Table Of ContentANDERSON-STARK UNITS FOR F [θ]
q
BRUNOANGLE`S1,FEDERICOPELLARIN2,3,ANDFLORICTAVARESRIBEIRO1
Abstract. WeinvestigatethearithmeticofspecialvaluesofanewclassofL-functionsrecently
introduced by the second author. We prove that these special values are encoded in some
particular polynomials whichwecall Anderson-Stark units. We then use these Anderson-Stark
unitstoprovethatL-functionscanbeexpressedassumsofpolylogarithms.
5
1
0
2
Contents
n
a 1. Introduction 1
J
2. Notation 3
7
3. Some functional analysis 4
2
3.1. General settings 4
] 3.2. The one variable case 5
T
3.3. The multivariable case 6
N
3.4. The Carlitz action 7
. 4. Multivariable log-algebraicity 9
h
t 4.1. The log-algebraicity theorem 9
a
4.2. The special polynomials 11
m
4.3. Converses of the log-algebraicity Theorem 14
[
5. Multivariable L-functions 16
1 5.1. Frobenius actions 16
v 5.2. Anderson-Stark units 17
4
6. Special L-values 18
0
References 19
8
6
0
.
1
1. Introduction
0
5 A major theme in the arithmetic theory of global function fields is the study of the arithmetic
1
properties of special values of D. Goss L-functions. A typical example of such a function is given
:
v by the Carlitz-Goss zeta function ζ (.), where A = F [θ] is the polynomial ring in the variable θ
A q
i with coefficients in a finite field F . Its special values are given by the following formula:
X q
1
r
a ∀n≥1, ζA(n)= an ∈K∞,
aX∈A+
where A is the set of monic elements in A and K = F ((1)). In 1990, G. Anderson and D.
+ ∞ q θ
Thakur proved the following fundamental result ([AT90, Theorem 3.8.3]): for n 1, there exists
≥
z Lie(C⊗n)(K ) such that exp (z ) C⊗n(A), and :
n ∈ ∞ n n ∈
Γ ζ (n)=e (z ),
n A n n
where exp is the exponentialmap associatedto the nth tensor power ofthe Carlitz module C⊗n,
n
e (z ) is the last coordinate of z Kn, and Γ A is the Carlitz factorial (we refer the reader
n n n ∈ ∞ n ∈
1NormandieUniversit´e,Universit´edeCaen,UMR6139,CampusII,BoulevardMar´echalJuin,B.P.5186,14032
CaenCedex,France.
2Institut Camille Jordan, UMR 5208 Site de Saint-Etienne, 23 rue du Dr. P. Michelon, 42023 Saint-Etienne,
France
3SupportedbytheANRHAMOT
Date:January28,2015.
2010 Mathematics Subject Classification. Primary: 11R58,11M38,Secondary: 11G09.
1
2 B.ANGLE`S,F.PELLARIN,ANDF.TAVARESRIBEIRO
to [BP] for the basic properties of C⊗n). This result has recently been generalized by M. A.
Papanikolasin[Pap]whoprovedalog-algebraicityTheoremforC⊗n inthespiritoftheworkofG.
Andersonin[And96]. M.A.Papanikolasappliesthislog-algebraicityTheoremtoobtainremarkable
explicit formulas for a large class of special values of D. Goss Dirichlet L-functions. Observe that
the t-motive associatedto the t-module C⊗n can be understood as the following object : (A[t],τ),
where t is an indeterminate over K =F (θ), and τ :A[t] A[t] is the F [t]-linear map defined as
q q
→
follows :
τ a tk =(t θ)n aqtk ,
k − k
k≥0 k≥0
X X
where a A.
k
∈
Lets 1beanintegerandlett ,...,t besindeterminatesoverK.Considerthefollowingobject:
1 s
(A[t ,≥...,t ],τ) where τ : A[t ,...,t ] A[t ,...,t ] is the morphism of F [t ,...,t ]-modules,
1 s 1 s 1 s q 1 s
→
semi-linear1 with respect to τ :A A,x xq, given by :
0
→ 7→
τ ai1,...,isti11···tiss=(t1−θ)···(ts−θ) aqi1,...,isti11···tiss,
i1,.X..,is∈N i1,.X..,is∈N
where a A. Note that we have a natural morphism of F [t ,...,t ]-algebras :
i1,...,is ∈ q 1 s
φ:A[t ,...,t ] End A[t ,...,t ]
1 s → Fq[t1,...,ts] 1 s
givenby φ =θ+τ. LetT (K )be theTate algebrainthe variablest ,...,t , withcoefficientsin
θ s ∞ 1 s
K . Thenτ extendsnaturallytoacontinuousmorphismofF [t ,...,t ]-moduleonT (K ).The
∞ q 1 s s ∞
second author introduced (see [Pel12], [Per14b], [Per14a], [AP14]) for integers N Z and s 0
∈ ≥
the following L-series
a(t ) a(t )
1 s
L(N,s)= ··· ,
aN
aX∈A+
which converge in T (K ). If z is another indeterminate, we also set :
s ∞
a(t )...a(t )
L(N,s,z)= zd 1 s K[t ,...,t ][[z]]
aN ∈ 1 s
Xd≥0 a∈XA+,d
These series converge at z =1 in T (K ) and we have the equality :
s ∞
L(N,s)=L(N,s,z) .
z=1
|
Our maingoalin this article is the study of the arithmetic properties ofthe L(N,s,z),N Z. Let
∈
us give a brief description of our principal results.
