Analysis on Arithmetic Schemes. I
Ivan Fesenko
Received: October 10, 2002 Revised: March 30, 2003
Dedicated to Kazuya Kato
Abstract. A shift invariant measure on a two dimensional local field, taking values in formal power series over reals, is introduced and discussed. Relevant elements of analysis, including analytic duality, are developed. As a two dimensional local generalization of the works of Tate and Iwasawa a local zeta integral on the topological Milnor Kt
2-group of the field is introduced and its properties are studied.
2000 Mathematics Subject Classification: 28B99, 28C99, 28E05, 28-99, 42-28-99, 11S28-99, 11M28-99, 11F28-99, 11-28-99, 19-99.
Two components are important for a two dimensional generalization of the works of Iwasawa and Tate (both locally and globally). The first component is appropriate objects from the right type of higher class field theory. In the local case these are so called topological MilnorKt-groups ([P2–P4], [F1–F3],
[IHLF]), for the local class field theory see [Ka1], [P3–P4], [F1–F3], [IHLF]. The second component is an appropriate theory of measure, integration and generalized Fourier transform on objects associated to arithmetic schemes of higher dimensions. In the local case such objects are a higher local field and its Kt-group. It is crucial that the additive and multiplicative groups of the field
are not locally compact in dimension > 1. The bare minimum of the theory for two dimensional local fields is described in part 1 of this work. Unlike the dimension one case, this theory on the additive group of the field is not enough to immediately define a local zeta integral: for the zeta integral one uses the topologicalKt
2-group of the field whose closed subgroups correspond (via class
field theory) to abelian extensions of the field. A local zeta integral on them with its properties is introduced and discussed in part 2.
We concentrate on the dimension two case, but as usual, the two dimensional case should lead relatively straightforward to the general case.
We briefly sketch the contents of each part, see also their introductions. Of course, throughout the text we use ideas and constructions of the one dimen-sional theory, whose knowledge is assumed.
Two dimensional local fields are self dual objects in appropriate sense, c.f. section 3. In part 1, in the absence of integration theory on non locally compact abelian groups and harmonic analysis on them, we define a new shift invariant measure
µ:A −→R((X))
on appropriate ringAof subsets of a two dimensional local field. This measure is not countably additive, but very close to such. The variable X can be viewed as an infinitesimally small positive element ofR((X)) with respect to the standard two dimensional topology, see section 1. The main motivation for this measure comes from nonstandard mathematics, but the exposition in this work does not use the latter. Elements of measure theory and integration theory are developed in sections 4–7 to the extent required for this work. We have basically all analogues of the one dimensional theory. Section 9 presents a generalization of the Fourier transform and a proof of a double transform formula (or transform inverse formula) for functions in a certain space. We comment on the case of formal power series over archimedean local fields in section 11.
In part 2, using a covering of a so called topological MilnorKt
2-group of a two
dimensional local field (which is the central object in explicit local class field theory) by the product of the group of units with itself we introduce integrals over theKt
2-group. In this part we use three maps
which in some sense generalize the one dimensional mapE\ {0} −→E on the module, additive and multiplicative level structures respectively. In section 17 we define the main object – a local zeta integralζ(f, χ) associated to a function f:F×F −→C continuous onT and quasi-characterχ:Kt
2(F)−→C
×
ζ(f, χ) =
Z
T
f(α)χt(α)|t(α)|−22dµT(α),
for the notations see section 17. The definition of the zeta integral is the result of several trials which included global tests. Its several first properties in analogy with the one dimensional case are discussed and proved. A local functional equation for appropriate class of functions is proved in section 19. In the ramified case one meets new difficulties in comparison to the dimension one case, they are discussed in section 21. Zeta integrals for formal power series fields over archimedean local fields are introduced in section 23, their values are not really new in comparison to the dimension one case; this agrees with the well known fact that the class field theory for such fields degenerate and does not really use MilnorK2.
For an extension of this work to adelic and K-delic theory on arithmetic schemes and applications to zeta integral and zeta functions of arithmetic schemes see [F4].
One of stimuli for this work was a suggestion by K. Kato in 1995 to collabo-rate on a two dimensional version of the work of Iwasawa and Tate, another was a talk of A.N. Parshin at the M¨unster conference in 1999, see [P6]. I am grate-ful to A.N. Parshin, K. Kato, T. Chinburg, J. Tate, M. Kurihara, P. Deligne, S.V. Vostokov, J.H. Coates, R. Hill, I.B. Zhukov, V. Snaith, S. Bloch for useful remarks on this part of the project or encouragement. This work was partially supported by the American Institute of Mathematics and EPSRC.
1. Measure and integration on higher local fields
We start with discussion of self duality of higher local fields which distinguishes them among other higher dimensional local objects. Then we introduce a non-trivial shift invariant measure on higher dimensional local fields, which may be viewed as a higher dimensional generalization of Haar measure on one di-mensional fields. We define and study properties of integrals of a certain class of functions on the field. Self duality leads to a higher dimensional transform on appropriate space of functions. One of the key properties, a double trans-form trans-formula for a class of functions in a space which generalizes familiar one dimensional spaces, is proved in section 9.
1. In higher dimensions sequential aspects of the topology are fundamental: for example, in dimensions greater than one the multiplication is not always continuous but is sequentially continuous. Certainly, in the one dimensional case one does not see the difference between the two.
