ON SPECIAL VALUES OF PELLARIN’S L-SERIES

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of

Philosophy in the Graduate School of the Ohio State University

By

Rudolph Bronson Perkins, BA

Graduate Program in Mathematics

The Ohio State University

2013

Dissertation Committee:

David Goss, Advisor

Wenzhi Luo

Warren Sinnott c Copyright by

Rudolph Bronson Perkins

2013 ABSTRACT

We consider various methods for explicitly computing the special values of Pel- larin’s L-series in several indeterminates, both at the positive integers and negative integers. Our premier result is the explicit calculation, in terms of the Goss-Carlitz zeta values, of the rational functions appearing in the paper by Pellarin in which these L-series were first introduced, as well as an explicit generalization of his result to s ≤ q indeterminates. We draw applications to new divisibility results for the numerators of the Bernoulli-Carlitz numbers by degree one irreducible polynomials and to explicit generating series and recursive relations for Pellarin’s series.

On the negative integers side, we deepen some work of Goss on the special poly- nomials associated to Pellarin’s series. In particular, we show that when certain parameters are fixed, the logarithmic growth in the degrees of the aforementioned special polynomials that is common in this area holds in great generality. We study the natural action of Goss’ group of digit permutations on a sub-class of these special polynomials and show that their degrees are an invariant of the action of Goss’ group.

Finally, we include a computation of the coefficients of the measure whose moments are the special polynomials associated to Pellarin’s L-series in one indeterminate.

The Wagner series for Pellarin’s evaluation character χt will play a central role in nearly all of the results of this dissertation. We will show that the Wagner series for χt, an object arising in the study of the completions of our ring of integers at the finite places, will also have meaning at the infinite place. In fact, at the infinite

ii place, this Wagner series is none other than Anderson’s generating function for the

Carlitz module, and we shall see that Pellarin’s L-series, the Wagner series for χt and Anderson’s generating function are inextricably tied.

iii to Chelsea Christine and Noah Benjamin

iv ACKNOWLEDGMENTS

David Goss treats the industry of function field arithmetic like a family business.

I’ve seen him greet his colleagues with a warm embrace, and evidence of his kind encouragement can be found in a significant proportion of the papers written in his

field. He has been an enthusiastic ear to all of my findings, sharing in my joy and urging me onward. He has encouraged me to write everything down since I first began working under his guidance, and if this dissertation is clear and well-written, it is a testament to his voice. Thank you so much, David!

Federico Pellarin is on the cutting edge of so many discoveries in function field arithmetic at the moment, and still he is a kind, generous and humble person. He has shared much of his recent work with me, and I have learned so much from him. He has been very generous with his time, offering insight into my work and suggesting we collaborate on some generalizations of results in this dissertation. Thanks to him,

I had the wonderful opportunity to share my results with the experts at a conference in Pisa, Italy. Federico, I am truly grateful and look forward to working with you in the coming year.

To all the mathematicians I’ve communicated with about my work - Tim All,

Bruno Angl`es,Francesc Bars, Gebhard B¨ockle, Alex Petrov, Lenny Taelman and

Dinesh Thakur - thank your for your support and insight. Thank to Matt Papanikolas for having me out to Texas to talk and for your generosity there and beyond! To

Adrian Wadsworth and Cristian Popescu, you are excellent teachers and I’m glad to

v have had you as mentors. It was an absolute pleasure to share my work with you in

San Diego. Thank you Cristian for giving me that opportunity.

Thanks to Chelsea Christine for coming along with me on this crazy journey. I’ve learned so much from you about being a gentle person, and I am glad to have you in my life. Noah Benjamin, you’ve added a new dimension to my life. You’ve changed everything in a wonderful way! To the rest of my family, my new family here in Ohio

- especially Bob Mauck - and those back on the west coast, I’ve been refreshed on many occasions by your presence and support.