We let τ act on K[t ,...,t ][[z]] by :
1 s
τ( f zk)= τ(f )zk,
k k
k≥0 k≥0
X X
where f K[t ,...,t ]. The exponential function associated to φ is defined by :
k 1 s
∈
1
exp = τi,
φ D
i
i≥0
X
where D =1, and for i 1, D =(θqi θ)Dq . We also set
0 ≥ i − i−1
zi
exp = τi.
φ,z D
i
i≥0
X
Aformulationofthes-variableversionofAnderson’slog-algebraicityTheoremis(seeTheorem4.6
and Proposition 5.4):
exp (L(1,s,z)) A[t , ,t ,z]
φ,z ∈ 1 ··· s
1We signal here to avoid confusion that in the rest of the article τ will denote more generally a morphism
semi-linearwithrespecttoτ0.
ANDERSON-STARK UNITS FOR Fq[θ] 3
from which we also deduce that, in T (K ) :
s ∞
exp (L(1,s)) A[t , ,t ].
φ ∈ 1 ··· s
This s-variable versionhas been provedin [APTR14] as a consequence ofa class formula. We give
here a more direct proof, close to Anderson’s original proof in [And96].
Thespecialelementsexp (L(1,s,z))andexp (L(1,s))playtheroleofStarkunitsinourcontext.
φ,z φ
Letusgiveanexample,for1 s q 1,by Proposition5.4wehavethe followingequalityinT :
s
≤ ≤ −
L(1,s)=log (1),
φ
wherelog = 1τi isthe Carlitzlogarithm,l =1andfori 1,l =(θ θqi)l ,. Wedefine
φ i≥0 li 0 ≥ i − i−1
for N >0, the Nth ”polylogarithm”
P
zi
log = τi.
φ,N,z lN
i≥0 i
X
Set b (t) = 1, and for n 1, b (t) = n1 (t θqk). Let N 1 be an integer and let n 1
0 ≥ n k=1 − ≥ ≥
be the unique integer such that qr N > qr−1. We can then prove (see Theorem 6.2 for the
≥ Q
precisestatement)thatthereexistsafinitesetofcompletelyexplicitelementsh A[t , ,t ,z],
j 1 s
∈ ···
0 j d that are built from the ”unit” exp (L(1,n+qr N,z)), such that :
≤ ≤ φ,z −
d
lqr−Nb (t ) b (t )L(N,n,z)= θjlog (h ).
r−1 r 1 ··· r n φ,N,z j
j=0
X
Thepaperisorganizedasfollows: wefirst( 3)introduceaBanachspaceB whichisacompletion
s
§
of an s-variable polynomial ring for a norm similar to the one consideredby Anderson in [And96].
The study of different natural Carlitz actions on B allows us to endow B with an action of the
s s
Tate algebra T and to translate some statements on B into statements on T . We then ( 4)
s s s
§
prove the s-variable log-algebraicity theorem, following the lines of Anderson’s proof in [And96],
and establish some properties of the special polynomials. We also state two ”converses” to the
log-algebraicity theorem (Propositions 4.15 and 4.16). In the next section ( 5) we translate the
preceding results in T , so that the L-functions L(1,s,z) and L(1,s) appear§naturally. The last
s
section ( 6) is devoted to the proof that L(N,n,z) can be expressed as a sum of polylogarithms.
§
2. Notation
Let F be a finite field with q elements, where q is a power of a prime p, θ an indeterminate
q
over F , A = F [θ], A∗ = A 0 and K = F (θ). The set of monic elements (respectively of
q q q
\{ }
degree j 0) of A is denoted by A (respectively A ). Let v be the valuation on K given by
+ +,j ∞
v (a) =≥deg b deg a. We identify with F ((1)) the completion K of K with respect to v .
L∞etCb bethθec−ompleθtionofanalgebraicclosqureθofK . Thenv ext∞endsuniquelytoavaluati∞on
∞ ∞ ∞
on C∞, still denoted by v∞, and we set for all α C∞, α∞ =q−v∞(α). The algebraicclosures of
K and F in C will be denoted by K and F . ∈ | |
q ∞ q
Let τ denote anoperator which we let act as the Frobenius on C : for all α C , τ(α)=αq. If
∞ ∞
R is a ring endowed with an actionof τ (for instance, R a subring of C stab∈le under τ), then we
∞
denote by R[[τ]] the ring of formal series in τ with coefficients in R subject to the commutation
rule : for all r R,τ.r =τ(r).τ. We also denote by R[τ] the subring of R[[τ]] of polynomials in τ.