In a two dimensional local fieldR((X)) a sequence of series¡Pai,nXi ¢
n is a
fundamental sequence if there isi0 such that for allncoefficientsai,n are zero
for all i < i0 and for every i the sequence (ai,n)n is a fundamental sequence;
similarly one defines convergence of sequences. Every fundamental sequence converges. Now consider onR((X)) the so called sequential saturation topology (sequential topology for short): a set is open if and only if every sequence which converges to any element of it has almost all its elements in the set.
The sequence (aX)n, n
∈ N, tends to zero for every a ∈ R. Hence, if a norm on R((X)) takes values in an extension of R and is the usual module on the coefficients, then one deduces that |aX|<1 and so|X|is smaller than any positive real number. In other words, for two dimensional fields the local parameter X plays the role of an infinitesimally small element, and therefore it is very natural to use hyperconstructions of nonstandard mathematics. Nev-ertheless, to simplify the reading the following text does not contain anything nonstandard.
2. LetF be a two dimensional local field with local parameters t1, t2 (t2 is
a local parameter with respect to the discrete valuation of rank 1) with finite last residue field having q elements. Denote its ring of integers by O and the group of units byU; denote byRthe set of multiplicative representatives of the last finite residue field. The multiplicative groupF× is the product of infinite cyclic groups generated byt2 andt1 and the group of unitsU.
Denote byOthe ring of integers ofF with respect to the discrete valuation of rank one (sot2generates the maximal idealMofO). There is a projection
p:O −→ O/M=E whereE is the (first) residue field ofF. Fractional ideals ofF are of two types: principalti
2t j
1Oand nonprincipalMi.
For the definition of the topology on F see [sect. 1 part I of IHLF]. We shall view F endowed with the sequential saturation topology (see [sect. 6 part I of IHLF]), so sequentially continuous maps from F are the same as continuous maps. If the reader prefers to use the former topology onF, then the word “continuous” should everywhere below be replaced with “sequentially continuous”.
3. For a field L((t)) denote by resi = resti:L((t)) −→ L the linear map
P
ajtj →ai. Similarly define res = res−1:L{{t}} −→Lin the case whereLis
a complete discrete valuation field of characteristic zero (for the definition of L{{t}} see [Z2]).
For a one dimensional local fieldLdenote byψL a character of the additive
group Lwith conductorOL. Introduce
where πL is a prime element of L. The conductor of ψ′ is the ring O of the
corresponding field: i.e. ψ′(O) = 16=ψ′(t−1
1 O). An arbitrary two dimensional
local fieldF of mixed characteristic has a finite extension of typeL{{t}}which is the compositum of F and a finite extension of Qp [sect. 3 of Z2], so the
restriction ofψ′ onF has conductorO
F.
Thus, in each case we have a character ψ0:F −→C×
with conductorOF.
The additive group F is self-dual:
Lemma. The group of continuous characters onF consists of characters of the formα7→ψ(aα)wherearuns through all elements ofF.
This assertion is easy, at least for the usual topology on F in the positive characteristic case where F is dual to appropriately topologized Ω2
F/Fq (c.f.
[P1], [Y]) which itself is noncanonically dual to F. Since we are working with a stronger, sequential saturation topology, we indicate a short proof of duality.
Proof. Given a continuous characterψ′, there arei, jsuch thatψ′(ti 2t
j 1O) = 1,
and so we may assume that the conductor ofψ′isO. In the equal characteristic case the restriction of ψ′ onO induces a continuous character onE which by the one dimensional theory is a “shift” of ψE, hence there isa0∈ Osuch that
ψ1(α) =ψ′(α)−ψ(a0α) is trivial onO. Similarly by induction, there isai∈ti2O
such that ψi+1(α) =ψi(α)−ψ(aiα) is trivial ont−2iO. Thenψ′(α) =ψ(aα)
witha=P+0∞ai.
In the mixed characteristic case it suffices to consider the case of Qp{{t}}.
The restriction of ψ′ ontiQ
p is of the formα7→ψ(aiα) with ai ∈t−i−1pZp,
andai→0 when i→+∞; henceψ′(α) =ψ(aα) fora=P+−∞∞ai.
Remark. Equip characters ofF with the topology of uniform convergence on compact subsets (with respect to the sequential topology) ofF; an example of such a set in the equal characteristic case is{Pi>i0ait
i
2:ai ∈Wi} whereWi
are compact subsets ofE. It is easy to verify that the mapa7→(α7→ψ(aα)) is a homeomorphism between F and its continuous characters.
4. To introduce a measure on F we first specify a nice class of measurable sets.
Definition. A distinguished subset is of the form α+ti 2t
j
1O. Denote by A
the minimal ring in the sense of [H] containing all distinguished sets.
It is easy to see that if the intersection of two distinguished sets is nonempty, then it equals to one of them. This implies that an element A of A can be written asSiAi\(SjBj) with distinguished disjoint setsAiand distinguished
each Bj is a subset of some Ai). One easily checks that if A is also of the
similar form SlCl\(SkDk) thenA=S(AiTCl)\¡S(AiTBjTClTDk) ¢
. Every element of A is a disjoint (maybe infinite countable) union of some distinguished subsets. Distinguished sets are closed but not open. For example, setsO\t2O=Sj>0tj1U andO×=
S j∈Zt
j
1U do not belong toA.
Alternatively,Ais the minimal ring which contains setsα+ti
2p−1(S),i∈Z,
whereS is a compact open subset of E.