Finally, to the many friends I’ve made since coming to Columbus - especially,

David and Jenny, Dan and Kati, Michael, Jonathan, Greg, and Neal - you’ve helped make the tough times in graduate school less so.

vi VITA

2009 ...... B.A. in Mathematics, University of California, San Diego

2009-2012 ...... Graduate Teaching / Research Assistant, The Ohio State University

2013 ...... Presidential Fellow, The Ohio State University

PUBLICATIONS

R. B. Perkins, On Pellarin’s L-series, Proc. Amer. Math. Soc. (2012), Accepted.

R. B. Perkins, Explicit formulae for L-series in finite characteristic, Submitted. arXiv:1207.1753v2 (2013).

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Special values of L-functions in positive characteristic

vii TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... v

Vita ...... vii

CHAPTER PAGE

1 Introduction ...... 1

1.1 Carlitz’ First Results ...... 2 1.1.1 The Carlitz-Euler Relations ...... 5 1.1.2 Carlitz’ von-Staudt Theorem ...... 6 1.2 Goss’ Extension of Carlitz’ zeta values ...... 8 1.2.1 Generating series for the Sd(k)...... 8 1.3 Non-archimedean measures ...... 11 1.4 The Tate Algebra, the Frobenius Endomorphism, and Twisting . . 16 1.4.1 The multivariate Tate Algebra ...... 17 1.4.2 The Frobenius endomorphism and Twisting ...... 18 1.5 Anderson and Thakur’s function ω ...... 20 1.6 The Anderson Generating Function ...... 23 1.6.1 The Wagner Series for χt and the Anderson Generating Function ...... 24

2 Pellarin’s L-series ...... 27

2.1 First Definitions and Properties ...... 27 2.1.1 The Analyticity Theorem of Angl`esand Pellarin ...... 29 2.1.2 The Rationality Theorem of Angl`esand Pellarin ...... 29 2.1.3 Deformation of Carlitz-Goss zeta values and Goss L-values 30

3 Pellarin’s special L-values at the negative integers ...... 32

3.1 Recursive Nature ...... 32

viii 3.2 Trivial zeros of Pellarin’s series ...... 34 3.3 Logarithmic growth ...... 35 3.3.1 Some preliminary bounds ...... 36 3.3.2 Exact Degree ...... 38 3.3.3 The action of Goss’ group ...... 39 3.4 Measure coefficients for Pellarin’s series ...... 41 3.4.1 The Wagner Series for χt on Av ...... 41 3.4.2 Calculation of the Measure coefficients for z(χt, x, −k) . . . 45 4 Explicit Formulae for Pellarin’s special L-values at the positive integers 53

4.1 Explicit formulae for power sums when s < q, k = 1 ...... 54 4.1.1 The general principle of interpolation and twisted power sums 54 4.1.2 Calculation of Sd(s, −1) ...... 55 4.1.3 Example: Calculation of Sd(q, −1) ...... 57 4.2 Explicit formulae when s ≤ q and k ≡ s mod (q − 1) ...... 60 4.2.1 A first proof of Theorem 4.2.2 by “unfolding” ...... 61 4.2.2 A second proof of Theorem 4.2.2 when s < q ...... 66 4.3 Applications ...... 68 4.3.1 Numerators of the Bernoulli-Carlitz numbers ...... 68 4.3.2 Recursive Relations for Pellarin’s L-Series ...... 72

Bibliography ...... 75

ix CHAPTER 1

INTRODUCTION

Definitions and Notation

We set the following notation for the remainder of this dissertation.

• Let Fq be the finite field with q elements of characteristic p.

• Let θ be an indeterminate and A := Fq[θ],K := Fq(θ) be the ring of polynomials

and field of rational functions over Fq, respectively.

−1 • Let K∞ := Fq((θ )) be the ring of formal Laurent series over Fq.

• Let C∞ be the completion of an algebraic closure of K∞ equipped with the absolute value | · |, normalized so that |θ| = q.

q−1 • Let ι be a fixed root in C∞ of the polynomial x + θ ∈ A[x].

• Let A+,A+(d) denote the monic elements of A and the monic elements in A of degree d, respectively.

• For j ≥ 1, let [j] = θqj − θ.

qj qj−1 qj q qj • Let D0 := 1 and for all j > 0, let Dj := (θ − θ ) ··· (θ − θ )(θ − θ) be the product of all monic polynomials of degree j.

1 q q2 qj • Let `0 := 1 and for j > 0, let `j := (θ − θ )(θ − θ ) ··· (θ − θ ). Observe that

j Lj := (−1) `j is the least common multiple of the set of polynomials of degree j.

X i • We shall say a non-negative integer k written as k = kib is written in base

b if 0 ≤ ki < b for all i ≥ 0. We call the number ki appearing in the base b expansion of k its i-th digit. We define the b-length of a non-negative integer to

be the sum of its digits when written in base b:

X lb(k) := ki. i≥0

We will work mostly in base q, and if l(·) is encountered, the reader may assume

it is the q-length function lq(·).

