Explicit Formulae for $L$-values in Finite Characteristic

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by  Rudolph Bronson Perkins

L
Explicit Formulae for -values in Finite
Characteristic
Rudolph Bronson Perkins
July 10, 2012
2
1
0
2 Abstract

l
u We give an elementary treatment making explicit Pellarin’s theorems
J regarding his new twist on Goss’ characteristic p valued L-series. Our
7 method extends to show that a slightly more general form of these se-
ries also turn out to be rational functions when multiplied by a certain

] canonicalfactor. WefurthergiveaclosedformforPellarin’sL-seriesat1,
T
andwereproducehisproofusingacertainoperatorformalisminspiredby
N Anderson’s t-motives. Along the way we prove what we call The Funda-
. mentalRelationrelatingthesemoregeneralseriestotheCarlitzlogarithm.
h
ThisrelationwillplayakeyroleinourexpositionofPellarin’sproof,and
t
a we shall also use it to give closed form formulas for a certain family of
m Dirichlet L-series at 1.
[

1 1 Introduction
v
3
In all that follows below, let q =pm0 be a fixed power of the odd prime p. Let
5
A=F [θ]andK =F (θ). Let|·| =|·|bethenon-archimedianabsolutevalue
7 q q ∞
1 on K which assigns |θ| = q. Let K∞ be the completion of K with respect to
. thisabsolutevalue,andletC bethecompletionofanalgebraicclosureofK
7 ∞ ∞
0 equipped with the canonical extension of |·|∞. We let A+ and A+(d) be the
2 collection of all monic polynomials and the collection of all monic polynomials
1 of degree d respectively. We let P be the collection of all monic irreducible
+
:
v polynomials in A.
i For each t ∈ C , let χ : A → C be the canonical F -algebra morphism
X ∞ t ∞ q
given by θ 7→ t. In [10], Pellarin introduces the formal characteristic p valued
r
a L-series
−1
χ (v)
L(χ ,k)= χ (a)a−k = 1− t .
t t vk
aX∈A+ vY∈P+(cid:18) (cid:19)
It is easy to see that both the product and series representations converge for
|t| < |θ| for all positive integers k. Pellarin’s series is a deformation of the
∞ ∞
characteristic p valued zeta function
ζ(k):= a−k
aX∈A+
first studied for positive integers k by Carlitz in the 30’s and 40’s and rediscov-
eredandvastlyextendedbyGossinthe70’sand80’s,see[3,4]and[8],Chapter
1
8. In relation to this zeta function Carlitz pioneered machinery which is abso-
lutely basic today to function field arithmetic’s analog of classical cyclotomic
theory.
Let ι be a fixed choice of (q − 1)-th root of −θ. Carlitz discovered the
following constant,
∞ θ −1
π˜ =ιθ 1− ,
θqi
i=1(cid:18) (cid:19)
Y
and this form is due to Anderson and Thakur in [2]. This constant is algebraic
over K , and its positive powers l ≡ 0 mod (q −1) lie in K . Using this
∞ ∞
number Carlitz showed ζ(l)/π˜l is an element of K. This is Carlitz’ analog of
Euler’s classical result for the Riemann zeta function and is now commonly
referred to as the Euler-Carlitz relations for the Goss-Carlitz zeta function.
AnotherdiscoveryofCarlitz isanA-module module structureonC which
∞
plays the role for function fields that the Z-module structure of exponentiation
onthemultiplicativegroupofthecomplexnumbersplaysclassically. TheCarlitz
module C is the F -algebra morphism from A into the twisted polynomial ring
q
A{τ} determined by sending θ 7→ τ + θτ0, where the elements of A{τ} are
subjecttothecommutationrelationτx=xqτ forallx∈A. Thereisanobvious
definition of evaluationfor elements x in an A-algebra L givenby a τi(x):=
i
a xqi. Via this definition the A-algebraL obtains a new A-module structure
i P
where the twisted polynomial C(a) acts on x∈L by evaluation.
P
In the 80’s Anderson constructed a category of t-modules, containing the
Carlitz module, and defined a notion of tensor product for this category. In
an important work [2] Anderson and Thakur investigated the tensor powers of
the Carlitz module in Anderson’s category of t-modules and related the n-th
tensor power of the Carlitz module to the value ζ(n) of the Goss-Carlitz zeta
function. This paved the way for results on the transcendence of [13] (and
later aclassificationofalgebraicrelationsamong[6])zeta values atthe positive
integers.