Definition–Lemma. There is a unique measureµonF with values inR((X)) which is a shift invariant finitely additive measure onAsuch thatµ(∅) = 0,
µ(ti
2tj1O) =q−jXi.
The proof immediately follows from the properties of the distinguished sets. The measureµdepends on the choice oft2 but not on the choice oft1.
We get µ(ti
2p−1(S)) = XiµE(S) where µE is the normalized Haar measure
onE such thatµE(OE) = 1.
Remarks.
1. This measure can be viewed as induced (in appropriate sense) from a measure which takes values in hyperreals ∗R. A field of hyperreal numbers ∗R introduced by Robinson [Ro1] (for a modern exposition see e.g. [Go1]) is a minimal field extension ofR which contains infinitesimally small elements and on which one has all reasonable analogues of analytic constructions. If one fixes a positive infinitesimal ω−1
∈ ∗R, then a surjective homomorphism from the fraction field of approachable polynomialsR[X]ap to R((X)), andω−1 7→X
determines an isomorphism of a subquotient of∗R ontoR((X)).
The first variant of this work employed hyperobjects, and its conceptual value ought to be emphasized.
2. The measure µ takes values in R((X)) (for its topology see section 1) which has the total ordering: Pn>n0anXn>0,an0∈R
×, iffa
n0 >0. Notice
two unusual properties: an = 1−q−n is smaller than 1−X, but the limit
of (an) is 1; not every subset bounded from below has an infimum. Thus, the
standard concepts in real valued (or Banach spaces valued) measure theory, e.g. the outer measure, do not seem to be useful here. In particular, the integral to be defined below will possess some unusual properties like those in section 8.
3. The set O∈ Aof measure 1 is the disjoint union of t2O∈ Aof measure
X, t2t−1jO\t2t−1j+1O of measureqj(1−q−1)X forj >0, andtl1O\tl+11 O of
measure q−l(1
−q−1) forl >0. Since P
j>0qj diverges, the measureµis not
σ-additive. It isσ-additive for those sets of countably many disjoint setsAn in
the algebra Awhich ”don’t accumulate at break points” from one horizontal line tm
2O to tm+12 O, i.e. for which the seriesµ(An) absolutely converges (see
5. ForA∈ A, α∈F× one hasαA∈ Aandµ(αA) =|α|µ(A), where| | is a two dimensional module: |0|= 0, |ti
2t j
1u|=q−jXi foru∈U. The module
is a generalization of the usual module on locally compact fields. For example, everyα∈F can be written as a convergent seriesPαi,j withαi,j∈ti2t
j 1Oand |αi,j| →0. This simplifies convergence conditions in previous use, e.g. [Z2]. 6. We introduce a space RF of complex valued functions on F and their
integrals against the measureµ.
Definition. Call a seriesPcn,cn ∈C((X)), absolutely convergent if it
con-verges and ifP|resXi(cn)|converges for everyi.
For an absolutely convergent series Pcn and every subsequencenm the series P
cnm absolutely converges and the limit does not depend on the terms order.
Lemma. Suppose that a function f:F −→ C can be written as PcncharAn
with countably many disjoint distinguished An, cn ∈ C, where charC is the characteristic function ofC, and suppose that the seriesPcnµ(An)absolutely converges. Iff has a second presentation of the same typePdmcharBm, then
P
cnµ(An) =Pdmµ(Bm).
Proof. Notice that if SAn = SBm for distinguished disjoint sets, then for
every neither An equals to the union of some of Bm, or the union ofAn and
possibly several otherA’s equals to one ofBm. It remains to use the following
property: if a distinguished setCis the disjoint union of distinguished setsCn
andPµ(Cn) absolutely converges, then for everya∈F and integericondition
C⊃a+ti2OimpliesCn⊃a+ti2Ofor alln; thereforeµ(C) =Pµ(Cn).
Definition. DefineRF as the vector space generated by functionsf as in the
previous lemma and by functions which are zero outside finitely many points. For anf as in the previous lemma define its integral
Z
f dµ=Xcnµ(An),
and forf which are zero outside finitely many points define its integral as zero. The previous definition implies that the integral is an additive function. Example. Let a functionf satisfy the following conditions:
there isi0such thatf(β) = 0 for allβ ∈ti2t
1 O as a disjoint union of finitely
If the series in the brackets absolutely converges, thenf ∈RF.
Redenotef asfi0. Then
P
i>i1fi also belongs toRF.
For another example see section 8.
7. Some important for harmonic analysis functions do not belong toRF, for
example, the functionα7→ψ(aα)charA(α) for, say,A=O∈ A,a6∈ O. In the
one dimensional case all such functions do belong to the analogue ofRF.
Recall that in the one dimensional case the integral over an open compact subgroup of a nontrivial character is zero. This leads to the following natural Definition. Denote ψ(C) = 0 is ψ takes more than one value on a distin-guished set C and = the value ofψif it is constant on C. For a distinguished set Aanda∈F× define
Z
ψ(aα)charA(α)dµ(α) =µ(A)ψ(aA).
So, if A⊂ O and is the preimage of a shift of a compact open subgroup of the residue field E with respect to p, and if a ∈ O \t2O, then the previous
definition agrees with the property of a character onE, mentioned above. Lemma. For a function f =Pcnψ(anα)charAn(α)with finite set {an} and
with countably many disjoint distinguished An such that the series Pcnµ(An) absolutely converges the sum Pcn
R
ψ(anα)charAn(α)dµ(α) does not depend
on the choice ofcn, an, An.
