Two-Variable Zeta Functions and Regularized Products

Two-Variable Zeta Functions and Regularized Products

Dedicated to Kazuya Kato

Christopher Deninger

Received: October 17, 2002 Revised: April 3, 2003

Abstract. In this paper we prove a regularized product expansion for the two-variable zeta functions of number fields introduced by van der Geer and Schoof. The proof is based on a general criterion for zeta-regularizability due to Illies. For number fields of non-zero unit rank our method involves a result of independent interest about the asymptotic behaviour of certain oscillatory integrals in the geometry of numbers. We also explain the cohomological motivation for the paper.

2000 Mathematics Subject Classification: 11M38; 11M41; 11H41;

14G40

Keywords and Phrases: zeta function, zeta regularization, oscillatory integral, metrized lattice

1 Introduction

In his paper [P] Pellikaan studied an interesting two-variable zeta function for algebraic curves over finite fields. Using notions from Arakelov theory of arithmetic curves, van der Geer and Schoof were led to introduce an analogous zeta function for number fields [GS].

In [LR] Lagarias and Rains investigated this two-variable zeta function thor- oughly for the special case of the rational number field. They also made some comments on the general case.

In earlier work we introduced a conjectural cohomological formalism to express Dedekind and more general zeta functions as regularized determinants of a

certain operator Θ on cohomology. In this framework it is not unreasonable to assume that the second variablewof the two-variable zeta function corresponds to an operator Θwdepending onw. These heuristics which are explained in the last section suggest a formula for the two-variable zeta function as a regularized product.

The main contribution of the paper is to prove this formula for the two-variable zeta function of any number field, Theorem 5.2. Our method is based on a powerful criterion of Illies for zeta-regularizability [I1], [I2]. We refer to section 5 for a short review of the relevant facts from the theory of regularization.

We also treat the much easier case of curves over finite fields. For number fields, our approach requires us to determine the asymptotic behaviour for Res→ ∞ of certain oscillatory integrals over spaces of lattices Γ. The function to be integrated isa⁻_Γ^swhere aΓ is the minimal length among the non-zero vectors in Γ. This is an interesting problem already for real quadratic fields in which case Don Zagier found a solution. The general case is treated in section 4.

The treatment in [GS] and [LR] of the two-variable zeta function for general number fields is somewhat brief. Also, the precise analogy with Pellikaan’s original zeta function is not written down. In the first two sections we therefore give a more detailed exposition of these topics. After this, some readers might wish to read the last section which motivated the paper.

I would like to thank Don Zagier very much for his help in the real quadratic case which was a great inspiration for me. I am also grateful to Eva Bayer and Georg Illies for useful remarks and to the CRM in Montreal for its support.

Finally I would like to thank the referees for their careful reading of the paper and their comments.

2 Background on two-variable zeta functions for curves over finite fields

Consider an algebraic curve X over the finite field F_q with q =p^r elements.

Let|X|be the set of closed points ofX and forx∈ |X|set

degx= (Fq(x) :F_q). The zeta function ofX is defined by the Euler product ZX(T) = Y

x∈|X|

(1−T^degx)⁻¹ inZ[|T|].

For a divisorD=P

x∈|X|nx·xwithnx∈Zwe set degD=P

nxdegx. Then we have

ZX(T) =X

D≥0

T^degD (1)

where the sum runs over all effective divisors i.e. those with nx ≥ 0 for all x∈ |X|. LetCH¹(X) denote the divisor class group ofX and forD= [D] set

hⁱ(D) :=hⁱ(D) = dimHⁱ(X,O(D)).

Summing over divisor classes in (1), one gets the formula:

ZX(T) =X

D

q^h0(D)−1

q−1 T^degD. (2)

Here it is enough to sum overD’s with degD:= degD≥0. In [P]§3 Pellikaan had the idea to replaceqby a variableuin this formula. His two-variable zeta function is defined by

ZX(T, u) =X

D

u^h0(D)−1

u−1 T^degD. (3)

Reconsidering classical proofs he obtained the following properties in the case whereX is smooth projective and geometrically irreducible:

ZX(T, u) = PX(T, u)

(1−T)(1−uT) withPX(T, u)∈Z[T, u] (4) PX(T, u) =

2g

X

i=0

Pi(u)Tⁱ withPi(u)∈Z[u], where (5) P0(u) = 1, P2g(u) =u^g, degPi(u)≤1 + i

2 and gis the genus ofX . (6) The two-variable zeta function enjoys the functional equation

ZX(T, u) =u^g−1T^2g−2ZX

µ 1 T u, u

¶

. (7)

In terms of thePi(u) it reads:

P2g−i(u) =u^g−iPi(u). (8) For example, for X = P¹ one has PX(T, u) = 1 and for X an elliptic curve PX(T, u) = 1 + (|X(Fq)| −1−u)T+uT².

Recently Naumann [N] proved the interesting fact that the polynomialPX(T, u) is irreducible in C[T, u].

In [GS]§7, van der Geer and Schoof consider the following variant of Pellikaan’s zeta function. They show that for complex sand t in Res <0,Ret <0 the series

ζ_X^GS(s, t) = X

D∈CH¹(X)

q^{sh0(D)+th1(D)} (9)

defines a holomorphic function with a meromorphic continuation toC×C. The explicit relation withZX(T, u) is not stated in [GS], so we give it here:

Proposition 2.1

ζ_X^GS(s, t) = (q^s+t−1)q^t(g−1)ZX(q^−t, q^s+t)

= (q^s+t−1)q^s(g−1)ZX(q^−s, q^s+t).