Thefollowingfunctionplaysacrucialroleinalltensorpowersofthe Carlitz
module. Let
∞ t −1
ω(t):=ι 1− .
θqi
i=0(cid:18) (cid:19)
Y
Thisproductconvergesfor|t|<|θ|,andoneobservesthatlim (θ−t)ω(t)=π˜.
t→θ
Our next result connects all of these stories together, and section 3 is devoted
to its proof.
Theorem 1.1 (Pellarin [10], Theorems 1 and 2). Let k be a positive integer
such that k≡1 mod (q−1). Then
L(χ ,k)ω(t)=λ (t)π˜k
t k
for some λ (t)∈K(t).
k
Furthermore, when k =1 we have λ (t)=1/(θ−t).
k
This is clearly a vast generalization of Euler’s classical theorem on zeta
values. Pellarin discovered this formula using the theory of deformations of
vectorial modular forms and this supplied the long desired connection between
characteristic p valued L-series and modular forms.
2
Pellarin has also given an elegant and elementary proof of the formula for
L(χ ,1) inspired by Anderson’s theory of t-motives, and we reproduce the es-
t
sentialideashere. Alongthe wayweintroduceandproveacertainfundamental
relation (see Theorem 1.2 just below) relating Pellarin’s L-series at k = 1 to
the Carlitz logarithm. Relations between this function and zeta values are well
known, see e.g. [1], [6], [13]. In a forthcoming work of the author and F. Pel-
larin relations of this type play a fundamental role in computing closed form
expressionsforPellarin’sL-seriesatvariousvaluesofk whereoneslightlyvaries
the characters involved as follows: Let s be a positive integer, and let t ∈C
i ∞
be such that |t |≤1 for all i=1,...,s. For positive integers k we define
i
χ (a)···χ (a) χ (v)···χ (v) −1
L(Πsχ ,k):= t1 ts = 1− t1 ts .
ti ak vk
aX∈A+ vY∈P+(cid:18) (cid:19)
See Remark 7 in [10] where these series were first defined, and where it is
sketched that these series are entire analytic in the variables t ,...,t .
1 s
The following three objects will appear in most results that follow.
1. Let b (t)=1, and for positive integers i≥1 let b (t)= i−1(t−θqj).
0 i j=0
2. Let D =1, and for positive integers i≥1 let D = i−Q1(θqi −θqj).
0 i j=0
3. Let L =1, and for positive integers i≥1 let L = Qi (θqj −θ).
0 i j=1
The polynomials b (t) are “universal” in the sense thatQthey give rise to the
i
basic objects D ,L (first defined by Carlitz in the 30’s) in a simple way. The
i i
connectionD =b (θqi)isclear,andwewillfurtherilluminateitfortheL when
i i i
wedeveloptheformalismofthe“partialFrobenius.” Thefollowingtheoremwill
proven in Section 4 and provides a concrete example of the above discussion.
Theorem 1.2 (The Generalized Fundamental Relation). Let 1 ≤ r < q be a
positive integer and |t |≤1 for i=1,...,r. Then
i
∞ r b (t )
L(Πrχ ,1)= (−1)i j=1 i j ∈C .
ti L ∞
i=0 Q i
X
As an application of this theorem, we will give explicit representations for
all Dirichlet L-series arising from primitive characters χ : A → F in terms of
q
the Carlitz’ logarithm and a monic uniformizer of the character’s conductor.
Our main result gives an explicit expression in terms of classical Carlitz-
Bernoulli numbers and Carlitz factorials (both of which will be defined in the
appropriate section) for the rational functions λ (t) defined in Pellarin’s theo-
k
rem. Ourproofiselementaryandusesseveralingredients: WewillusePellarin’s
formulaforL(χ ,1),andcertainexpansionsofbasicfunctions(suchasthechar-
t
acter χ ). Furthermore, the vanishing of Goss’ zeta function (which has yet to
t
be defined in this paper) at the negative “even” integers, and the vanishing of
the powersums ak forcertainnon-negativesintegersd,k willbe used
a∈A+(d)
in a crucial way.