• For x a rational number, let bxc denote the greatest integer less than or equal

to x.

• Let logb(·) denote the classical logarithm in base b.

1.1 Carlitz’ First Results

Definition 1.1.1. Let d, k be non-negative integers. The power sum of degree d and exponent k is defined as an element of A by the formula

X k Sd(k) := a . (1.1.1)

a∈A+(d)

Through an analysis of the generating functions for the power sums Sd(k), Carlitz

[7] was lead to the discovery of an explicit transcendental constant πe ∈ K∞(ι) and a function eC (z) ∈ K{{z}} such that

Y 0  z  π−1e (πz) = z 1 − , e C e a a∈A 2 where the prime indicates that the product is taken over the non-zero elements of A.

This product is easily seen to define an entire function of z on C∞. Later we briefly −1 sketch how the function πe eC (πze ) gives rise to generating series for the Carlitz zeta values X ζ(l(q − 1)) := a−l(q−1)

a∈A+ which represent well defined elements of K∞ for all positive integers l. These sums give rise to elements in K that contain interesting arithmetic information about K, and we still have a great deal to learn about them. On the one hand, while much was known about the numbers ζ(l(q − 1)), Pellarin’s series - the main topic of this dissertation - has been instrumental to the further understanding of these numbers through new approaches, and on the other, in the final chapter of this dissertation we shall see that both the power sums Sd(k) as well as the zeta values ζ(l(q − 1)) play a crucial role in obtaining our explicit formulas for Pellarin’s series.

We now guide the reader through the theory relevant to the special values of the

Goss zeta function for the ring A in hopes that the relevance of our results in later chapters will be more easily seen.

The Carlitz Module

Definition 1.1.2. Let L be an Fq-algebra. Define the ring of twisted power series X qi over L, written L{{z}}, to be the set of power series of the form liz with i≥0 coefficients li in L, with ordinary addition of power series, and with product given by composition.

We will denote by L{z} the subring of L{{z}} whose elements consist of those series such that all but finitely many coefficients equal zero. We call L{z} the ring of twisted polynomials over L.

3 Definition 1.1.3. The Carlitz module C is the Fq-algebra monomorphism from A into the twisted polynomial ring A{z} determined by

q θ 7→ Cθ(z) := z + θz.

Given any A-algebra R, the twisted polynomial ring A{z} acts on the underlying additive group of R via evaluation. The Carlitz module determines a new A-module structure on R via the monomorphism C. In this way we may also think of the Carlitz module as a functor from A-algebras to A-modules, and from this point of view the

Carlitz module plays the role for function field arithmetic that the multiplicative group functor plays classically. See Rosen [25], Chapter 12 or Goss [16], Chapter 3 for how deep this analogy actually goes.

The Fundamental Period of the Carlitz Module

Definition 1.1.4. We refer to Carlitz’ constant

∞ −1 q Y  qj  πe = −ι 1 − θ/θ j=1 as the fundamental period of the Carlitz module.

Note that this constant is unique up to the choice of ι.

Anderson and Thakur discovered the form for πe given above, see [1]. Carlitz’ student Wade was the first to prove the transcendence of this constant [31], [32].

The Carlitz Exponential

Definition 1.1.5. We refer to the formal twisted power series

X −1 qj eC (z) = Dj z ∈ K{{z}}, j≥0 as the Carlitz exponential.

4 Theorem 1.1.6 (Carlitz). For all z ∈ C∞ the following equality holds,

Y 0  z  X j z 1 − = π−1 D−1(πz)q . a e j e a∈A j≥0 The fundamental property of the Carlitz exponential is that it is the unique entire

Fq-linear function with derivative equal to 1 that takes the linear action of multipli- cation by elements of A, for any A-algebra R ⊆ C∞, to the action of the Carlitz module. This is summarized by the equality

eC (θz) = Cθ(eC (z)). (1.1.2)

The analogy with Lie groups is the reason we refer to this function as an exponential.

The Carlitz Logarithm

Definition 1.1.7. The Carlitz logarithm logC is defined as the formal twisted power series inverse of the Carlitz exponential function.

Using the functional equation (1.1.2) one may find the coefficients of the Carlitz logarithm explicitly. We have,

X −1 qj logC (z) = `j z . j≥0

One may easily check that the radius of convergence of this power series is |πe|.

1.1.1 The Carlitz-Euler Relations

In analogy with the special values of the Riemann zeta function at the positive inte- gers, Carlitz studied the following elements of K∞.