Proof Using the Riemann–Roch theorem one obtains, c.f. [GS] proof of prop. 5:

ζ_X^GS(s, t) =q^t(g−1) X

0≤degD≤2g−2

q^(s+t)h0(D)q^−tdegD+h µ q^sg

1−q^s+ q^tg 1−q^t

¶ .

Here his the order of CH¹(X)⁰, the group of degree zero divisor classes on X. This gives the meromorphic continuation to C×C. On the other hand according to [P], p. 181 settingu=q^s+t, T =q^−twe have:

(q^s+t−1)ZX(q^−t, q^s+t) = X

0≤degD≤2g−2

q^(s+t)h0(D)q^−tdegD

+h

µq^sg+t(1−g) 1−q^s − 1

1−q^−t

¶ .

This implies the first equality in the proposition. The second follows from the

functional equation (7) ofZX(T, u). 2

In particular the second relation in the proposition shows that fors+t= 1 we have

ζ_X^GS(s,1−s) = (q−1)q^s(g−1)ζX(s) whereζX(s) =ZX(q^−s) (10) as stated in [GS] proposition 5. Note that forζ_X^GS(s, t) the functional equation takes the simple form:

ζ_X^GS(s, t) =ζ_X^GS(t, s). (11) In the number field case, Lagarias and Rains introduced the substitutiont = w−s. Thus we define here as well

ζX(s, w) =ζ_X^GS(s, w−s) = (q^w−1)q^−s(1−g)ZX(q^−s, q^w). (12) This meromorphic function ofsand wsatisfies the functional equation

ζX(s, w) =ζX(w−s, w) (13)

and forw= 1 we have:

ζX(s,1) = (q−1)q^−s(1−g)ζX(s). (14) The rest of this section contains observations of a tentative nature which are not necessary for the sequel. It is unknown whether ZX(T, u) has a natural cohomological interpretation. The properties ofZX(T, u) are compatible with the following conjectural setup. Let K be a field of characteristic zero con- taining Q(u). For varieties over finite fields there might exist a cohomology theoryQHⁱconsisting of finite-dimensionalK-vector spaces with the following

property: Theq-linear Frobenius Frq acting on a varietyX/Fq should induce a K-linear map Fr^∗_q such that we have:

ZX(T, u) =

2

Y

i=0

detK(1−TFr^∗_q|QHⁱ(X))^(−1)i+1. (15) We get the correct denominator in (4) if

QH⁰(X) =K , Fr^∗_q = id and QH²(X)∼=Kwith Fr^∗_q =u·id and

QHⁱ(X) = 0 fori >2.

Then P(T, u) would be the characteristic polynomial of Fr^∗_q onQH¹(X) and therefore we would have

dimKQH¹(X) = 2g .

The functional equation (7) would be a consequence of Poincar´e duality – a perfect Fr^∗_q-equivariant pairing of K-vector spaces:

QHⁱ(X)×QH²⁻ⁱ(X)−→QH²(X)∼=K . Moreover Poincar´e duality would imply

det(Fr^∗_q|QH¹(X)) =u^g.

For an elliptic curve X/F_q, comparing the logarithmic derivatives of (4) and (15) atT = 0 gives

2

X

i=0

(−1)ⁱTr(Fr^∗_q|QHⁱ(X)) =|X(Fq)|. (16) However, if in (16) we replace Fr^∗_q by its power Fr^ν_q^∗we do not obtain|X(F_q^ν)| forν ≥2.

3 Background on two-variable zeta functions of number fields We begin by collecting some notions from one-dimensional Arakelov theory following [GS].

For a number fieldk/Qleto_k be its ring of integers. Bypwe denote the prime ideals ino_k and byvthe infinite places ofk. Consider the “arithmetic curve”

Xk = speco_k∪ {v| ∞}. The elements of the group

Z¹(Xk) =M

p

Z·p⊕M

v| ∞

R·v

are called Arakelov divisors. Define a map div :k^∗−→Z¹(Xk) by the formula

div (f) =X

p

ordp(f)p−X

v

evlog|f|^vv .

Here |f|^v=|σv(f)| for any embeddingσv in the class v andev= 1 ifv is real and ev = 2 ifv is complex. The cokernel of div is called the Arakelov Chow group CH¹(Xk) ofXk.

With the evident topologies the groupsk^∗, Z¹(Xk) and CH¹(Xk) become lo- cally compact topological groups. The counting measure on L

pZ·p and the Lebesgue measure onL

v| ∞R·vinduce Haar measuresdDonZ¹(Xk) anddD onCH¹(Xk).

For an Arakelov divisor D=X

p

νp·p+X

v

xv·v inZ¹(Xk) define a fractional ideal inkby the formula

I(D) =Y

p

p^−νp.

The infinite components ofD determine a normk k^Donk⊗R=L

vkv by the formula

k(zv)k²D=X

v

|zv|²||1||²v .

Here ||1||²v=e^−2xv ifvis real and||1||²v= 2e^−xv ifv is complex.

Forf ∈k ,→k⊗Rwe then have

||f||²D= X

vreal

|f|²ve^−2xv+ 2 X

vcomplex

|f|²ve^−xv . (17)

The embedding I(D),→k⊗Rand the norm || ||^D turn I(D) into a metrized lattice. The latticesI(D) andI(D⁰) are isometric (by anok-linear isometry) if and only if [D] = [D⁰] inCH¹(Xk).

Letκbe the Arakelov divisor with zeroes at the infinite components andI(κ) = d⁻¹, whered=d_k/Qis the different ofk/Q.