P
3
Theorem 1.3 (Main Result). 1. Let k,s be positive integers such that 1 ≤
s≤q, k ≥s and k ≡s mod (q−1), and let t ∈C be such that |t |≤1
i ∞ i
for i=1,...,s. Then
s s s π˜qki
L(Πsχ ,k) ω(t )= ζ k− qki , (1)
ti Yi=1 i k1X,...,ks" Xi=1 !Yi=1Dki(θqki −ti)#
where the sum is over all combinations of s positive integers k ,...,k
1 s
such that k− s qki ≥0, and ζ is Goss’ zeta function.
i=1
2. With the abovePassumptions on k,s,t ,
i
L(Πsχ ,k) s ω(t )=π˜k BCk−Psi=1qki s 1 , (2)
ti Yi=1 i k1X,...,ks"Π(k− si=1qki)iY=1Dki(θqki −ti)#
P
where the sum is over all combinations of s positive integers k ,...,k
1 s
such that k− si=1qki ≥ 0, and the BCj ∈ K are the Bernoulli-Carlitz
numbersandΠ(j)∈Ais thej-thCarlitz factorial bothdefinedfor integers
P
j ≥0.
Remark 1.4. 1. Inaprivatecorrespondence,fors>0andk ≡s mod (q−1)
F.Pellarinhadconjecturedtheexistenceofrationalfunctionsλ (t ,...,t )∈
k 1 s
K(t ,...,t ) such that
1 s
s
L(Πsχ ,k) ω(t )=π˜kλ (t ,...,t ).
ti i k 1 s
i=1
Y
Our main theorem verifies this for 1≤s≤q, k ≥s, and k ≡s mod (q−
1). Our method for computing these rational functions λ is naturally
k
limited by these restrictions on s and k as can be seen in the proof of the
main result.
2. As the righthandsides ofthe two equationsin the statementof ourmain
theorem are finite sums with simple poles agreeing with those of ω(t )
i
weobtaintherigid-analyticcontinuationofthefunctionL(Πsχ ,k)toan
ti Q
entire function in the variables t ,...,t . See [10] Remark 7.
1 s
3. As the denominators of the Bernoulli-Carlitz numbers have been given
explicitly, see [7], we obtain a complete description of the denominator of
L(Πsχ ,k).
ti
4. We also give some discussion of possible generalizations of these formulas
and some of the roadblocks to going in those directions following their
proofs.
2 First Consequences
In this first section we derive some pleasing first consequences of Pellarin’s for-
mula at k =s=1.
As has been observed by Pellarin, if one starts with an element λ∈F and
q
lets t = λ in L(χ ,k), then one obtains a characteristic p valued Dirichlet L-
t
function. Indeed, under this substitution χ becomes a map A→A/(f), where
t
4
f ∈ A is the minimal polynomial of λ. In this way, χ may be thought of as a
t
continuous parameterizationof characters χ:A→F .
q
As an immediate consequence of Pellarin’s theorem we recover a product
formula, this time over primes of A, for the fundamental period of the Carlitz
moduleπ˜. M.PapanikolasandhisstudentB.Luteshaveshownthisbydifferent
means by in [9].
Recall that we let P be the set of monic irreducible polynomials of A.
+
Proposition 2.1. We have
−1
f(0)
π˜ =ιθ 1− .
f
fY∈P+(cid:18) (cid:19)
Proof. We simply use the product formula for ω first given by Anderson and
Thakur
∞ t −1
ω(t)=ι 1− ,
θqi
i=0(cid:18) (cid:19)
Y
the Euler product for L(χ ,s), and Pellarin’s formula at s = 1. Evaluating at
t
t=0 finishes the proof.
Corollary 2.2. We have
∞
f(0) θ
1− = 1− .
f θqi
fY∈P+(cid:18) (cid:19) iY=1(cid:18) (cid:19)
Proof. This is immediate.
The next consequence of Pellarin’s result has some features of an analytic
classnumberformulafortheextensionK(((−1)df)q−11),wherep∈Spec(A)and
f ∈p is a monic generator. Recall that the Galois group G=Gal((A/p)/F ) is
q
cyclic generated by the Frobenius τ :x7→xq. We begin with a lemma.