Definition 1.1.8. For positive integers k, define the k-th Carlitz zeta special value as an element of K∞ by, X ζ(k) := a−k.

a∈A+

5 Carlitz used the function eC to prove the following beautiful theorem, in perfect analogy with Euler’s theorem for the special values of the Riemann zeta function at the positive even integers.

Theorem 1.1.9 (The Carlitz-Euler relations). Suppose k is a positive integer such that k ≡ 0 mod (q − 1). Then

k ζ(k)/πe ∈ K.

−1 Proof. The essential idea is to examine the logarithmic derivative of πe eC (πze ) and to expand about the origin. Since eC has coefficients in K, the result follows. See [7] for the complete proof.

Remark 1.1.10. Chang and Yu have determined the exact transcendence degree over K of the field K(π,e ζ(1), ··· , ζ(n)) for all n ≥ 1, [11]. Their calculation implies that the only K-algebraic relations among π,e ζ(1), . . . , ζ(n) are those coming from the trivial relations ζ(k)pn = ζ(pnk) and the Carlitz-Euler relations.

Remark 1.1.11. L. Taelman has proven an analog of the Herbrand-Ribet theorem

k relating the numerators of the rational numbers ζ(k)/πe occurring in the last theo- rem to the vanishing and non-vanishing of certain χ parts of his finite class module associated to the integral extension A[eC (π/Pe )]. Here P ∈ A is a monic irreducible polynomial, and χ ranges over the characteristic p valued characters of the Galois group of the extension K(eC (π/Pe ))/K. See [28] for the precise statement.

1.1.2 Carlitz’ von-Staudt Theorem

It proved useful to Carlitz to normalize power series in positive characteristic with certain “factorials” now bearing his name.

6 The Carlitz Factorial

For j ≥ 0, recall that we have defined Dj as the product of all monic polynomials of degree j. The Carlitz factorials are now defined for all non-negative integers by a process known as the digit principle, see [12].

d Definition 1.1.12. For non-negative integers k = kdq + ··· + k1q + k0 written in base-q with kd 6= 0. Define the k-th Carlitz factorial by d Y ki Π(k) := Di . (1.1.3) i=0 These are factorials for the ring A in the very general sense discovered by Bhargava

[4]. They may be interpolated to functions from Zp → K∞, and this interpolation is referred to as the arithmetic gamma function. For the definition of this function and some of its connections with πe, see [16] Section 9.4. The transcendence and algebraic independence of the special values of the arithmetic gamma function (even in combination with Carlitz zeta values) is well understood, see [10].

As an example of the factorial like properties of these numbers, we recall one such property that we will use later. For a proof, see [16] Section 9.1, and for many more factorial-like properties of these elements, see [4].

Proposition 1.1.13. Let j, k be non-negative integers. Then Π(j)Π(k) divides Π(j + k).

The Bernoulli-Carlitz numbers

Definition 1.1.14. The Bernoulli Carlitz numbers are defined by the generating series relation, z X zk := BC(k) . (1.1.4) eC (z) Π(k) k≥0

λz z × Remark 1.1.15. Since = for all λ ∈ Fq , we observe that BC(k) = 0 eC (λz) eC (z) if k 6≡ 0 mod (q − 1). 7 As mentioned above, for k divisible by q − 1 we have

πkBC(k) ζ(k) = e . (1.1.5) Π(k)

Theorem 1.1.16 (Carlitz’ von-Staudt. [16] Theorem 9.2.2). Suppose q = pm > 2.

d Let k = k0 + k1p + ··· + kdp be a positive multiple of q − 1 written in base-p (i.e.

0 ≤ ki < p for all i ≥ 0). Suppose that (p − 1)m divides lp(k) := k0 + k1 + ··· + kd,

h and let h := lp(k)/ ((p − 1)m). Suppose q − 1 divides k. Then the denominator of BC(k) (in reduced form) is the product of all monic irreducible polynomials in A of degree h.

Otherwise, BC(k) is an element of A.

Remark 1.1.17. For the case q = 2, see [16] Remark 9.2.3 where references to the correct statement are given.

We shall use this theorem later in Theorem 4.3.5 in combination with our explicit calculation of Pellarin’s L-series in q − 1 indeterminates to give a lower bound on divisibility of the Bernoulli-Carlitz numbers by degree one primes of A.