Proof. To show correctness, one can reduce to sets on which |f| is constant, then use a simple fact that if a distinguished setCis the disjoint union of distin-guished sets Cn, andψ(aα)charC(α) =Pdnψ(bnα)charCn(α) with|dn|= 1,
absolutely convergent series Pdnµ(Cn) and finitely many distinct bn, then
ψ(aC) =Pdnψ(bnCn)µ(Cn).
Definition. Put
Z
f dµ=Xcn Z
ψ(anα)charAn(α)dµ(α).
Denote by R′
F the space generated by functions f(α)ψ(aα) with f ∈ RF,
a∈F, and functions which are zero outside a point; all of them are integrable. This space will be enough for the purposes of this work.
Remark. Here is a slightly different approach to the extension ofRF: Suppose
that a function f:F → C is zero outside a distinguished subgroup A of F. Suppose that there are finitely many a1, . . . , am ∈ A such that the function
g(x) =Pif(ai+x) belongs toRF. Thendefine Z
f = 1 m
Z
First of all, this is well defined: if h(x) = Pjf(bj +x) belongs toRF for
b1, . . . , bn∈A, thenPmi=1h(ai+x) =Pnj=1g(bj+x), and fromh(ai+x)∈RF
and shift invariant property, and the similar property for shifts of g one gets mRh dµ=nRg dµ.
Third, this definition is compatible with all the properties of RF in the
previous section: iff already belongs toRF then R
f =R f dµ. Also,R f(x) =
R
f(a+x) fora∈A.
Finally, this definition is compatible with the preceding definitions of this section: of course, for nontrivial characters on a distinguished sub-group one has g = 0. The space R′
otherwise (since thenψ(aα) is a nontrivial character onti 2t
their restriction toO \t2O coincide, then Z
From the previous we deduceRt2Oc dµ= 0, and therefore, similarly,
Remark. One hasRZ
ldµ=
P
−l6j6lq−jwhereZl= S
−l6j6lt j
1U, andO\t2O
is the “limit” ofZl whenl→ ∞. Compare with “equality”Pn∈Zzn= 0 used by L. Euler. One of interpretations of it is to viewzas a complex variable, then for every integermthe sum of analytic continuations of two rational functions
P
n∈Z,n<mzn and P
n∈Z,n>mzn is zero. Euler’s equality can be applied to
show equivalence of the Riemann–Roch theorem and the functional equation of the zeta function of a one dimensional global field of positive characteristic (cf. [Rq, sect. 4.3.3]).
From the definitions we immediately get
Lemma. Suppose that g:E−→C is integrable overE with respect to the nor-malized Haar measure µE as in section4. Then the functiong◦pextended by zero outside O is inRF and
Z
O
g◦p dµ=
Z
E
g dµE.
One can say that the Haar measureµE equals p∗(µ′) where the measureµ′
onOis induced by µby extending functions onOby zero toF. 9. Definition. Forf ∈RF introduce the transform function
F(f)(β) =
Z
F
f(α)ψ(αβ)dµ(α).
In particular,F(charO) =charO.
Definition. Denote byQF the subspace ofRF consisting of functionsf with
support in O and such that f|O = g◦p|O for a function g:E −→ C which belongs to the Bruhat space generated by characteristic functions of shifts of proper fractional ideals of E.
Theorem. Given f ∈ QF, the function F(f) belongs to QF and we have a double transform formula
F2(f)(α) =f( −α).
Proof. First note thatψ(α) =ψE◦p(α) for allα∈ O where ψE is an
appro-priate character onE with conductorOE.
Using sections 7 and 8 we shall verify that ifβ6∈ O thenF(f)(β) = 0;
ifβ∈ OthenF(f)(β) =F(g)◦p(β) (whereF(g) denotes the Fourier transform ofg with respect toψE andµE).
Forβ6∈ O the definitions imply
F(f)(β) =
Z
O
Forβ∈ O \ O×
It remains to use the one dimensional double transform formula.
As a corollary, we deduce that if a functionf:F −→C with support inO× coincides onO× with a functionh
∈QF, then
F2(f)(α) =
½ f(
−α) ifα∈ O×S(F\ O), h(0) ifα∈ O \ O×.
Remark. It is more natural to integrateC((X))-valued functions onF: for a functionf =Pi>i0fiX
Similarly to the previous text one checks the correctness of the definition and properties of the integral. In particular, the previous theorem remains true for functionsf =Pi>i0fiX
gn belong to QF (or even just ”lifts” of functions on E for which the double
11. In the case of two dimensional local fields of type F = K((t)), where K is an archimedean local field, O, U are not defined. Define a character ψ:K((t)) −→ C× to be the composite of the res0:K((t)) −→ K and the
archimedean character ψK(α) = exp(2πiTrK/R(α)) onK. The role of distin-guished sets is played by A = a+tiD+ti+1K[[t]] where D is an open ball
in K. In this case if the intersection of two distinguished sets is nonempty then it equals either to one of them, or to a smaller distinguished set. The measure is the shift invariant additive measure µ on the ring generated by distinguished sets and such that µ(A) = µK(D)Xi. It can be extended to
µ(a+tiC+ti+1K[[t]]) = µ
K(C)Xi, where C ⊂K is a Lebesgue measurable
set.