Lemma 2.3. Let λ∈F , and let f ∈A be its minimal polynomial. Suppose f
q
has degree d. Then
d
ω(τiλ)q−1 =(−1)df.
i=1
Y
Proof. With the notations above we have f = d (θ − τiλ). It is a basic
i=1
property of limits that if a → a and b → b, then a b → ab. With this and
n n n n
Q
5
the continuity of the map x7→xq−1 we have
q−1
d d n τiλ −1
ω(τiλ)q−1 = lim ι 1−
n→∞ θqj 
i=1 i=1 j=0(cid:18) (cid:19)
Y Y Y
 d n τiλ−(q−1)
= ι(q−1)d lim 1−
n→∞ θqj
i=1j=0(cid:18) (cid:19)
YY
n d τiλ −(q−1)
= (−θ)d lim 1−
n→∞ θqj
j=0i=1(cid:18) (cid:19)
YY
n
= (−θ)d lim (f(θ)/θd)−qj(q−1)
n→∞
j=0
Y
= (−θ)d lim (f/θd)1−qn+1
n→∞
= (−1)df.
The last equality follows since f/θd =1+g, where g ∈K is such that |g|< 1,
and hence
lim (f/θd)−qn+1 = lim (1+g)−qn+1 = lim (1+gqn+1)−1 =1.
n→∞ n→∞ n→∞
Let f,λ be as in the lemma above, and let us fix an isomorphism A/(f) ∼=
F (λ) ⊆ C given by θ 7→ λ. We quickly recall some facts about the function
q ∞
field analog of classical cyclotomic extensions. We defined the Carlitz module
in the introductionto be the F -algebramorphismC fromA into C {τ} given
q ∞
by θ 7→ τ +θτ0. For a ∈ A let us denote the image of a under C by C . We
a
examinethepolynomialC (x)∈A[x],whereagainweemploythemapfromthe
f
twisted polynomial ring to A[x] given by τ(x) = xq. Let us write C[f] for the
roots of C (x). Then it is well known,see [8] or [12], that as an A-module C[f]
f
is isomorphicto A/(f). Furthermore,adjoiningthe elementsofC[f]toK gives
anabelian extensionwhose GaloisgroupG is isomorphicto the multiplicative
f
group of A/(f), and in this extension (f) totally ramifies while all other finite
primes remain unramified.
Remark 2.4. Let f be as in the lemma above, and let ξ be a root of the poly-
nomial C (x). It can be shown, see Rosen [12] Ch. 12, Ex. 1,2, that
f
(−1)df = C (ξ)q−1,
a
a∈AY+(<d)
wherewe define A (<d) tobe the setofmonicpolynomialsofdegreelessthan
+
d. Thus ω(τiλ) and C (ξ) are both (q−1)-th roots of (−1)df, and hence
a
they differ by an element of F∗. This implies that ω(τjλ)∈K(C[f]).
Q Q q
Before stating the next theorem we make a connection with the characters
Q
that will appear in its statement and a certain symbol introduced by Carlitz.
Viathe isomorphism(A/(f))∗ ∼=Gf wemayviewthe charactersχτiλ andtheir
L-series as associated to the extension K(C[f])/K. Now the characters arising
6
from evaluation at the roots of f, as in the lemma above, are special in that
they arise from elements of the Galois group Gal((A/(f))/F ). Indeed, given
q
an element σ :x7→xqj ∈Gal((A/(f))/F ) we extend this to a character on A
j q
via composition with the natural projection A → A/(f) ∼= F [λ]. We observe
q
further thatfor j =1,...,d we have χ =χqj. Hence multiplying all ofthese
τjλ λ
characters together we obtain
d χ =d−1χqj =χqqd−−11.
τjλ λ λ
j=1 j=0
Y Y
We observe that this final character has order q−1 and hence takes values in
F . In fact, this final character is exactly one of Carlitz’ “quadradic symbols”
q
(a/f) which is defined for all a∈A to be the unique element of F such that
q
a qd−1
≡a q−1 (mod f).
f
(cid:18) (cid:19)
One can show that a has a (q−1)-th root modulo f if and only if (a/f) = 1.
And as we have already sketched there are connections between this symbol
and the field K(C[f]). There is even a reciprocity law which holds for these
characters. M.Rosengivesadelightfulaccountofallofthesethingsinhisbook
[12].