The Carlitz mo∈dule is the unique morphism C : A A[τ] of F -algebras determined by C =
q θ
θ +τ. If M is an A-module endowed with a s·emi-li→near endomorphism τ ( m M, a A,
M
∀ ∈ ∀ ∈
τ (am)=τ(a)τ (m)), then C induces a new action of A on M ; endowed with this action, the
M M
A-module M is denoted by C(M·).
The Carlitz exponential is the formal series
1
exp = τi K[[τ]]
C D ∈
i
i≥0
X
whereD =1andfori 1,D =(θqi θ)Dq . Theevaluationexp :C C ;x exp (x)=
0 ≥ i − i−1 C ∞ → ∞ 7→ C
1 τi(x) defines an entire F -linear function on C and ker(exp :C C )=π˜A where
i≥0 Di q ∞ C ∞ → ∞
P
4 B.ANGLE`S,F.PELLARIN,ANDF.TAVARESRIBEIRO
π˜ is the Carlitz period defined by (see [Gos96, Chapter 3]):
θqi θ 1
π˜ = q−√1θ θq 1 − q−√1 θ(θ+F [[ ]]).
− − θqi+1 θ!∈ − q θ
i≥1 −
Y
The Carlitz logarithm is the formal series
1
log = τi K[[τ]]
C l ∈
i
i≥0
X
where l =1 and for i 1,l =(θ θqi)l . It satisfies in K[[τ]] the equality log .exp =1. It
defines0a functionx ≥log (ix) onC− coni−ve1rgingfor v (x)> q . Moreover,ifCv (x)C> q ,
7→ C ∞ ∞ −q−1 ∞ −q−1
then v (x) = v (exp (x)) = v (log (x)) and exp log (x) = x = log exp (x). We have
∞ ∞ C ∞ C C◦ C C◦ C
the formal identities in K[[τ]] for all a A:
∈
exp a=C exp and log C =alog .
C a C C a C
The identity exp (ax)=C (exp (x)) holds for all x C ,a A.
The set of A-torCsion pointsa of CC(C ) is denoted by∈Λ ∞ C∈(K). Let a A with deg a > 0,
∞ C ⊂ ∈ θ
the a-torsion points are precisely the elements exp (bπ˜) K with b A and deg b < deg a.
C a ∈ ∈ θ θ
Therefore, Λ =exp (Kπ˜). Since exp is continuous for the topology defined by v , the closure
C C C ∞
of Λ in C is the compact set :
C ∞
1 1 1
K=Λ =exp (K π˜)=exp F [[ ]]π˜ = q−√1 θF [[ ]]
C C ∞ C θ q θ − q θ
(cid:18) (cid:19)
where the lastequality comes from the facts that π˜ q−√1 θ(θ+F [[1]]), and that for λ F∗ and
∈ − q θ ∈ q
n 1, exp λπ˜ λπ˜ mod π˜ F [[1]]. It is customary to consider K π˜ as an analogue of the
≥ C θn ≡ θn θn+1 q θ ∞
imaginaryline ; the compact K is then an analogueof the unit circle. Remark that exp and log
(cid:0) (cid:1) C C
define reciprocal automorphisms of K.
3. Some functional analysis
3.1. General settings. Let s 1 be a fixed integer and X = (X ,...,X ) be a set of indeter-
1 s
minates over C . We want to≥consider polynomials F C [X] as polynomial functions on Ks.
∞ ∞
Thus we introduce the following norm on C [X] : ∈
∞
F =max F(x ,...,x ) ; x ,...,x K .
1 s ∞ 1 s
k k {| | ∈ }
Since K is compactand infinite, this is a well-defined, ultrametric normof C -algebra. (In partic-
∞
ular,for all F,G C [X], FG F G . Moreover, F =0 F =0 is a consequenceof the
∞
∈ k k≤k kk k k k ⇒
fact that K is infinite.)
Ifi=(i1,...,is)wheretheij ≥0areintegers,thenwewriteXiforX1i1...Xsis and|i|=i1+···+is.
Lemma 3.1. (1) If i Ns, then Xi =qq|−i|1.
(2) Write for n 1, λ∈ =exp (kπ˜ )k Λ and let W Λ be the F -vector space spanned by
the λ , n ≥1. Thθenn W isCdeθnnse∈in KC. In particul⊂ar, fCor all F qC [X],
θn ∞
≥ ∈
F =sup F(x) ; x Λs =sup F(x) ; x Ws .
k k {| |∞ ∈ C} {| |∞ ∈ }
Proof. (1) This is a consequence of the fact that if a,b A∗ with deg a<deg b, then
∈ θ θ
aπ˜ aπ˜ q 1
v exp =v =deg b deg a − .
∞ C b ∞ b θ − θ − q 1 ≥ q 1
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19) − −
(2) This follows from the fact that the F -vector space spanned by the 1 for n 1 is 1F [1]
q θn ≥ θ q θ
which is dense in 1F [[1]].