1.2 Goss’ Extension of Carlitz’ zeta values

In order to discuss Goss’ extension of Carlitz’ zeta values to the negative integers, we must first discuss some basic properties of the power sums Sd(k). We will examine their generating series and some well known results about them due to Carlitz.

1.2.1 Generating series for the Sd(k)

Definition 1.2.1. Let A(0) = {0}, and for d ≥ 1 define A(d) := {a ∈ A : deg(a) < d}. The sets A(d) are all Fq-vector spaces and give a filtration of A. Y Definition 1.2.2. For d ≥ 0 define the polynomial ed(z) := (z − a) ∈ A[z]. a∈A(d) 8 One knows that a polynomial whose roots form an Fq-vector space is Fq-linear, see [16]. Hence, for all d ≥ 0 the polynomials ed are Fq-linear. Carlitz computed their coefficients explicitly.

Lemma 1.2.3 (Carlitz). Let d be a non-negative integer. Then

d X Dd i e (z) = zq . (1.2.1) d qi i=0 Di`d−i Proof. See [16], Chapter 3.

Power sums and Fq-vector spaces We shall be interested in power sums of the form

X k SV (k) := λ , λ∈V where the elements λ come from some finite-dimensional Fq-vector space V (or an d affine set obtained by linearly shifting V , e.g. A+(d) = θ + A(d)). As we have Y remarked above, any polynomial of the form (z−λ) is Fq-linear. A basic technique λ∈V for us is to examine the logarithmic derivative of some Fq-linear polynomial (or more generally, power series) and to expand it about either the origin or the point at infinity on the projective line over C∞. These techniques are well-known, and we do not develop them in full generality. Rather, we apply them in the specific instances related to our interests. The following proposition is an example of this.

Proposition 1.2.4. For all d ≥ 1 and z ∈ C∞ sufficiently large,

Dd X −(k+1) d = z Sd(k). (1.2.2) `ded(z − θ ) k≥0

d Proof. By the previous lemma, the derivative with respect to z of ed(z − θ ) equals

d 0 d Dd/`d. Hence the left side equals the logarithmic derivative ed(z − θ ) /ed(z − θ )

9 d d Y of ed(z − θ ). Further, one sees easily that ed(z − θ ) = (z − a), so that the

a∈A+(d) X left side of the equality to be proved equals (z − a)−1. The expansion of this

a∈A+(d) function at ∞ on the projective line over C∞ equals the right side of the equality to be proved.

d Corollary 1.2.5. Let d ≥ 1. The power sums Sd(k) vanish whenever 0 ≤ k < q − 1,

d and Sd(q − 1) = Dd/`d.

Proof. The order of vanishing at ∞ of the left side of the previous proposition is qd.

Hence the same must be true of the right.

For the second claim, one multiplies both sides of the previous proposition by zqd

d and lets z → ∞. Noting that ed(z − θ ) is monic finishes the proof.

The previous proof is due originally to Carlitz, see [7]. We obtain a strengthening of this classical vanishing result through an analysis of the power sums arising in connection with Pellarin’s L-series, see Proposition 3.3.1.

Special polynomials for Goss’ zeta

Now we know that when k is fixed the power sums Sd(k) vanish for d sufficiently large, we make the following definition.

Definition 1.2.6. For non-negative integers k we define the k-th special polynomial associated to Goss’ zeta function by

X −d −1 z(x, −k) := x Sd(k) ∈ A[x ]. (1.2.3) d≥0 There are many interesting properties of these special polynomials. As Corollary

1.2.5 implies, the degree in x−1 of the polynomial z(x, −k) grows logarithmically in k. This should be compared with the growth of the number of zeros of the Riemann zeta function in the critical strip and up to a fixed height. The zeros of z(x, −k) are 10 also all simple and lie in K∞. This is a weak version of the Riemann hypothesis for the Goss zeta function, a result which is due in full generality to J. Sheats. See [26].

The specializations at x−1 = 1 of z(x, −k) give rise to Goss’ zeta values at the negative integers, also known as the Bernoulli-Goss numbers for A.

Special values of the Goss zeta at the negative integers

Definition 1.2.7. For non-negative integers k we define the −k-th special value of the Goss zeta function by

ζ(−k) := z(1, −k) ∈ A. (1.2.4)

Lemma 1.2.8. Suppose k is a positive integer such that k ≡ 0(q − 1), then ζ(−k) = z(1, −k) = 0.

This result is remarkable in comparison with the vanishing of the Riemann zeta function at the negative even integers. For proofs see [16] or [30]. This result is also implied by our Proposition 3.2.1 below.