One final reason the characters χ are special is because once we have
τjλ
fixed an isomorphism from A/(f) ≡ F (λ) all characters χ : A → A/(f) may
q
be realized as χn for some n = 1,...,qd −1. Thus if one were to define new
λ
L-values in terms of “digit expansions,” as is common in the subject, the L-
seriesassociatedto these characterswouldbe the base. The readershouldkeep
this in mind when viewing the next result as it may be a possible instance
of such constructions. Taking the product over the L-series associated to the
specialcharactersdiscussedaboveincombinationwithPellarin’stheoremyields
aformulawhichisreminiscentofclassicalanalyticclassnumberformulas. With
the lemma above we are justified in writing d ω(τjλ)=((−1)df)q−11.
j=1
Theorem2.5. Letλ∈F andletf beitsmiQnimalpolynomial whichweassume
q
is of degree d. Then
d (−1)dπ˜d
L(χ ,1)= .
τiλ q
((−1)df)q−1
i=1
Y
Proof. From Pellarin’s theorem we have
π˜
L(χ )ω(τiλ)= .
τiλ,1 θ−τiλ
Multiplying over i=1,...,d we obtain
d π˜d
L(χ )ω(τiλ)= .
τiλ,1 f
i=1
Y
Applying the lemma above and doing a little algebra finishes the proof.
7
Remark 2.6. The author speculates that this L-value should be related to the
extension L:=K(((−1)df)q−11)⊆K(C[f]). Recall again that f is totally ram-
ified in the extension K(C[f]), while all other finite primes remain unramified.
Thus we have an appearance of a (q −1)-th root of the discriminant of this
extension, as well as the appearance of the fundamental period of the Carlitz
module π˜. It may be possible that the class number, q-regulator, and number
of roots of unity (which are all integers) for the ring of integers of L are all
sitting in the formula above, but because our L-series take values in a field of
characteristic p there is a congruence which is disguising them. For example,
it’s quite clear that modulo p the number of roots of unity in L is −1! There is
muchleft tothink aboutinthis direction,butsinceitwouldtakeusafieldfrom
the purposes of this note, we leave it for future investigation.
3 Pellarin’s Result at 1
In this section we indicate how to derive the formula
π˜
ω(t)L(χ ,1)=
t
θ−t
by elementary means modulo a certain “fundamental identity.” For a similar
account see the last section of [10]. We begin by developing the necessary
formalism.
Definition 3.1. Let L/K be a finite extension of fields equipped with the
∞
canonicalextensionof|·| . TheTate algebra overLisdefinedastheL-algebra
∞
T = a tj :a ∈L for all j, and a →0 .
L  j j j 
Xj≥0 
Observe that by our assumptions on L it is complete, and hence TL is also
complete with respect to the norm defined by
∞
kfk:=max|a |,
i
i=0
for all f = a ti ∈T . For now, we fix L and drop the subscript.
i L
We will be interested in understanding certain families of linear endomor-
P
phisms of the Tate algebra. Throughout this section we view the universal
character χ as an embedding of A into T under the identification θ 7→ t. Via
t
this embedding one obtains the “trivial action” of A on T where A acts by left
multiplication. More precisely, for all a∈A and f ∈T we define
a∗ f :=χ (a)f.
1 t
Now we may also extend the arithmetic Frobenius τ : L → L sending x 7→ xq
trivially to an F [t]-linear endomorphism of T by having it act as follows: For
q
f = a ti ∈T we define
i
τ ∗f := aqti.
P i
Thetwistedpolynomial ring L{τ}overLXisthenon-commutativeringofpolyno-
mialsinτ satisfyingthe commutationrelationτa=aqτ forall a∈L. With the
8
action of τ on T defined above we extend the action of the twisted polynomial
ring L{τ} from L to T , and thus we obtain a second action of A on the Tate
L
algebra, this time via the Carlitz module which we now recall.