In the number field case, van der Geer and Schoof replace the orderq^hi(D)of Hⁱ(X,O(D)) forX/Fq by the Theta series:

k⁰(D) = X

f∈I(D)

e^−π||f||2D (18)

and

k¹(D) =k⁰([κ]− D) (19) forD= [D] inCH¹(Xk). For quadratic number fields the behaviour ofk⁰(D) is studied in some detail in [F].

According to [GS] proposition 1, the Poisson summation formula gives the Riemann–Roch type formula

k⁰(D)k¹(D)⁻¹=N(D)d⁻_k^1/2. (20) Here dk =|d_k/Q|is the absolute value of the discriminant ofk/Qand

N :CH¹(Xk)−→R^∗₊ is the Arakelov norm induced by the map

N :Z¹(Xk)−→R^∗₊, N(D) =Y

p

Np^νpY

v

e^xv .

LetZ¹(Xk)⁰ be the kernel of this map and set

CH¹(Xk)⁰=Z¹(Xk)⁰/div (k^∗).

This is a compact topological group which fits into the exact sequence

0−→CH¹(Xk)⁰−→CH¹(Xk)−→^N R^∗₊−→1. (21) Letd⁰Dbe the Haar measure onCH¹(Xk)⁰ with

vol (CH¹(Xk)⁰) =hkRk (22) where hk =|CH¹(speco_k)| is the class number of k andRk is the regulator.

Then we have

dD=d⁰Ddt

t . (23)

Fortin R^∗₊ consider the Arakelov divisor, wheren= (k:Q) Dt=n⁻¹ X

vreal

logt·v+n⁻¹ X

vcomplex

2 logt·v .

Setting D^t = [Dt] we have N(D^t) = t, so that the homomorphism t 7→ D^t provides a splitting of (21).

We need the following estimates:

Proposition 3.1 For every number fieldkand everyR≥0there are positive constants c1, c2, αsuch that uniformly inD ∈CH¹(Xk)⁰ and|w| ≤R we have the estimates

a)|k⁰(D+D^t)^w−1| ≤c1|w|exp(−πnt^−2/n) for all 0< t≤√ dk. b) |k⁰(D+D^t)^w−t^wd⁻_k^w/2| ≤c2|w|exp(−αt^2/n) for allt≥√

dk.

ProofAccording to [GS] corollary 1 there is a constantβ >0 depending only on the fieldksuch that for allDinCH¹(Xk)⁰ and all 0< t≤d^1/2_k we have

0< k⁰(D+D^t)−1≤βexp(−πnt^−2/n). (24) We may assume thatR≥1. For every−¹2 ≤x≤ ¹2 and|w| ≤Rsetting

(1 +x)^w= 1 +wx+wx²ϑ(x, w) (25) we have

|ϑ(x, w)| ≤e^2R. (26)

Namely, writing

(1 +x)^w=e^wlog(1+x)=e^wx(1+ηx)

we haveη =−¹2+^x₃ −^x4² +−. . . and hence|η| ≤ 1. Expandinge^wx(1+ηx) as a Taylor series and estimating gives inequality (26). For the moment we only need the following consequence of (26):

|(1 +x)^w−1| ≤x|w|³ 1 +1

2e^2R´

for 0≤x≤1/2 and|w| ≤R≥1. (27) Ifε=ε(k)>0 is sufficiently small, (24) implies that

x=k⁰(D+D^t)−1

lies in (0,1/2) for all 0< t ≤ε and allD. Using (24) and (27) we therefore find a constantc⁰₁ such that a) holds for all 0< t≤ε. By compactness of

CH¹(Xk)⁰× {|w| ≤R} ×[ε,p dk]

and continuity of _w¹(k⁰(D+D^t)^w−1) as a function of D, w and t there is a constantc⁰⁰₁ such that a) holds inε≤t≤√

dk. Thus we get the estimate a) by taking c1= max(c⁰₁, c⁰⁰₁). The estimate b) follows from a) using the Riemann–

Roch formula (20) and observing thatN([κ]) =dk. 2

The two-variable zeta function of van der Geer and Schoof is defined by an integral analogous to the series (9)

ζ_X^GS_k(s, t) = Z

CH¹(Xk)

k⁰(D)^sk¹(D)^tdD in Res <0,Ret <0. (28) According to [GS] proposition 6, this integral defines a holomorphic function in Res <0,Ret <0. This also follows from the considerations below.

We refer the reader to the introduction of [LR] for further motivation to consider this two-variable zeta function.

Making the substitutionD 7→[κ]− Din the integral we find the formula ζ_X^GS_k(s, t) =

Z

CH¹(Xk)

k⁰(D)^tk¹(D)^sdD in Res <0,Ret <0. (29) We will use the Lagarias–Rains variabless andw=t+s and concentrate on the function

ζXk(s, w) = ζ_X^GS_k(s, w−s) = Z

CH¹(Xk)

k⁰(D)^w−sk¹(D)^sdD (30)

(20)= d^s/2_k Z

CH¹(Xk)

k⁰(D)^wN(D)^−sdD. (31) It is holomorphic in the region Rew <Res <0.

Most of the following proposition is stated in [GS] and proved in [LR] Appendix using references to Ch. XIII of Serge Lang’s book on algebraic number theory.

Below we will write down the direct proof which is implicit in [GS].

Proposition 3.2 The function ζXk(s, w)has a meromorphic continuation to C² and it satisfies the functional equation

ζXk(s, w) =ζXk(w−s, w). Moreover the function

w⁻¹s(w−s)ζXk(s, w) is holomorphic in C². More precisely, the integral

J(s, w) = Z ^√dk

0

Z

CH¹(Xk)⁰

w⁻¹(k⁰(D+D^t)^w−1)d⁰Dt^−sdt t defines an entire function in C² and we have the formula

ζXk(s, w) =w³

d^s/2_k J(s, w) +d^(w_k ^−s)/2J(w−s, w)´

− µ1

s+ 1 w−s

¶ hkRk.