The module is|Pi>i0ait
i
|=|ai0|KX
i0, where the module on the real Kis
the absolute value and on the complexKis the square of the absolute value. The fields of this type are not used in the global theory, but we briefly sketch the analogues of the previous constructions.
The definitions of spacesRF andR′F follow the general pattern of sections 6
and 7: for disjoint distinguished setsAnsuch thatPcnµ(An) absolutely
con-verges put
Z X
cncharAndµ=
X
cnµ(An).
For a function f = Pcnψ(anα)charAn(α) with finite set {an} and with
countably many disjoint distinguishedAn such that the seriesPcnµ(An)
ab-solutely converges put
Z
f dµ=Xcn Z
An
ψ(anα)dµ(α),
where
Z
A
ψ(cα)dµ(α) =ψ(ca)Xi
Z
D
ψK(c−iβ)dµK(β), c−i= rest−ic,
A, a, D, iare defined at the beginning of this section.
The analogues of the definitions and results of sections 8 and 9 hold. The transform of a functionf ∈RF is defined by the same formula
F(f)(β) =
Z
F
f(α)ψ(αβ)dµ(α).
The spaceQF consists of functions f ∈RF such that f =g◦pfor a function
g in the Schwartz space onK, where p= rest0. The double transform formula
isF2(f)(α) =f(−α) forf ∈Q F.
12. In general, for the global theory, we need to work with normalized mea-sures corresponding to shifted characters. Similar to the one dimensional case, if we start with a characterψwith conductoraOthen the dual toµmeasureµ′ onF is normalized such that|a|µ(O)µ′(O) = 1. The double transform formula of section 9 holds.
For example, if µ′ =µthen µ(O) =|a|−1/2. If aO =ti2O then in the last
formula of section 8 one should insert the coefficientX−i/2 on the right hand
side. Iff0(α) =f(aα) belongs toQF orQ′FthenF(f) =|a|1/2F0(f0) whereF0
is the transform with respect to character ψ0(α) =ψ(aα) of conductorO and
measureµ0 such thatµ0(O) = 1. In particular,F(charO) =|a|−1/2charaO. 13. Generally, given an integral domain A with principal ideal P =tAand projection A −→ A/P = B and a shift invariant measure on B, similar to the previous one defines a measure and integration on A. For example, the analogue of the ring A is the minimal ring which contains setsα+tip−1(S),
where S is from a class of measurable subsets of B; the space of integrable functions is generated as a vector space by functionsα→g◦p(t−iα) extended
by zero outsidetiA, wheregis an integrable function onB. Such a measure and
integration is natural to call lifts of the corresponding measure and integration on the baseB.
In particular, if B =Ak for a global fieldk of positive characteristic, then
one has a liftµ1 of the measureµAk, a liftµ2 of the counting measure ofk(so
µ2(a+tA) = 1), and a lift µ3 of the measure onAk/k. If the measures are
normalized such thatµAk=µk⊗µAk/k (of course,µk is a counting measure),
thenµ1=µ2⊗µ3.
Remarks.
1. T. Satoh [S], A.N. Parshin [P6], and M. Kapranov [Kp] suggested two other very different approaches to define a measure on two dimensional fields (notice that Satoh’s work was an attempt to use Wiener’s measure). The work [P6] suggested to use an ind-pro description for a generalization of Haar measure to two dimensional local fields, see also [Kp]. The works [P6] and [Kp] don’t introduce a measure, and deal with distributions and functions; local zeta integrals are not discussed.
After this part had been written, the author was informed by A.N. Parshin that the formula µ(ti
2t j
1O) = q−jXi was briefly discussed in a short message
[P5], however, it led to unresolved at that stage paradoxes.
2. In this work we do not need a general theory of analysis on non locally compact abelian groups, since the case of higher dimensional local fields and natural adelic objects composed of them is enough for applications in number theory. Undoubtedly, there is a more general theory of harmonic analysis on non locally compact groups than presented above, see also the next remark.
induced from the counting measure on a covering hyperfinite abelian group (e.g. [Gor1]); the same is also true for the Fourier transform [Gor2].
Conceptually, the integration theory of this work is supposed to be induced by integration on hyper locally compact abelian groups (or hyper hyper finite groups). This would also give an extension of this theory to a more general class of non locally compact groups than those of this part.
2. Dimension two local zeta integral
Using the theory of the previous part we explain how to integrate over topolog-ical K-groups of higher local fields, define a local zeta integral and discuss its properties and new phenomena of the dimension two situation. Using a cover-ing of a so called topological MilnorKt
2-group of a two dimensional local field
(which is the central object in explicit local class field theory) by the product of the group of units with itself we introduce integrals over theKt
2-group. For
this we use additional mapsr,t,o whose meaning can only be finally clarified in constructions of the global part [F4]. In section 17 we define the main object – a higher dimensional local zeta integral associated to a function on the field and a continuous character of the Kt
2-group, followed by concrete examples.
Several first properties of the zeta integral, in analogy with the one dimen-sional case, are discussed and proved. In the case of formal power series over archimedean local fields zeta integrals introduced in section 23 are not really new in comparison to one dimensional integrals; this agrees with the fact that the class field theory for such fields does not really requireKt
2. 14. OnF× one has the induced fromF
⊕F topology with respect to F×−→F⊕F, α7→(α, α−1),
and the shift-invariant measure
µF× = (1−q−1)−1µ/| |.
