg Germany

(Received February 26, 1989; in rev i sed form March 20, 1989)

502