Recall that volCH¹(Xk)⁰=hkRk. Finally, forw= 1 one has

ζXk(s,1) =|µ(k)|d^s/2_k 2^−r1/2ζˆk(s). (32) Here ˆζk(s) is the completed Dedekind zeta function ofk

ζˆk(s) =ζk(s)ΓR(s)^r1ΓC(s)^r2 where we have set

ΓR(s) = 2^−1/2π^−s/2Γ(s/2) and ΓC(s) = (2π)^−sΓ(s).

Thus ΓR(s)ΓR(s+ 1) = ΓC(s). Here r1 and r2 are the numbers of real resp.

complex places ofk. Moreover µ(k) is the group of roots of unity ink.

Remarks 1 Formula (32) coincides with the corresponding formula in [GS]

proposition 6 after correcting two small misprints in that paper: We have p|∆|^s instead of p

|∆|^s/2 in [GS] proposition 6 and 2⁻¹π^−s/2. . . instead of 2π^−s/2. . .in the third equality on p. 388 of [GS].

2 The reason for our normalization of ΓR(s) comes from the theory of zeta- regularization, c.f. section 5.

ProofWe write the integral representation (31) forζXk(s, w) as a sum of two contributions:

ζXk(s, w) =I(s, w) +II(s, w) (33) where

I(s, w) =d^s/2_k Z ^√dk

0

Z

CH¹(Xk)⁰

k⁰(D+D^t)^wd⁰Dt^−sdt t and

II(s, w) =d^s/2_k Z ∞

√dk

Z

CH¹(Xk)⁰

k⁰(D+D^t)^wd⁰Dt^−sdt t .

The estimate in proposition 3.1 a) shows that the first integral defines a holo- morphic function of (s, w) in the region {Res < 0} ×C. Here and in the following we use the following well known fact. Consider a function f(s, x) holomorphic in several complex variablessandµ-integrable inxwhich locally in sis bounded by integrable functions ofx. Then the integral R

f(s, x)dµ(x) is holomorphic ins.

WritingI(s, w) in the form I(s, w) =d^s/2_k

Z √

d_k 0

Z

CH¹(X_k)⁰

(k⁰(D+Dt)^w−1)d⁰Dt^−sdt

t −hkRk

s (34)

the same estimate gives its meromorphic continuation to C². Note that, even divided bywthe first term is holomorphic inC².

Using Riemann–Roch (20) a short calculation shows that for Res >Rewwe have

II(s, w) =I(w−s, w). (35)

In particular the integral (31) defines a holomorphic function in Rew <Res <

0 as asserted earlier. Using (34) we find the formula:

II(s, w) =d^(w_k ^−s)/2 Z √

dk 0

Z

CH¹(Xk)⁰

(k⁰(D+Dt)^w−1)d⁰Dt^−(w−s)dt

t −hkRk

w−s (36) which gives the meromorphic continuation ofII(s, w) toC²: Again, even after division by wthe first term is holomorphic in C². This implies the assertions

of the proposition except for formula (32) which requires a lemma that will be

useful in the next section as well: 2

Lemma 3.3 In the region Res > Rew,Res > 0 the following integral repre- sentation holds, the integral defining a holomorphic function even after division by w:

ζXk(s, w) =d^s/2_k Z

CH¹(Xk)

(k⁰(D)^w−1)N D^−sdD. (37)

Proof of formula (32) Using (37) we find forw= 1<Resthat

|µ(k)|⁻¹d⁻_k^s/2ζXk(s,1) =|µ(k)|⁻¹ Z

CH¹(Xk)

(k⁰(D)−1)N D^−sdD. Now on p. 388 of [GS] this integral is shown to equal

(2⁻¹π^−s/2Γ(s/2))^r1((2π)^−sΓ(s))^r2ζk(s)

c.f. remark 1 above. 2

Proof of the lemma The estimate in proposition 3.1, b) shows that the following formula is valid in the region Res >Rew,Res >0:

II(s, w) =d^s/2_k Z ∞

√dk

Z

CH¹(Xk)⁰

(k⁰(D+Dt)^w−1)d⁰Dt^−sdt

t +hkRk

s . (38)

Note here that the double integral with integrand 1−t^wd⁻_k^w/2 is absolutely convergent when Res >Rew,Res >0.

The integral in formula (38) defines a holomorphic function in this region even after division by w. As the integral in formula (34) for w⁻¹I(s, w) gives a holomorphic function inC²the assertion follows by adding equations (34) and

(38). 2

Remark For k = Q a more elaborate version of the lemma is given in [LR]

Theorem 2.2.

Proposition 3.2 and formula (32) in particular suggest that a better definition of a two variable zeta function might be the following

ζ(Xk, s, w) =w⁻¹2^r1/2

|µ(k)|d⁻_k^s/2ζXk(s, w).

This is a meromorphic function onC² which satisfies the equations

ζ(Xk, w−s, w) =d^s_k^−w/2ζ(Xk, s, w) and ζ(Xk, s,1) = ˆζk(s). (39)

In section 5 we will see that ζ(Xk, s, w) is the “_2π¹ -zeta regularized version”

of ζXk(s, w). We also consider an entire version of this function which in the one variable case and in [LR] is called theξ-function. Because of our different normalization we give it another name which is suggested by the cohomological arguments in section 6.