Explicit higher class field theory describes abelian extensions ofF via closed subgroups in the topological Milnor K-group Kt
2(F) = K2(F)/Λ2(F) where
Λ2(F) is the intersection of all open neighborhoods of zero in the strongest
topology onK2(F) in which the subtraction inK2(F) and the map
F××F×−→K2(F), (a, b)7→ {a, b}
are sequentially continuous. For more details see [IHLF, sect. 6 part I] and [F3]. Denote byU Kt
15. Introduce a module onKt 2(F):
| |2:K2t(F)−→R×+, α7→q−v(α),
wherev:Kt
2(F)−→K0(Fq) =Z is the composite of two boundary
homomor-phisms.
Definition. Introduce a subgroup
T =O×× O×={(tj1u1, tl1u2) :j, l∈Z, ui∈U}
ofF×
×F×. The closure of T in F
×F equals O × O.
A specific feature of the two dimensional theory is that one needs to use auxiliary mapso,o′ to modify the integral in such a way that later on, in the adelic work in [F4] one gets the right factors for transformed functions and their zeta integrals.
Definition. Introduce a map (morphism of multiplicative structures) o′:T −→F×F, (tj
1u1, tl1u2)7→(t2j1 u1, t2l1u2)
and denote byothe bijectiono′(T)−→T.
For a complex valued continuous functionf whose domain includesT and is a subset ofF×F formf◦o:o′(T)−→C, then extend it by continuity to the closure ofo′(T) inF×F and by zero outside the closure, denote the result by fo:F×F →C.
To be able to apply the transformFintroduce an extension e(g) of a function g=fo:F×F−→C as the continuous extension onF×F of
e(g)(α1, α2) =g(α1, α2) + X
16i63
g((α1, α2)νi), (α1, α2)∈F××F×,
whereν1= (t−11, t
−1
1 ),ν2= (t−11,1),ν3= (1, t−11).
For example, iff =char(tj
1O,tl1O)thenf
o(α1, α2) = 1 for (α1, α2)∈F××F×
only if α1 ∈ t2k1 U, α2 ∈ t2m1 U for k >j, m > l, hence fo is not a continuous
function; but e(fo) =char(t2j
1O,t 2l
1O)is.
Definition. For a function f:T −→ C such that e(fo) ∈ RF×F define its
transform
b
16. Due to the well known useful equality
{1−α,1−β}={α,1 +αβ/(1−α)}+{1−β,1 +αβ/(1−α)}, and a topological argument we have surjective maps ([IHLF, sect. 6 part I]):
r:T −→K2t(F), r¡(tj1, tl1)(u1, u2)¢= min(j, l){t1, t2}+{t1, u1}+{u2, t2}, t:T −→K2t(F), t¡(tj1, tl1)(u1, u2)¢= (j+l){t1, t2}+{t1, u1}+{u2, t2}. For (α1, α2)∈T we have
|t(α1, α2)|2=|α1| |α2|=|(α1, α2)|.
The homomorphism t plays in important role in the study of topological MilnorK-groups of higher local fields ([P3], [F3]). If we denoteH ={(α, β) : α, β ∈ ht1iU, αβ−1 ∈ U}, then r(H) = K2t(F). Note that if K2t(F) is
re-placed by the usual K2(F) then none of the previous maps is surjective. The
topology ofK2t(F) is the quotient topology of the sequential saturation of the
multiplicative topology onH (cf. [F3], [IHLF, sect. 6 part I]).
The maps r,t depend on the choice of t1, t2 but the induced map to
Kt
2(F)/ker(∂) (see section 18 for the definition) does not depend on the choice
of t2. The homomorphism t (which depends on the choice of t1, t2) is much
more convenient to use for integration onKt
2(F) than the canonically defined
mapF×
×F×
−→Kt 2(F).
For a functionh:Kt
2(F)−→C denote byhr:F×F −→C (ht:F×F −→C)
the compositeh◦r(resp. the compositeh◦t) extended by zero outsideT.
17. Let χ:Kt
2(F)−→ C× be a continuous homomorphism. Similarly to the
one dimensional case one easily proves thatχ=χ0| |s2(s∈C), whereχ0is a lift
of a character ofU Kt
2(F) such thatχ0({t1, t2}) = 1. The groupV = 1 +t1O
of principal units has the property: every open subgroup contains Vpn
for sufficiently large n, c.f. [Z1, Lemma 1.6]. This property and the definitions imply that |χ0(K2t(F))|= 1. Put s=s(χ).
Definition. For a functionf such thatcharo′(T)fo∈RF×F introduce Z
T
f dµT = (1−q−1)−2 Z
o′(T)
fodµF×F.
Notice thatRTf dµT = (1−q−1)−2 R
F×FcharTf| |dµF×F.
For a function f:F ×F −→ C continuous on T and a continuous quasi-characterχ:Kt
2(F)−→C× introduce a zeta function
ζ(f, χ) =
Z
T
f(α)χt(α)|t(α)|−22dµT(α)
From the definitions we obtainζ(f, χ)
which converges for Re(s)>0, and has a single valued meromorphic continu-ation to the whole complex place. Note that for this specific choice of f the local zeta integraldoes not involve X.
2. More generally, for a function f ∈QF×F the local zeta integral ζ(f, χ)
converges for Re(s)>0 to a rational function inqs and therefore has a
mero-morphic continuation to the whole complex plane.