These zeta values at the negative integers were also shown by Goss and Sinnott

[19] to be directly related to the vanishing and non-vanishing of the irreducible pieces

(after decomposition as a Galois module) of the classical class group for extensions obtained by adjoining Carlitz torsion.

1.3 Non-archimedean measures

Here we recall the basic notation and theorems regarding non-Archimedean measures in positive characteristic. See [16] Section 8.22 and also Katz’ paper [21] for a more thorough discussion of these things. Our intention is to describe a beautiful result of

Thakur and to motivate a result of our own appearing in a later chapter.

Let v be a finite place of A, and denote the completion of A at v by Av. Denote the completion of K at v by Kv. Write | · |v the absolute value under which both 11 Av and Kv are complete. In order to give a basis for the continuous maps from Av to itself we follow a construction of Carlitz, again appealing to the so called digit principle.

Definition 1.3.1. For i ≥ 0, let Ei(z) := ei(z)/Di ∈ K[z].

2 For non-negative integers k = k0 + k1q + k2q + ··· , written in base q, define

Y ki Gk(z) := Ei(z) ∈ K[z]. i≥0

By construction, Gk is a polynomial of degree k in z, and hence any polynomial f ∈ K[z] of degree d may be expanded as

d X f = fiGk (1.3.1) k=0 for some fi ∈ K. The following theorem, due to Carlitz, already makes a strong case for comparing the polynomials Gk with the classical binomial polynomials

z z(z − 1) ··· (z − (k − 1)) := . k k!

Theorem 1.3.2 (Carlitz [9]). Let f ∈ K[z] be a polynomial of degree d, expanded as in (1.3.1). Then f maps A to A if and only if fi ∈ A for i = 0, . . . , d.

Definition 1.3.3. For all continuous f : Av → Av we define

||f||v = sup |f(x)|v. x∈Av

The space of all continuous functions f : Av → Av is complete with respect to the norm || · ||v. We also have the following classical theorem of Wagner which should be compared with Mahler’s theorem for continuous functions on the p-adic integers.

12 Theorem 1.3.4 (Wagner [34]). A function f : Av → Av is continuous if and only if X there exist coefficients fi ∈ Av such that fi → 0 as i → ∞ and f = fiGi on Av. i≥0 Further, the coefficient sequence (fi) determined by f is unique, and

||f||v = max |fi|v. i≥0

Definition 1.3.5. We call the coefficient sequence (fi)i≥0 determined by a continuous function f : Av → Av, as in the last theorem, the Wagner coefficients of f.

Definition 1.3.6. An Av-valued measure on Av (or just measure) will be any contin- uous Av-linear functional on the space of continuous functions f : Av → Av equipped Z Z with the norm || · ||v. We will write f dµ (or just f dµ) for the image of f Av under µ.

Remark 1.3.7. If a continuous Av-valued function f on Av is given by a formula Z Z z 7→ p(z) we will sometimes write p(z) dµ(z) in place of f dµ. For example, it Z will be convenient to write zn dµ(z) rather than giving the function z 7→ zn an additional name.

Remark 1.3.8. Our definition may be shown to be equivalent to Goss’ definition

[16] 8.22.11 of an Av-valued measure on Av, that is, as a finitely additive Av-valued function on the compact-open subsets of Av.

Definition 1.3.9. For each non-negative integer k and measure µ, we define the k-th measure coefficient associated to µ by the value

Z µ(k) := Gk dµ ∈ Av. Av

Observe that the numbers µ(k) are bounded in absolute value, as they lie in Av. Clearly then, by continuity any measure µ is uniquely determined by its associated sequence of measure coefficients. 13 Remark 1.3.10. The collection of Av-valued measures on Av forms an algebra under natural definitions of addition and convolution. Given a measure µ one can associate a unique divided power series to µ via

X zk p (z) := µ(k) . µ k! k≥0 zk Here the basis elements are to be understood as formal symbols (i.e. not taken k! mod p) and are to be multiplied according to the rule zk zj j + k zj+k = . k! j! j (j + k)! Goss computed the algebra of such measures, and it was Anderson who observed that the algebra of measures is isomorphic via the map just defined to the algebra of divided power series over Av, see [13].