Definition 3.4 The two-variableL-function of Xk is defined by the formula L(H¹(Xk), s, w) = s

2π s−w

2π ζ(Xk, s, w)

= 1

4π²

s(s−w) w

2^r1/2

|µ(k)|d⁻_k^s/2ζXk(s, w).

According to proposition 3.2 it is holomorphic inC²and satisfies the functional equation

L(H¹(Xk), w−s, w) =d^s−w/2_k L(H¹(Xk), s, w). Proposition 3.5 For any k/Qand every fixedw the entire function L(H¹(Xk), s, w)of shas order at most one.

ProofProposition 3.2 implies the formula L(H¹(Xk), s, w) =s(s−w)(T(s, w) +d

w 2−s

k T(w−s, w)) +d⁻_k^s/2 4π²

2^r1/2

|µ(k)|hkRk

whereT(s, w) is the entire function inC² defined by the integral T(s, w) = 1

4π² 2^r1/2

|µ(k)| Z ^√dk

0

Z

CH¹(Xk)⁰

w⁻¹(k⁰(D+D^t)^w−1)d⁰Dt^−sdt t .

Using the estimate in proposition 3.1, a) we find for some c(w)>0:

|T(s, w)| ≤ c(w) Z

√d_k

0

exp(−πnt^−2/n)t^−Resdt t

= c(w)d⁻_k^Res/2 Z ∞

1

exp(−πnd⁻_k^1/nt^2/n)t^Resdt t .

For Res≤1 the latter integral is bounded. For Res >1 we have

|T(s, w)| ≤ c(w)d^−Re_k ^s/2 Z ∞

0

exp(−πnd^−1/n_k t^2/n)t^Resdt t

= nc(w)

2 (πn)^−nRe2sΓ

µnRes 2

¶

= O³ exp³n

2Res´

log(Res)´

where theO-constant depends onw. Hence for alls∈Cwe have

|T(s, w)|=O³ exp³n

2|s|log|s|´´

.

Thus for everyε >0 the required estimate holds:

|L(H¹(Xk), s, w)|=O(exp(|s|^1+ε)) fors∈C.

2

Remark Fork =QLagarias and Rains prove that L(H¹(XQ), s, w) is entire of order at most one as a function of two variables, [LR] Theorem 4.1. They also mention that this assertion holds for generalk as well.

4 An oscillatory integral in the geometry of numbers

Recall that an Arakelov divisor D in Z¹(Xk) may be viewed as the lattice (I(D),|| ||^D). Two divisors define the same class Din CH¹(Xk) if and only if the corresponding metrized lattices are isometric by ano_k-linear isometry. In particular the following numbers are well defined forD= [D]:

a(D) = min{||f||²D|06=f ∈I(D)}

b(D) = min{||f||²D|f ∈I(D) such that||f||²D> a(D)} ν(D) = |{f ∈I(D)| ||f||²D=a_D}|.

By definitionb(D)> a(D)>0 are positive real numbers andν(D) is a positive integer – the so called kissing number of the lattice class.

These numbers arise naturally in the study of theta functions: Ordering terms, we may write

k⁰(D+D^t) = X

f∈I(D)

exp(−πt^−2/n||f||²D)

= 1 +ν(D)e^{−πt−2/na(D)}+. . . Here the next term is e^{−πt−2/nb(D)} with its multiplicity.

Proposition 4.1 OnCH¹(Xk)the functionais continuous whereas bandν are only upper semicontinuous. In particular a, b and ν are measurable. We have b(D) ≤ 4a(D) for all D, and ν is locally bounded. On CH¹(Xk)⁰ the functionsa, b, ν are bounded.

Points of discontinuity for b and ν arise as follows. Already for k = Q(√ 2) there exist convergent sequences Dⁿ → D even in CH¹(Xk)⁰ such that

b(Dⁿ) → a(D). Thus at the point D we have limn→∞b(Dⁿ) < b(D) and also the multiplicityν jumps up.

ProofFix an elementf ∈I(D) with ||f||²D=a_D. Then ||2f||²D= 4a_D. Thus b_D ≤4a_D. The continuity properties may be checked locally. So let us fix a classD⁰= [D⁰] inCH¹(Xk) and write:

D⁰=X

p

ν_p⁰·p+X

v

x⁰_v·v inZ¹(Xk). LetV be an open neighborhood ofx⁰= (x⁰_v)v| ∞in L

v| ∞Rand consider the continuous map:

V −→CH¹(Xk), x7−→ D^x= [Dx] whereDx=X

p

ν⁰_p·p+X

v| ∞

xv·v .

For V small enough this map is a homeomorphism of V onto an open neigh- borhood U of D⁰ in CH¹(Xk). Fix some R >0 such that for all xin V we have

R⁻¹≤e^xv ≤Rifvis real and R⁻²≤e^xv ≤R²ifv is complex. It follows that forx∈V and allf ∈I(Dx) =I(D⁰) we have the estimate

R⁻²||f||²≤ ||f||²Dx= X

vreal

|f|²ve^−2xv+ 2 X

vcomplex

|f|²ve^−xv ≤2R²||f||². (40) Here

||f||=³ X

v| ∞

|f|²v

´1/2

is the Euclidean norm in k⊗Rapplied to the elementf ∈k⊂k⊗R.