To show this, one can assume that f(α1, α2) =charA1(α1)charA2(α2) with nal function ofqson the whole plane.
3. Forf ∈QF×Fand (α1, α2)∈Tdefine, in analogy with A. Weil’s definition
in [W1]
W(f)(α1, α2) =f(α1, α2)−f(t−11α1, α2)−f(α1, t−11α2) +f(t1−1α1, t−11α2).
18. Let ∂:K2(F) −→ K1(E) be the boundary homomorphism. Note that
∂(Λ2(F)) = 1. Define
λ:K1(E)−→K2t(F)/ker(∂), α7→ {α, t˜ 2}
where ˜α∈ O is a lifting ofα∈E×. Then∂(λ(α)) =α. Let f:Kt
2(F)−→C be a continuous function which factorizes through the
quotientKt
2(F)/ker(∂). Letχ:K2t(F)−→C×be a weakly ramified continuous
quasi-character, i.e. χ0(ker(∂)) = 1, so χ is induced by a (not necessarily
unramified) quasi-character of the first residue fieldEofF. Then using Lemma in section 8 one immediately deduces that
ζ(fr, χ) =ζE(f◦λ, χ◦λ)2
where ζE is the one dimensional zeta integral onE which corresponds to the
normalized Haar measure on E.
Similarly, if F is a mixed characteristic field, andK is the algebraic closure of Qp in F, then one has a residue map d:K2t(F)−→ K1(K), see e.g. [Ka2,
sect. 2], the mapdis−res defined there.
Assume that a prime element ofKis at2-local parameter ofF, then we can
identify F=K{{t1}}. Define
l:K1(K)−→K2t(F), α7→ {t1, α},
thend(l(α) =α. Let f:Kt
2(F)−→ C and a quasi-character χ have analogous to the above
properties but with respect tod. Then similarly one has ζ(fr, χ) =ζK(f ◦l, χ◦l)2.
Thus, the two dimensional zeta integral for F links zeta integrals for both E andK, local fields of characteristicpand 0.
19. Put χb=χ−1| |2 2.
Proposition. Letg, h∈QF×Fbe continuous functions, and let χ:K2t(F)−→
C× be a continuous quasi-character. For0<Res(χ)<2 (and therefore for all
s)one has a local functional equation
ζ(g, χ)ζ(h,b χ) =b ζ(bg,χ)b ζ(h, χ).
Proof. One proof is similar to Example 2 in the previous section and re-duces the verification to the case where both g and h are of the simple form charA1charA2. We indicate another, similar to dimension one, with one new
For α = (α1, α2) denote α−1 = (α−11, α2−1), αβ = (α1β1, α2β2), |α| = |α1| |α2|,µ(α) =µF×F(α1, α2),ψ(α) =ψ(α1)ψ(α2). Put
k(β) =ψ(β) +q−2ψ(ν1−1β) +q−1ψ(ν2−1β) +q−1ψ(ν3−1β),
νi defined in section 15. We will use|to(α)|22=|α|forα∈o′(T).
We have
ζ(g, χ)ζ(h,b χ) =b
Z Z
T×T
g(α)h(βb )χt(αβ−1)|β|2|α|−2|β|−2dµT(α)dµT(β)
=
Z
F×F Z
F×F
go(α)F(e(ho))(β)χto(αβ−1)|α|−1dµ(α)dµ(β)
=
Z Z Z
F6
go(α) e(ho)(γ)ψ(βγ)χto(αβ−1)|α|−1dµ(α)dµ(β)dµ(γ)
=
Z Z Z
F6
go(α)ho(γ)ψ(βγ)χto(αβ−1)|α|−1dµ(α)dµ(β)dµ(γ)
+ X
16i63 Z Z Z
F6
go(α)ho(γ)ψ(βνi−1γ)χto(αβ−1)|α|−1dµ(α)dµ(β)dµ(νi−1γ)
=
Z Z Z
F6
go(α)ho(γ)k(βγ)χto(αβ−1)|α|−1dµ(α)dµ(β)dµ(γ)
(β→γ−1β) =
Z Z Z
F6
go(α)ho(γ)χto(αγ)|αγ|−1dµ(α)dµ(γ)k(β)χto(β−1)dµ(β)
(due to tothe integrals are actually taken overo′(T) where one can apply the Fubini property). The symmetry inα, γimplies the local functional equation. Remark. Undoubtedly, one can extend this proposition to a larger class of continuous functionsg, h:F×F −→C.
20. Similar to section 12, we need to take care of zeta integrals which involve normalized measures corresponding to shifted characters. The zeta integral should be more correctly written as ζ(f, χ| |s
2, µ) since it depends on the
nor-malization of the measure µ. Above we used the normalized measure µ such thatµ(O) = 1 (and the conductor ofψisO). More generally, let the conductor of a characterψ′betd
1O, soψ′(α) =ψ0(t−1duα) for some unitu,ψ0was defined
in section 3. Putψ(α) =ψ0(t−12duα) and useψandψ(α1, α2) for the transform F of functions inQF andQF×F as in sections 9–10. Then self dual measures
µonF andF×F with respect to ψsatisfyµ(O) =qd,µ(O, O) =q2d.