θ q θ
(cid:3)
Remark 3.2. Note that the norm . is not multiplicative. We shall give an example in the one
kk
variable case. We have
1
Cθ(X) = X =qq−1
k k k k
but Cθ(X)= λ∈Fq X −λexpC(πθ˜) , where for all λ∈Fq, kX−λexpC(πθ˜)k=qq−11.
Q (cid:0) (cid:1)
ANDERSON-STARK UNITS FOR Fq[θ] 5
Since Λ is the torsion set of C(C ), it is naturally endowed with the Carlitz action of A : if
C ∞
x Λ ,a A, then C (x) Λ , which extends by continuity to K. Thus, we get a natural action
C a C
∈ ∈ ∈
of the multiplicative monoid of A on the polynomial functions on Ks :
(3.1) F(X) C [X], a A ; a F(X)=F(C (X ),...,C (X )).
∞ a 1 a s
∀ ∈ ∀ ∈ ∗
Observe that since a A∗,C : Λ Λ is surjective, this action is isometric with respect to
a C C
∀ ∈ →
the norm .
k·k
3.2. The one variable case. Set L=K (π˜) and π = θ. Since v (π)= 1 , the valuationring
∞ π˜ ∞ q−1
of L is
q−2
1
O =F [[π]]= πkF [[ ]],
L q q
θ
k=0
X
and its maximal ideal is
P =πO .
L L
Recall that since K =A 1F [[1]], we have
∞ ⊕ θ q θ
1 1
exp (π˜K )= F [[ ]] P−1.
C ∞ π q θ ⊂ L
Let N N = 0,1,... and let N = r N qi, with for all i, 0 N q 1 be its base-q
∈ { } r i=0 i ≤ i ≤ −
decomposition. Set l (N)= N . We define the polynomial G (X) by
q i=0 i P N
r r
P
GN(X)=πlq(N) (θi X)Ni =πlq(N) Cθi(X)Ni L[X].
∗ ! !∈
i=0 i=0
Y Y
Lemma 3.3. (1) The set G (X),N 0 generates L[X] as an L-vector space.
N
(2) For N N, we have : { ≥ }
∈
G (X) =1.
N
k k
Proof. (1) It follows from the fact that N 0,deg (G (X))=N.
∀ ≥ X N
(2) We remark that λ Λ , v (G (λ)) 0 and that if α F∗ and λ = exp π˜ ,
∀ ∈ C ∞ N ≥ ∈ q θ+α C θ+α
then v∞(GN(λθ+α))=0. (cid:16) (cid:17)
(cid:3)
If β =(β ) is a sequence of elements in F , we set :
i i≥1 q
π˜
λ(β)= β exp P−1.
i C θi ∈ L
i≥1 (cid:18) (cid:19)
X
Remark that if we set µ(β)= βi K , then we have λ(β)=exp (π˜µ(β)).
i≥1 θi ∈ ∞ C
Lemma 3.4. Let β = (βi)i≥P1 be a sequence of elements in Fq, and N = ri=0Niqi be a non
negative integer written in base q. Then
P
r
G (λ(β)) βNi mod P .
N ≡ i+1 L
i=0
Y
Proof. Observe that
π˜ 1
exp mod O .
C θ ≡ π L
(cid:18) (cid:19)
Thus, for j 0,
≥
πC (λ(β)) β mod P .
θj ≡ j+1 L
Whence the result. (cid:3)
Lemma 3.5. Let k,r be two integers such that r 1 and 1 k qr, let α ,...,α F∗ and let
≥ ≤ ≤ 1 k ∈ q
N ,...,N be k distinct integers in 0,...,qr 1 . Write N = r−1n qi in base q. Then, there
ex1ists β ,.k..,β F such that { − } i j=0 i,j
1 r q
∈ P
k r−1
α βni,j =0
i j+1 6
i=1 j=0
X Y
with the convention that 00 =1.
6 B.ANGLE`S,F.PELLARIN,ANDF.TAVARESRIBEIRO
Proof. We proceed by induction on r.
If r =1, then k q and for 1 j k, N =n 0,...,q 1 . Since the N ’s are distinct, the
i i,1 i
≤ ≤ ≤ ∈{ − }
polynomial ki=1αiXNi is not divisible by Xq−X,andthis implies the assertionofthe lemma in
this case.
P
We assume now that the lemma is proved for all integers less than r 1 1, and we also assume
− ≥
that at least one N is qr−1. We define an equivalence relation over the set 1,...,k : for all
i
≥ { }
1 i,i′ k, i i′ if and only if n =n for all 1 j r 2 (that is, if N N mod qr−1).
i,j i′,j i i′
≤ ≤ ∼ ≤ ≤ − ≡
WedenotebyI ,...,I theequivalenceclassesandifi I wedefinefor1 j r 2,n(m) =n
the common va1lue. Lett β ,...,β F∗, then ∈ m ≤ ≤ − j i,j
1 r ∈ q
k r−1 t r−2
αi βjn+i,1j = αiβrni,r−1 βjn+j(m1).