SinceI(D⁰) is discrete ink⊗Rit follows that for anyC >0 the set F^C={f ∈I(D⁰)|0<||f||²Dx ≤C for somex∈V}

is finite. If V is bounded it also follows that the map D 7→a(D) is bounded onU and so is bsince b(D)≤4a(D). Thus for large enough C >0 the finite subsetF=F^C⊂I(D) has the following properties: For allx∈V we have:

a(D^x) = min{||f||²Dx|f ∈ F}

b(D^x) = min{||f||²Dx|f ∈ F such that||f||²Dx > a(D^x)}

ν(D^x) = |{f ∈ F | ||f||²Dx =a(D^x)}|. (41) The functionsx7→ ||f||²Dx forf ∈ F being continuous it is now clear thata(D) is continuous nearD⁰, hence everywhere sinceD⁰ was arbitrary. (This fact is already mentioned in [GS].)

To check upper semicontinuity of b and ν at D⁰ let F⁰ be the subset of F consisting of allf with||f||²D⁰> a(D⁰). For small enoughV we then have

||f||²Dx > a(D^x) for allf ∈ F⁰ andx∈V (42) since both sides are continuous inx. It follows that

b(D^x)≤min{||f||²Dx|f ∈ F⁰}=:µ(x).

Then µis continuous and µ(x⁰) =b(D⁰). Hence, for every ε >0 there exists an open neighborhoodV⁰ ofx⁰in V such that

µ(V⁰)⊂(µ(x⁰)−ε, µ(x⁰) +ε) = (b(D⁰)−ε, b(D⁰) +ε).

Thus b(D^x) ≤ b(D⁰) +ε for all x ∈ V⁰ and hence b(D) ≤ b(D⁰) +ε for all D in a neighborhood (the image of V⁰) of D⁰ in CH¹(Xk). Henceb is upper semicontinuous atD⁰.

As forν, the representation (41) shows thatν(D^x)≤ |F| for allx∈V. Hence ν is a locally bounded function onCH¹(Xk).

With notations as above we have by (41) that

ν(D^x)≤ |FrF⁰|=ν(D⁰) for allx∈V .

This implies thatν is upper semicontinuous atD⁰. 2 The following theorem shows that onCH¹(Xk)⁰the functiona=a(D) acquires a unique global minimum at D = 0. We also describe a(D) explicitly in a neighborhood ofD= 0.

Set

amin= min{a(D)| D ∈CH¹(Xk)⁰}>0 and

binf = inf{b(D)| D ∈CH¹(Xk)⁰}.

Theorem 4.2 Setn= (k:Q) and let the notations be as above.

1 amin=n.

2 ForD ∈CH¹(Xk)⁰ we have a(D) =amin if and only if D= 0.

3 For the representative D = 0 of D = 0 and f ∈ok =I(0) we have ||f||²0 = a(0) =amin if and only iff ∈µ(k).

4 ForD ∈CH¹(Xk)⁰ with I(D)non-principal there is the estimate a(D)≥ √ⁿ

4amin=n√ⁿ 4.

5 For every open neighborhood U of D= 0 inCH¹(Xk)⁰ there is a positiveε such that a(D)< amin+εfor someD ∈CH¹(Xk)⁰ impliesD ∈U.

6 There is a neighborhood U of D= 0 in CH¹(Xk)⁰ with the following prop- erties: Every D ∈ U has the form D = [D] with D = P

v| ∞xv ·v. For f ∈I(D) =ok we have:

||f||²D=a(D) if and only iff ∈µ(k).

Moreover:

a(D) = X

vreal

e^−2xv + 2 X

vcomplex

e^−xv andν(D) =|µ(k)|.

7 We havebinf> amin.

Proof The main tool is the inequality between the arithmetic and the geo- metric mean. This inequality was already used in [GS]. Let|| ||^v=| |^ev^v be the normalized absolute value at the infinite placev.

1ForD= [D] inCH¹(Xk)⁰ andf ∈I(D) we have

||f||²D = X

vreal

(||f||^ve^−xv)²+ X

vcomplex

||f||^ve^−xv+ X

vcomplex

||f||^ve^−xv

(a)

≥ n Ã

Y

v

||f||^v

!2/nà Y

v

e^−xv

!2/n

=n|N(f)|^2/n Ã

Y

v

e^xv

!−2/n

= n(|N(f)|/N(I(D)))^2/n.

Here (a) is the arithmetic-geometric mean inequality and we have used that 1 =N(D) =Y

p

Np^νpY

v

e^xv =N(I(D))⁻¹Y

v

e^xv .

NowI(D) divides (f) and forf 6= 0 we therefore have

|N(f)|/N(I(D))≥1.

It follows that ||f||²D ≥n, so that a(D)≥n and therefore amin ≥n. On the other hand forD = 0 andf ∈ µ(k) we have||f||²0 =r1+ 2r2 =n. Therefore a(0) =nand henceamin=n.

2 We have seen thata(0) =amin. Now assume thata(D) =amin. Then there is somef ∈I(D) with||f||²D=n. It follows that|N(f)|=N(I(D)) hence that I(D) = (f) is principal and that we have equality in (a) above. Now in the arithmetic-geometric mean inequality, equality is achived precisely if all terms are equal. Thus there is a positive realξ such that

ξ= (||f||^ve^−xv)² for realv andξ=||f||^ve^−xv for complexv . Hence

ξⁿ=ξ^r1ξ^2r2 = Ã

Y

v

||f||^ve^−xv

!2

= (|N(f)|N(I(D))⁻¹)²= 1

since N(D) = 1 and |N(f)|= N(I(D)) as observed above. Thus ξ = 1 and therefore

||f||^v=e^xv for allv| ∞. (43)

It follows that

divf⁻¹ = X

p

ordpf⁻¹·p−X

v

log||f⁻¹||^v·v

= X

p

ordpI(D)⁻¹·p+X

v

log||f||^v·v

= X

p

νp·p+X

v

xv·v=D . Hence D= [D] = 0 inCH¹(Xk)⁰ and2is proved.