The Carlitz module C is the F -algebra morphism A → A{τ} determined
q
by sending θ 7→θτ0+τ. We denote the image of the element a∈A under this
map by C . Via this embedding, any A-algebra J obtains a new A-action by
a
letting C , which is a polynomial in τ, act on the elements of J by evaluation.
a
Thus, to be explicit, the second action of A on the Tate algebra is defined
by
a∗ f :=C ∗f = C (a )ti,
2 a a i
where f = a ti. X
i
The natural question now arises: Is there a subset of T on which these
P
two actions agree? The answer is yes, and this subset is a free 1-dimensional
F [t]-submodule of T generated by Anderson and Thakur’s function ω. This
q
functionoriginallyappearedin[2]asacanonicalgeneratorofthe 1-dimensional
F [t]-module of solutions to the difference equation for φ∈T
q
τφ=(t−θ)φ. (3)
Rewriting(3)asθω+τ∗ω =tω,Pellarinobserves[10]thatthisimpliesC ∗ ω =
a
χ (a)ωforalla∈Aaswehavedescribedabove. Onecouldsaythatthefunction
t
ω “trivializes” the Carlitz action on T, and in fact ω is closely related to the
rigid analytic trivialization of the Carlitz module in the sense of Anderson. In
light of the L{τ} action on T it is convenient to have a special names for the
eigenvalues of the powers of τ acting on ω for all positive integers j.
Definition 3.2. Let b (t)=1, and for i≥1, let
0
i−1
b (t)= (t−θqi).
i
i=0
Y
Remark 3.3. 1. One observes immediately that τj ∗ ω =b (t)ω.
j
2. For j ≥ 1, one can recover D as b (θqj) and (−1)jL as (τb (t))| .
j j j j t=θ
Trivially D =L =b (t).
0 0 0
3. We will see the polynomials b (t) arise in a surprising way in the next
j
section as the coefficients of the “Wagner expansion” of the character χ .
t
Remark 3.4. As another way of looking at things, each element of A gives
rise to an F [t]-linear endomorphism of T via the Carlitz module, and ω is a
q
simultaneous eigenvector for this family of operators. With knowledge of the
numerous applications of which the function ω has been a part, it is now very
interestingtolookforothereigenvectorsofthetheCarlitzmoduleactionofAon
T, or indeed to look for a basis of simultaneous eigenvectorsfor the commuting
family of operators {C } .
a a∈A
We arealmostready to dive into the proof ofPellarin’sresult. Before doing
so, we must observe that the Carlitz exponential and logarithm act on suitable
subspaces of the Tate algebra in a natural way.
For an A-algebra L we define the ring of twisted power series L{{τ}} to be
the set of experssions of the form
∞
a τj,(a ∈L for all j)
j j
j=0
X
9
with addition and multiplication given by
a+b:= (a +b )τi and ab:= a bqi τk
i i  i j 
i k i+j=k
X X X
 
respectively, where a= a τi and b= b τj. Note that this is compatible
i i j j
with the definition of multiplication given above for A{τ}⊆K{{τ}}.
P P
The Carlitz exponential is defined to be the series
∞ τj
e := ,
C
D
j
j=0
X
which is convergenton all of C , with the obvious definition of evaluation:
∞
∞ zqj
e (z):= .
C
D
j
j=0
X
We also have the composition inverse (see just below for the definition of com-
position of two such series) of this map called the Carlitz logarithm:
∞ τj
log := (−1)j ,
C L
j
j=0
X
with the analogous definition of evaluation, valid for those z ∈ C such that
∞
|z|< q .
q−1
We alsonote thatif z lies in anextensionfieldL ofK,where Lis complete,
then so do e (z) and log (z) whenever they are defined.
C C
For all elements f = a ti ∈ T (resp. f such that kfk < q/(q −1)) we
i
extend e (resp. log ) to F [t]-linear endomorphisms of T by
C C Pq
e ∗f := e (a )ti (resp. log ∗f := log (a )ti).
C C i C C i
X X
Withallofthedefinitionsinorder,wemaynowstateTheFirstFundamental
Relation:
∞
b (t)
L(χ ,1)= (−1)i i .
t
L
i
i=0
X
Theproofofamoregeneralresultwhichhasthisequalityasaspecialcaseap-
pears asTheorem(1.2) inthe next section. Take note ofthe similaritybetween
the fundamental relationand the expressiongivenabovefor Carlitz’logarithm.
We exploit this now in the proof of Pellarin’s Theorem.
Proof of Theorem 1.1. We recall a simple observation of Pellarin [10]. Observe
that
∞
π˜
ω(t)= e tj,
C θj+1
j=0 (cid:18) (cid:19)
X
and that
∞
π˜ π˜
= tj.
θ−t θj+1
j=0
X
10