!
Xi=1 jY=0 mX=1 iX∈Im jY=0
Now, by the case r =1, we can find β such that the sum α βni,r−1 is not zero and we can
apply the induction hypothesis to concrlude the proof. i∈I1 i r (cid:3)
P
Let K E C be a field complete with respect to , and B(E) denote the completion of
∞ ∞ ∞
⊆ ⊆ |·|
E[X] with respect to .
k·k
Theorem 3.6. The family G (X),N 0 forms an orthonormal basis of the E-Banach space
N
B(E), that is : { ≥ }
(i) Any F B(E) can be written in a unique way as a convergent series F = f G (X)
∈ N≥0 N N
with f E,N 0 ; lim f =0 ;
N N→∞ N
∈ ≥ P
(ii) if F is written as above, F =max f .
N≥0 N ∞
k k | |
Proof. It is enough to prove the above properties (i) and (ii) for F E[X]. Note that property
∈
(i) is a consequence of the fact that deg G (X) = N for all N 0. Let us prove property (ii).
r X N ≥
It is enough to consider F = x G with for all 0 i r, v (x ) = 0. We are reduced to
i=0 r Ni ≤ ≤ ∞ i
proving that F = 1, that is, F 1 since we already know the converse inequality. And the
existence of λk kΛ such thatPvk(Fk(≥λ))=0 is a consequence of Lemmas 3.4 and 3.5. (cid:3)
C ∞
∈
3.3. The multivariable case. Let s 1 be an integer, we define for a field K E C
∞ ∞
complete with respect to , B (E) to≥be the completionof E[X] with respectto ⊆. We⊆write
∞ s
also for short B =B (C |·)|. Observe that for N ,...,N N, we have : k·k
s s ∞ 1 s
∈
G (X ) G (X ) =1.
k N1 1 ··· Ns s k
Theorem 3.7. Let L E C be complete with respect to , then the family
∞ ∞
⊆ ⊆ |·|
G (X ) G (X ),N ,...,N N
{ N1 1 ··· Ns s 1 s ∈ }
forms an orthonormal basis of the E-Banach space B (E), that is :
s
(i) Any F B (E) can be written in a unique way as the sum of a summable family
s
∈
F = f G (X ) G (X )
N1,...,Ns N1 1 ··· Ns s
(N1,..X.,Ns)∈Ns
with N ,...,N N,f E ; f goes to 0 with respect to the Fr´echet filter ;
(ii) if F i∀s w1ritten ass∈above,N1,F...,N=s ∈max fN1,...,Ns ,N ,...,N N .
k k {| N1,...,Ns|∞ 1 s ∈ }
Proof. Weproceedbyinductionons 1. Thecases=1isthestatementofTheorem3.6. Assume
≥
now that s 2 and that the theorem is true for s 1. It will be enough to prove (i) and (ii)
≥ −
for polynomials, and (i) is still an easy consequence of deg G (X) = N for all N 0. Write a
X N ≥
polynomial
r
F = α G (X ) E[X], where 1 i r,α E[X ,...,X ].
r Ni s ∈ ∀ ≤ ≤ i ∈ 1 s−1
i=0
X
Write for 1 i r, the polynomial
≤ ≤
α = α(i) G (X ) G (X )
i i1,...,is−1 i1 1 ··· is−1 s
i1,.X..,is−1
ANDERSON-STARK UNITS FOR Fq[θ] 7
(i)
with α E. Then the induction hypothesis tells that for all i :
i1,...,is−1 ∈
α =max α(i) ,i ,...,i N .
k ik i1,...,is−1 ∞ 1 s−1 ∈
n(cid:12) (cid:12) o
Thus (cid:12) (cid:12)
F max α =m(cid:12)ax α(i)(cid:12) ,i ,...,i ,i N .
k k≤1≤i≤rk ik i1,...,is−1 ∞ 1 s−1 ∈
n(cid:12) (cid:12) o
Lλe,t.1..≤,λi0 ≤KrsbseucshucthhatthaFt(kλαi,0.k.=.,λma)x1≤=i≤(cid:12)(cid:12)rαkαik.,Ltoet(cid:12)(cid:12)pλro,v.e.t.h,λe conveKrsse−i1nesquuchalitthya,twe will find
1 s ∈ | 1 s |∞ k i0k 1 s−1 ∈
α (λ ,...,λ ) = α .
| i0 1 s−1 |∞ k i0k
Then, by the case s=1,
F(λ ,...,λ ,X ) =max α (λ ,...,λ ) = α
k 1 s−1 s k | i0 1 s−1 |∞ k i0k
Therefore, we can find λ K such that F(λ ,...,λ ,λ) = α , proving that F = α
and the theorem. ∈ | 1 s−1 |∞ k i0k k k k i0(cid:3)k
For all N = r N qi 0, define
i=0 i ≥
r
P
HN(X)= (θi X)Ni =π−lq(N)GN(X) K∞[X].