3 For D = 0 we have I(D) = ok. For f ∈ I(D) = ok the equation ||f||²0 = a(0) = nimplies ||f||^v = 1 for allv| ∞by (43). Since ||f||^p ≤1 for all finite primesp it follows by a theorem of Kronecker thatf is a root of unity.

4 IfI(D) is non-principal and 06=f ∈I(D), then we have (f) =I(D)·a for some integral ideal a6=ok. Hence|N(f)| ≥2N(I(D)) and4 follows from the above estimate for||f||²D.

5LetaU be the minimum of the continuous functiona=a(D) on the compact set CH¹(Xk)⁰ rU. For D 6= 0 we have a(D) > amin by 2. Hence ε :=

aU−amin>0. It is clear thata(D)< amin+εimplies that D ∈U.

6 As in the proof of proposition 4.1 there exists an open neighborhood V⁰ of x⁰= 0 in{x∈L

v| ∞R|P

xv= 0}such that firstly the map V⁰−→CH¹(Xk)⁰, x7−→ D^x= [Dx] whereDx= X

v| ∞

xv·v

is a homeomorphism onto an open neighborhoodU⁰ofD= 0 inCH¹(Xk)⁰. In particularI(Dx) =o_kfor allx∈V⁰. Secondly there is a finite subsetF ⊃µ(k) ofo_k such that for allx∈V⁰ we have:

a(D^x) = min{||f||²Dx|f ∈ F}

b(D^x) = min{||f||²Dx|f ∈ Fsuch that||f||²Dx> a(D^x)} (44) and

ν(D^x) =|{f ∈ F | ||f||²Dx =a(D^x)}|. Now, according to3we have

||f||²D0 =a(D⁰) forf ∈µ(k) and

||f||²D0 > a(D⁰) forf ∈ Frµ(k).

Choose someε >0, such that ||f||²D0−a(D⁰)≥2ε for allf ∈ F rµ(k). By a continuity argument we may find an open neighborhood 0∈V ⊂V⁰ such that for allx∈V we have

||f||²Dx−a(D^x)< ε iff ∈µ(k)

and

||f||²Dx−a(D^x)≥ε iff ∈ Frµ(k).

As||f||²Dx=||1||²Dx for allf ∈µ(k) it follows that forx∈V we have

||f||²Dx=a(D^x) if and only iff ∈µ(k). Moreoverν(D^x) =|µ(k)|and

a(D^x) =||1||²Dx = X

vreal

e^−2xv+ 2 X

vcomplex

e^−xv .

Therefore, in 6we may takeU to be the image ofV inCH¹(Xk)⁰.

7 Assume that binf = amin and let (Dⁿ) be a sequence of Dⁿ ∈ CH¹(Xk)⁰ with b(Dⁿ) → amin. Since CH¹(Xk)⁰ is compact we may assume that (Dⁿ) is convergent, Dⁿ → D⁰. Because of amin ≤ a(Dⁿ) ≤ b(Dⁿ) it follows that a(Dⁿ) → amin. On the other hand since a is continuous we have a(Dⁿ) → a(D⁰). Hencea(D⁰) =amin and by2 this implies thatD⁰ = 0. Thus we have Dⁿ→0 andb(Dⁿ)→amin.

LetV, U andF be as in the proof of6. Then forf ∈ F andx∈V we have

||f||²Dx> a(D^x) if and only iff /∈µ(k). By (44) this gives

b(D^x) = min{||f||²Dx|f ∈ Frµ(k)} for allx∈V .

In particular b(D^x) is a continuous function of x ∈ V and therefore b|^U is continuous. Let ˜U ⊂U be a compact neighborhood ofD = 0 inCH¹(Xk)⁰. Then there is some ˜D ∈U˜ withb(D)≥b( ˜D)> a( ˜D)≥aminfor allDin ˜U. On the other hand, fornlarge enough we haveDⁿ∈U˜ and henceb(Dⁿ)≥b( ˜D)>

amin. Henceb(Dⁿ) cannot converge toamin, Contradiction. 2

RemarkSince µ(k) acts isometrically on (I(D),|| ||^D) and sinceν(0) =|µ(k)| the minimal value of the function ν = ν(D) is |µ(k)|. As ν is upper semi- continuous it follows that the set of D in CH¹(Xk) resp. CH¹(Xk)⁰ with ν(D) = |µ(k)| is open. It should be possible to show that the complements have measure zero.

In the following we will deal with the asymptotic behaviour of certain functions defined at least in Res >0 as Restends to infinity. For such functionsf and g we will write

f ∼g to signify that lim

Res→∞f(s)/g(s) = 1. The following theorem is the main result of the present section:

Theorem 4.3 For a number field k/Q let r =r1+r2−1 be the unit rank.

Then the entire function C(s) =

Z

CH¹(Xk)⁰

ν(D)a(D)^−sd⁰D has the following asymptotic behaviour as Res→ ∞

C(s)∼ |µ(k)|αks^−r/2n^−s. Here we have set:

αk = (πn)^r/22^−r1/2p 2/n .

Proof If r = 0 then αk = 1 and CH¹(Xk)⁰ = CH¹(speco_k) is the class group ofk. HenceC(s) is a finite Dirichlet series. Fork=Qwe haveC(s) = ν(0)a(0)^−s=|µ(Q)|= 2. Forkimaginary quadratic the main contribution as Res→ ∞comes from the term corresponding toD= 0 which isν(0)a(0)^−s=

|µ(k)|2^−s. These assertions follow from theorem 4.2 parts1and 2(or 4) and 3.