Now on the power series ring Av[[x]], we may consider the family of hyper- derivatives {∂j}j≥0 which are defined by their action on the powers of x (and then extended Av-linearly to all of Av[[x]]) by the rule n ∂ xn = xn−j j j for all non-negative integers j, n. The ∂j are a good replacement in positive charac- teristic for the powers of the usual derivative with respect to x. One may show that for all f ∈ Av[[x]] and for all non-negative integers j, k one has j + k ∂ ∂ f = ∂ ∂ f = ∂ f. (1.3.2) j k k j j j+k

Thus one may consider the Av-algebra generated by the hyper-derivatives with mul- tiplication defined as in (1.3.2) and addition defined in the natural way. Goss was the

first to observe in [14] that the algebra of Av-valued measures on Av is canonically isomorphic to the algebra of hyper-derivatives as just defined via the map

X µ 7→ µ(j)δj. j≥0 14 This remains an area in need of much development.

Definition 1.3.11. For each non-negative integer k and measure µ, we define the Z k k-th moment of µ to be the value of the integral z dµ(z) ∈ Av. Av

Remark 1.3.12. An Av-valued measure on Av is uniquely determined by its mo- ments, since one can reconstruct the measure coefficients from its sequence of mo- ments.

Conversely, not every sequence in Av gives rise to an Av-valued measure whose moments arise from this sequence. Given a sequence from Av, one can formally compute the measure coefficients by expanding out Gk(z) in powers of z and using one’s desired moment sequence. If all of these measure coefficients lie in Av, then one obtains a measure from the given sequence.

For any finite extension L of K, Goss has defined a Dedekind zeta function for the ring of integers of L, and he has shown in Section 3.3.1 of [14] that there exists a unique measure whose moments are the special polynomials associated to this zeta function. The next definition is an instance of his result.

Definition 1.3.13. For each x ∈ Kv such that |x|v ≥ 1, let µx be the Av-valued mea- Z k sure whose k-th moment z dµx(z) is the value at x of the k-th special polynomial Av z(x, −k) associated to Goss’ zeta function.

Theorem 1.3.14 (Thakur, [16] Theorem 8.22.12 ). For x ∈ Kv such that |x|v ≥ 1, let µx be as in the previous definition. Then for all j ≥ 0, the j-th measure coefficient

15 µx(j) of µx is given by µx(0) = 1, and   if j = cql + ql − 1  −1 l  (−x )  for some l ≥ 0 and 0 < c < q − 1,   l l µx(j) = if j = cq + q − 1 −1 l −1 l+1  (−x ) + (−x )   for some l ≥ 0 and c = q − 1,    0 otherwise .

Remark 1.3.15. Notice that the measure coefficients above are independent of the place v used to compute them!

We will use Thakur’s calculation in Section 3.4 to outline a method for determining the measure coefficients arising from the special polynomials associated to Pellarin’s

L-series. We shall actually compute these coefficients in the special case of Pellarin’s

L-series in one indeterminate.

1.4 The Tate Algebra, the Frobenius Endomorphism, and

Twisting

The ideas of this section are basic in all applications of Anderson’s theory of t- motives. Pellarin has discovered how these ideas fit into the realm of L-series in positive characteristic, and in particular, in [22] he uses these ideas to prove an explicit formula for a value of his L-series in one indeterminate introduced in that paper. We introduce these ideas here since it is becoming apparent that they hold a basic places in the future of function field arithmetic. Anderson and Thakur’s function ω fits perfectly within the ideas developed in this section and is fundamentally related to

Pellarin’s L-series as we shall see in later chapters. The ideas of this section also allow us to quickly and easily compute the Wagner series for the Fq-algebra evaluation map

16 χt : A → Fq[t] determined by the replacement θ 7→ t which we will study thoroughly in later chapters.

1.4.1 The multivariate Tate Algebra

Let I = (i1, . . . , is) be a multi-index. As usual, we will write |I| := i1 + ··· + is,

I i1 is aI := ai1,...,is , and t := t1 ··· ts .

Definition 1.4.1. For positive integers s we define the Tate algebra in s-variables over C∞ to be    X I  Ts := aI t ∈ C∞[[t1, . . . , ts]] : aI → 0 as |I| → ∞ . s I∈(N∪{0}) 

The subalgebra Ts of C∞[[t1, . . . , ts]] is complete with respect to the norm given for all φ ∈ Ts by ||φ|| = max |a |. (1.4.1) s I I∈(N∪{0})

Remark 1.4.2. The space Ts may be equivalently described as those power series in

C∞[[t1, . . . , ts]] that converge on the closed unit polydisc

s s O := {(l1, . . . , ls) ∈ C∞ : |li| ≤ 1 for i = 1, . . . , s} .

s One shows that this algebra of functions on the closed unit polydisc in C∞ is complete with respect to the sup-norm

0 ||φ|| := sup |φ(t1, . . . , ts)| s (t1,...,ts)∈O and that the numbers assigned by the a priori different norms || · || and || · ||0 are actually equal.