∗ ! ∈
i=0
Y
lq(N)
Then the HN’s generate K∞[X] and HN(X) =q q−1 . If E does not contain L, in particular if
k k
E =K , then G has no longer coefficients in E and there might not exist an orthonormalbasis
∞ N
of B(E). However, Theorem 3.7 still implies :
Corollary 3.8. Let K E C be complete with respect to , then the family
∞ ∞ ∞
⊆ ⊆ |·|
H (X ) H (X ),N ,...,N N
{ N1 1 ··· Ns s 1 s ∈ }
forms an orthogonal basis of the E-Banach space B (E), that is :
s
(i) Any F B (E) can be written in a unique way as the sum of a summable family
s
∈
F = f H (X ) H (X )
N1,...,Ns N1 1 ··· Ns s
(N1,..X.,Ns)∈Ns
with ∀N1,...,Ns ∈ N,fN1,...,Ns ∈ E ; |fN1,...,Ns|∞qlq(N1)+q·−··1+lq(Ns) goes to 0 with respect to
the Fr´echet filter ;
(ii) if F is written as above,
F = max f H (X ) H (X ) ,N ,...,N N
k k {| N1,...,Ns N1 1 ··· Ns s |∞ 1 s ∈ }
= max{|fN1,...,Ns|∞qlq(N1)+q·−··1+lq(Ns),N1,...,Ns ∈N}.
3.4. The Carlitz action. In this section, K E C is a field complete with respect to
∞ ∞
⊆ ⊆
. Note that the action of A on E[X] defined in (3.1) satisfies that for all a A∗, the map
∞
|·| ∗ ∈
F a F is an isometry on E[X]. Thus, the action extends to an action, still denoted , of A
on7→B (E∗), such that for all a A∗, the map F a F∗ is an isometry on B (E). ∗
s s
∈ 7→ ∗
Now, instead of considering the simultaneous action of A on each of the X , we will separate this
j
action into actions on a single variable X , namely, for 1 j s, F B (E) and a A, we set :
j s
≤ ≤ ∈ ∈
(3.2) a F(X)=F(X ,...,X ,C (X ),X ,...,X )
j 1 j−1 a j j+1 s
∗
This is still an action of monoid, but if we restrict this action to the set of polynomials in E[X]
which are F -linear in the variable X , the action induces a structure of A-module. Thus we
q j j
∗
define :
E[X]lin = F E[X];F is linear with respect to each of the variables X ,...,X
1 s
{ ∈ }
whichisjustthesub-E-vectorspaceofE[X]spannedbythemonomialsXqi1 Xqis,i ,...,i N.
1 ··· s 1 s ∈
Since the actions and commute and commute with the linear action of E, E[X]lin has a
j i
∗ ∗
structure of module over E A⊗s, that is, if t ,...,t are new indeterminates, we identify
⊗Fq 1 s
E A⊗s with E[t ,...,t ] and E[X]lin has a structure of E[t ,...,t ]-module given by :
⊗Fq 1 s 1 s
(3.3) 1 j s, t .F(X ,...,X )=F(X ,...,X ,C (X ),X ,...,X ).
j 1 s 1 j−1 θ j j+1 s
∀ ≤ ≤
8 B.ANGLE`S,F.PELLARIN,ANDF.TAVARESRIBEIRO
We write t for the set of variable t1,...,ts and if i=(i1,...,is)∈Ns, ti =ti11···tiss.
Theactiondefinedbyformula3.3extendstoanactiononE[X],turningE[X]intoanE[t]-algebra.
We define the subordinate norm . on E[t] by :
∞
kk
f.F
f = sup k k.
∞
k k F∈E[X]\{0} kFk
Lemma 3.9. Let f E[t], f = ifiti, then for all F E[X] 0 ,
∈ ∈ \{ }
f.F
P
f ∞ =max fi ∞ = k k.
k k i | | F
k k
Remark 3.10. The lemma says in particular that the norm coincides with the Gauss norm
∞
k·k
on E[t], which is known to be multiplicative. This property also follows easily from the lemma.
Proof of the lemma. Write F = N1,...,NsFN1,...,NsHN1(X1)···HNs(Xs) and M = maxi|fi|∞.
Remark that for all N 1 and for all 1 i s, t .H (X )=H (X ). Since l (N)=l (qN), we
i N i qN i q q
≥ P ≤ ≤
deduce from Corollary 3.8 that f.F M F .
Conversely,consider(N ,...,Nk )kth≤e indkex,kminimalforthe lexicographicorderingonNs, such
1,0 s,0
that
lq(N1,0)+···+lq(Ns,0)
|FN1,0,...,Ns,0|∞q q−1 =kFk
and i = (i ,...,i ) the index, minimal for the lexicographic ordering on Ns, such that M =
0 1,0 s,0
|fi0|∞. Then, the coefficient of
H (X ) H (X )
qi1,0N1,0 1 ··· qis,0Ns,0 s
in the expansion of f.F in the basis of Corollary 3.8 is equal to
fi0FN1,0,...,Ns,0 + terms of lower norm,
whence the result. (cid:3)
We define
Blin(E) the adherence of E[X]lin in B (E),
• Ts(E) the completion of E[t] for the Gs auss norm . .
s ∞
• kk
Recallthat T (E)is the standardTate algebrain s variablesoverE (see [FvdP04, II.1.]), thatis,
s
the algebra of formal series i∈Nsfiti with fi ∈E going to zero with respect to the§Fr´echetfilter.