Letf =charc{t1,t2}+OKt2(F)r, sof(t j
1u1, tl1u2) = 1 ifj, l>c and = 0
other-wise. Now F(e(fo))(tj1u1, tl1u2) =q2d−4c ifj, l >2d−2c and = 0 otherwise;
onT. We get
21. The previous constructions of the mapso′,r,t,e are suitable for unramified characters, but are not good enough for ramified characters.
Let a weakly ramified characterχofKt
2(F) be induced by a characterχof the
fieldEwith conductor 1 +t1 r
OE,r >0. By analogy with the one dimensional
case it seems reasonable to calculate the zeta integral ζ(f, χ) for f:T −→ C which is defined by (tj1u1, tl1u2)7→1 if and only if j >0, l= 0, u2 ∈1 +tr1O.
Using the dimension one calculation in [T] one easily obtains ζ(f, χ) = 1
Notice that unlike the case of unramified characters in section 20,fbbdiffers fromf onT due to the difference between e(fbo) andF(e(fo)); so for the current
choice ofoand e as above there is no canonical local constant associated toχ. This may be related to the well known difficulty of constructing a general higher ramification theory which is still not available. It is expected that one can refine o, e, r,t to get canonical local constants for ramified characters as well.
22. One can slightly modify the definition of the zeta integral by extending T to the set
T′={(tm2t j
1u1, tn2tl1u2)}, wherej, l∈Z,ui∈U,m, n>0,mn= 0.
The setT′ will be useful in the study of global zeta integrals in [F4]. Introduce a set Γ ={(tm
2,1) :m>0}S{(1, tm2) :m>1}, thenT′=T×Γ.
Extendr,tsymmetrically toT′ by (m >0)
Extendo′ to o′:T′ −→F××F× by (tm2tj1u1, t2ntl1u2)7→(tm2t2j1 u1, tn2t2l1u2) and denote byothe inverse bijection o′(T′)−→T′.
For a functionf:T′ −→C define fo:F ×F −→ C as the composite f ◦o
extended by zero outside o′(T′). Introduce
Z
T′
f dµT′ = (1−q−1)−2
Z
F×F
fodµF×F.
Let a functionf:F×F −→C be continuous onT′ and let f(tm
2 α1, α2) =f(α1, tm2α2) =f(α1, α2)
for every (α1, α2) ∈ T, m > 0. Then for a continuous quasi-character
χ:Kt
2(F)−→C× such thatχ(U K2t(F)) = 1 we have Z
T′
f(α)χt(α)|t(α)|−22dµT′(α) =ζ(f, χ).
This follows from the following observation: the function f(α)χt(α)|t(α)|−22
on the set (tm2t j
1u1, tl1u2),m >0, depends ontl1u2 only, and therefore one can
write the integral RT′f(α)χt(α)|t(α)|
−2
2 dµT′(α) as the sum ofζ(f, χ) and the
sum of the product of two integrals one of which is the integral of a constant function overtm
2O \tm+12 O, and hence is zero by Example 8. 23. We describe elements of the theory for Kt
2(K((t))) where K is an
archimedean local field. Class field theory for such fields is described in [P2] and in [KS]. In this case class field theory does not really match nicely the structure of theKt
2-group of the field, and this is reflected in the constructions
of this section.
Introduce the sequential topology onL=K((t)) as in section 1. Define K2t(L) =K2(L)/Λ2(L), Λ2(L) = Λ2′(L)+{K×,1+tK[[t]]}+{t,1+tK[[t]]},
Λ′
2(L) is the intersection of all neighborhoods of zero in the strongest topology
onK2(L) in which the subtraction inK2(L) and the map
L××L×−→K2(L), (a, b)7→ {a, b}
are sequentially continuous. Then Kt
2(L) is generated by {a, t}, a ∈K× and {−1,−1} (which is zero if√−1 ∈K). We have∂(Λ2(L)) = 1 where∂ is the
boundary map. ∂inducesKt
2(L)−→K1(K).
Introduce a module onKt
2(L): |α|2=|∂(α)|K.
A continuous homomorphismχ:Kt
2(L)−→C× can be written asχ=χ0| |s2
(s ∈ C), where χ0({K×, t2}) = 1 and|χ0(K2t(L))| = 1 (so χ0 is determined
Let
r:K×−→K2t(L), α7→ {α, t}
t:T =K××K×−→K2t(L), (α, β)7→ {αβ, t} (totally ramified characters with respect to L/K are ignored).
For a function f:L×L −→ C continuous on T and rapidly decaying at (±∞,±∞) and a continuous quasi-character χ:Kt
2(L) −→ C× introduce a
zeta function
ζ(f, χ) =
Z
K×K
f(α, β)χt(α, β)
dµ(α, β)
|α|K|β|K.
Define λ:K1(K) −→ K2t(L)/ker(∂) by α 7→ {α, t˜ }. Let f:K2t(L) −→ C
factorize throughKt
2(L)/ker(∂). Letχ:K2t(L)−→C
×
be a continuous quasi-character such that χ0(ker(∂)) = 1 (i.e. χ is an unramified character with
respect to L/K). Then
ζ(g, χ) =ζ1(f◦λ, χ◦λ)2,
whereg(α, β) =fr(α)fr(β), andζ1is the one dimensional zeta integral.
Remark. The well known technique in dimension one which extends the local integrals associated to representations of the multiplicative group to represen-tations of algebraic groups is likely to give a similar extension of this work.
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Ivan Fesenko
School of Mathematics Univ. of Nottingham Nottingham NG7 2RD England