Now assume that r≥1. The functionν =ν(D) is measurable and bounded on CH¹(Xk)⁰ by proposition 4.1. The function a = a(D) is continuous and CH¹(Xk)⁰is compact. HenceC(s) is an entire function ofs. We will compare C(s) with certain integrals over unbounded domains which can be evaluated explicitly in terms of Γ-functions. It is not obvious that these integrals converge.

For this we require the following lemma where for x ∈ R^N we set ||x||∞ = max|xi|.

After a series of auxiliary results the proof of theorem 4.3 is concluded after the proof of corollary 4.3.4 below.

Lemma 4.3.1 AssumeN ≥2 and consider the hyperplane HN ={x| P

xi= 0} inR^N. For everyxinHN we have

maxxi≥(N−1)⁻¹||x||∞ and minxi≤ −(N−1)⁻¹||x||∞.

Proof We may assume that x1 ≤ . . . ≤ xN, so that x1 = minxi and xN = maxxi. As x ∈ HN we have x1 ≤ 0 ≤ xN. It is clear that

||x||∞= max(−x1, xN).

If||x||∞=xN the first estimate is clear. If||x||∞=−x1then

(N−1) maxxi = (N−1)xN ≥xN+xN−1+. . .+x2=−x1=||x||∞. Hence the first estimate holds in this case as well. The second estimate follows

by replacingxwith −x. 2

We can now evaluate a certain class of integrals which are useful for our pur- poses.

Proposition 4.3.2 For N ≥ 2 let dλ be the Lebesgue measure on HN. Fix positive real numbers c1, . . . , cN and positive integers ν1, . . . , νN. Then for Res >0 we have the following formula where q= 1/PN

i=1ν_i⁻¹ I:=

Z

HN

³X^N

i=1

cie^−νixi´−s

dλ= q ν1· · ·νN

³Y^N

i=1

c^q/ν_i ⁱ´−s

Γ(s)⁻¹

N

Y

i=1

Γ(qs/νi).

ProofFirst we show that the integral exists. Using lemma 4.3.1 and the fact that min(xi)≤0 forx∈HN, we find with c= min(ci):

N

X

i=1

cie^−νixi ≥c e^−min(xi)≥cexp((N−1)⁻¹||x||∞) forx∈HN . (45) Thus the function

³X^N

i=1

cie^−νixi´−Res

is integrable over HN. In order to evaluate the integral we recall the Mellin transform of a (suitable) functionhonR^∗₊:

(M h)(s) = Z ∞

0

h(t)t^sdt

t for Res≥1 and the convolution of twoL¹-functions h1 andh2onR^∗₊:

(h1∗h2)(t) = Z ∞

0

h1(t1)h2(tt⁻₁¹)dt1

t1

.

For suitableh1 andh2Fubini’s theorem implies the basic formula M(h1∗h2) = (M h1)·(M h2) for Res≥1.

For t > 0 let dµ be the image of Lebesgue measure under the exponential isomorphism:

{x∈R^N| X

xi= logt}−→ {^∼ (t1,· · · , tN)∈(R^∗₊)^N|t1· · ·tN =t}. TheN-fold convolution of L¹-functionsh1, . . . , hN onR^∗₊ is given by the for- mula

(h1∗. . .∗hN)(t) = Z

t1···tN=t

h1(t1)· · ·hN(tN)dµ . Note that convolution is associative.

We may rewriteI as follows I=

Z

t1···tN=1

³X^N

i=1

cit^ν_iⁱ´−s

dµ .

Thus

Γ(s)·I = Z ∞

0

³Z

t1···tN=1

exp³

−t

N

X

i=1

cit^ν_iⁱ´ dµ´

t^sdt

t (46)

= Z ∞

0

³Z

t1···tN=t^1/q

exp³

−

N

X

i=1

cit^ν_iⁱ´ dµ´

t^sdt t

= qM(e^−c1tν1∗. . .∗e^−cNtνN)(qs)

= qM(e^−c1tν1)(qs)· · ·M(e^−cNtνN)(qs)

= q

N

Y

i=1

ν_i⁻¹c⁻_i^qs/νiΓ(qs/νi).

2 We may now use the complex Stirling asymptotics

Γ(s)∼√

2πe^−se(^s−¹2)^logs for|s| → ∞in −π <args < π (47) to draw the following consequence of proposition 4.3.2.

Corollary 4.3.3 Let k/Q be a number field of degree n with unit rankr = r1+r2−1≥1. Then we have the following asymptotic formula forRes→ ∞, the integral being defined for Res >0:

Z

P

v| ∞xv=0

³ X

vreal

e^−2xv + 2 X

vcomplex

e^−xv´−s

dλ∼αks^−r/2n^−s. Here

αk = (πn)^r/22^−r1/2p 2/n .

Proof Applying proposition 4.3.2 withN =r1+r2 and the obvious choices ofci’s andνi’s the integral is seen to equal:

n⁻¹2^1−r12^−2sr2/nΓ(s)⁻¹Γ(s/n)^r1Γ(2s/n)^r2 .

Applying the Stirling asymptotics gives the result after some calculation. 2 Corollary 4.3.4 Assumptions as in corollary 4.3.3. For anyε >0 set

Vε=n

x∈ M

v| ∞

R| X

v| ∞

xv= 0 and ||x||∞< εo .

Then we have the asymptotic formula forRes→ ∞: Z

Vε

³ X

vreal

e^−2xv + 2 X

vcomplex

e^−xv´−s

dλ∼αks^−r/2n^−s.