The action of E[t] extends naturally to an action of T (E) on B (E) and on Blin(E).
P s s s
Lemmaba3s.i1s1o.f elem(1en)tsThoef tfhaemsialym{eHnqonr1m(Xq1q)−s·1··oHfqtnhse(XEs-B),ann1a,c.h..s,pnasce∈BNlsi}n(Efo)r,ms an orthogonal
T (E) B (E)
(2) The map sf → f.(X1s Xs) is injective, with for all f ∈Ts(E),
(cid:26) 7→ ···
s
f.(X1 Xs) =qq−1 f ∞.
k ··· k k k
(3) E[X]lin =E[t].X X ,
1 s
(4) Blin(E)=T (E).X··· X .
s s 1··· s
Proof. Since for all 1 i s and all n 0, H (X ) is an F -linear polynomial of degree qn,
qn i q
tfohlelofwams firlyom{HCqonr1o(lXla1r)y·≤·3·.8H.q≤Tnsh(eXrse)l,anti1o,n..t.ni,≥.nXsi∈=NH}qfno(rXmis)athbeansiismopfliEes[Xth]leinoatnhdertahsesefirrtsitonass.sertio(cid:3)n
s
As a consequence, the map f f.X1 Xs defines, up to the normalisation constant qq−1, an
isometric immersion of T (E) in7→to B (E·)·.·Write A[X]lin =A[X] E[X]lin, we have :
s s
∩
Lemma 3.12. Let f E[t], then f.(X X ) A[X]lin if, and only if, f A[t]. In particular,
1 s
∈ ··· ∈ ∈
A[X]lin =A[t].X X .
1 s
···
Proof. It is clear that if f A[t], then f.(X X ) A[X]. Note that, since
1 s
∈ ··· ∈
ti11···tiss.HN1(X1)···HNs(Xs)=Hqi1N1(X1)···HqisNs(Xs),
a consequenceof Corollary3.8is thatB (E) is a torsion-freeT (E)-module. Then, the converseis
s s
an easy consequence of the fact that t .X is a monic polynomial in A[X ]. (cid:3)
i i i
ANDERSON-STARK UNITS FOR Fq[θ] 9
4. Multivariable log-algebraicity
4.1. The log-algebraicity theorem. Let Z be another indeterminate over C . We let τ act on
∞
C [X][[Z]] (or in the one variable case on C [X][[Z]]) via τ(F)=Fq.
∞ ∞
Let F A[X] ; we form the series
∈
a F
Zqd ∗ K[X][[Z]]
a ∈
Xd≥0 a∈XA+,d
and take exp of this series which makes sense in K[X][[Z]]. Anderson’s log-algebraicity theorem
C
[And96, Theorem 3] for A states then
Theorem 4.1 (Anderson). For all F A[X]
∈
a F
exp Zqd ∗ A[X,Z].
C a ∈
Xd≥0 a∈XA+,d
The aim of this section is to give a multivariable generalisationof this result. But first, let us give
a simple proof of Theorem 4.1 in the case of F =X and Z =1.
Example 4.2. Write X = exp Y, where Y = log X K[[X]]. Then a X = exp (aY) =
C C ∈ ∗ C
aqjYqj. Thus,
j≥0 Dj
P a X aqj−1Yqj Yqj
Zqd ∗ = Zqd = Zqd aqj−1.
a D D
j j
Xd≥0 a∈XA+,d Xd≥0 a∈XA+,dXj≥0 Xj≥0 Xd≥0 a∈XA+,d
But one can evaluate at Z = 1 since (see [Gos96, Example 8.13.9]) aqj−1 = 0 for d j,
a∈A+,d ≫
andmoreover aqj−1 =0forallj >0whilethissumequals1whenj =0. Therefore,
d≥0 a∈A+,d P
we get :
P P
a X
∗ =Y =log X.
a C
Xd≥0a∈XA+,d
Lemma 4.3. If F A[X] satisfies F 1, then F F .
q
∈ k k≤ ∈
Proof. Ifλ ,...,λ Λ ,thenF(λ ,...,λ )isintegraloverA,andthecondition F 1implies
1 s C 1 s
thatforallλ ,...,λ∈ Λ ,F(λ ,...,λ ) F . ButF isalgebraicallyclosedinKk(λk,≤...,λ )(see
1 s C 1 s q q 1 s
[Ros02, Corollary to T∈heorem 12.14]), so∈that F(λ ,...,λ ) F . Now, for any λ ,...,λ
1 s q 1 s−1
∈ ∈
Λ , the polynomial F(λ ,...,λ ,X ) takes at least one value infinitely many times. An easy
C 1 s−1 s
induction on s then implies that F is constant, that is F F . (cid:3)
q
∈
We define an action of the multiplicative monoid A∗ over C [X][[Z]] by letting for F(X,Z)
∞
C [X][[Z]] and a A∗ : ∈
∞
∈
a F =F C (X ),...,C (X ),Zqdegθa .
a 1 a s
∗
Observe that exp gives rise to a well-(cid:16)defined endomorphism of C (cid:17)[X][[Z]] and that
C ∞
exp (K[X][[Z]]) K[X][[Z]].
C ⊂
Let F C [X] ; following Anderson, we set for k <0 :
∞
∈
L (F)=Z (F)=0
k k
and for k 0 :
≥
a F
L (F) = ∗ C [X],
k ∞
a ∈
a∈XA+,k
L (F)qj
Z (F) = k−j C [X].
k ∞
D ∈
j
j≥0
X
10 B.ANGLE`S,F.PELLARIN,ANDF.TAVARESRIBEIRO
Define moreover :
a (FZ)
l(F,Z) = ∗ = ZqkL (F) C [X][[Z]],
k ∞
a ∈
aX∈A+ kX≥0
L(F,Z) = exp (l(F,Z))= Z (F)Zqk C [X][[Z]].
C k ∈ ∞
k≥0
X
Lemma 4.4. Let F C [X] and k 0.
∞
∈ ≥
(1) L (F) F q−k,
k
k k≤k k
(2) Z (F) max F qjq−kqj.
k 0≤j≤k
k k≤ k k
Proof. This comes from the definitions and the fact that for all a A∗, a F = F . (cid:3)
∈ k ∗ k k k
We call a monic irreducible polynomial of A a prime of A. Let P be a prime of A. Let F K[X]
andletI beafinitesubsetofNs suchthatF = i∈IαiXi ∈K[X],LetvP betheP-adicva∈luation
on K normalized by v (P)=1, we set :
P
P
vP(F)=inf vP(αi),i I .
{ ∈ }
Recall that we have for F,G K[X], and λ K :
∈ ∈
v (F +G) inf(v (F),v (G)), v (FG)=v (F)+v (G),
P P P P P P
• ≥
v (λF)=v (λ)+v (F),
p P P
•
v (F)=+ if and only if F =0.
P
• ∞
Lemma 4.5. Let P be a prime of A. Let F K[X] be such that v (F) 0. Then for all k 0,
P
∈ ≥ ≥
v (Z (F)) 0.
P k
≥
Proof. The proof is essentially the same as [And96, Proposition 6]. We recall it because some
details will be needed in the proof of Proposition 4.17.
Set A = x K,v (x) 0 . Let d be the degree of P, we have in A[τ] : C τd mod PA[τ].
(P) P P
{ ∈ ≥ } ≡
We prove that if G= G Zqk K[X][[Z]] satisfies
k≥0 k ∈
P (CP P )(G) PA(P)[X][[Z]],
− ∗ ∈
then k 0,G A [X]. Set G = 0 if k < 0 and write C = d [P] τi where [P] = P,
∀ ≥ k ∈ (P) k P i=0 i 0
[P] PA if i<d and [P] =1. We have (C P )(G)= H Zqk with for all k 0,
i ∈ d P − ∗ k≥0 kP ≥
d d−1 P
H = [P] τi(G ) P G =PG + [P] τi(G )+τd(G ) P G PA [X].
k i k−i k−d k i k−i k−d k−d (P)
− ∗ − ∗ ∈
i=0 i=1
X X
In particular, H = PG PA [X] so that G A [X]. Now, by induction on k, if we know
0 0 (P) 0 (P)
∈ ∈
that G A [X] for i =1,...,d, then τd(G ) P G PA [X] and we deduce that
k−i (P) k−d k−d (P)
∈ − ∗ ∈
G PA [X].
k (P)
∈
Define l∗(F,Z)= a∗(FZ) A [X][[Z]], we have
a∈A+,P∤a a ∈ (P)
P a (FZ) a (FZ)
l(F,Z) = ∗ + ∗
a a
a∈AX+,P|a a∈AX+,P∤a
(aP) (FZ) 1
= ∗ +l∗(F,Z)= (P l(F,Z))+l∗(F,Z)
aP P ∗
aX∈A+
which yields the relation :
Pl(F,Z) P l(F,Z)=Pl∗(F,Z).
− ∗
Remark that the action commutes with τ, and thus with exp , thus if we apply exp , we get
∗ c C
Pqj
(C P )(L(F,Z))=exp (Pl∗(F,Z))= l∗(F,Z)qj
P − ∗ C D
j
j≥0
X
and since j 0, v (Pqj) 1, we get (C P )(L(F,Z)) PA [X][[Z]] whence L(F,Z)
∀ ≥ P Dj ≥ P − ∗ ∈ (P) ∈
A [X][[Z]]. (cid:3)
(P)