We also mention that ||φ1φ2|| = ||φ1|| · ||φ2|| for all φ1, φ2 ∈ Ts.

Convention 1.4.3. As often as possible, we shall denote the elements of the Tate algebra by lowercase Greek letters.

17 1.4.2 The Frobenius endomorphism and Twisting

In an effort to assist the reader, we describe a ring isomorphic to the twisted power series ring whose elements we will let act as linear transformations on subsets of the

Tate algebra.

Twisted power series in τ

The ring of twisted power series C∞{{z}} is naturally isomorphic to the non - com- mutative ring of twisted power series in τ, written C∞{{τ}}. The isomorphism is given by “evaluation”,

τ i(z) := zqi .

X i X j The elements a = aiτ and b := bjτ in C∞{{τ}} are power series in the i≥0 j≥0 variable τ with coefficients in the ring R subject to usual addition of series and the non-commutative product

X X qi k ab := aibj τ . (1.4.2) k≥0 i+j=k

Convention 1.4.4. Elements of C∞{{τ}} shall be written in Gothic script.

Example 1.4.5. Important elements in C∞{{τ}} (whose coefficients actually lie in K) include cθ := θ + τ (and the images of all elements of A under the Fq-algebra map X −1 i determined by cθ), the Carlitz exponential eC := Di τ , and the Carlitz logarithm i≥0 X −1 i log := `i τ which is the formal twisted power series inverse of eC . i≥0 We obtain a notion of evaluation by twisted polynomials in τ on elements of the

Tate algebra from the following definition.

i i Definition 1.4.6 (Twisting). The Fq-linear maps τ : C∞ → C∞ defined by τ (z) := qi X I z may be extended to Fq[t1, . . . , ts]-linear maps for all aI t ∈ Ts and all i ≥ 0 via the twisting operation

i X I  X i I τ aI t := τ (aI )t . 18 Thus τ is a map of Fq[t1, . . . , ts]- algebras. This action extends linearly to all poly- nomials in τ with coefficients in C∞.

Twisting by C∞{{τ}}

We wish to allow series such as eC and log to act on elements of the Tate algebra as well. We need two preliminary definitions.

X i Definition 1.4.7. The radius of convergence of an element aiτ ∈ C∞{{τ}} is i≥0 X qi the usual radius of convergence of the image aiz ∈ C∞{{z}} as a power series. i≥0

Definition 1.4.8. For r > 0, we define Bs(r) as the Fq[t1, . . . , ts]-submodule of Ts whose elements consist of those φ ∈ Ts such that ||φ|| < r.

Definition 1.4.9. A twisted power series a over C∞ with radius of convergence r > 0 X I gives a well-defined Fq[t1, . . . , ts]-linear map from Bs(r) into Ts for all φ = φI t ∈

Bs(r) via X I a(φ) := a(φI )t .

Remark 1.4.10. Pellarin writes Ea(φ) for what we write simply as a(φ). We hope that the change of notation, from Roman to Gothic will be enough to clue the reader

s into the fact that we are using these series in τ as operators on T . Also, note that when C∞ is viewed as the subspace of constants in Ts, the operator action of X i X qi f = fiτ on z ∈ C∞ is simply evaluation f(z) = fiz .

Example 1.4.11. The operator eC associated to the Carlitz exponential has infinite radius of convergence, and hence gives a well-defined action on all of Ts(C∞). Further, eC is an isometry on Bs(|π|) with inverse log.

19 1.5 Anderson and Thakur’s function ω

The material from this section is a summary of the author’s exposition in [24]. The reader may look there for more explanation and proofs.

As a first, and perhaps most important, example of an element of the Tate algebra

T1, we have a function discovered by Anderson and Thakur in connection with the Carlitz module and its tensor powers, see [1]. This function will be of fundamental importance in all that follows.

Definition 1.5.1. For t ∈ C∞ such that |t| < |θ|, we define the Anderson-Thakur function ω(t) by the convergent product

−1 Y  t  ω(t) := ι 1 − . θqj j≥0 Remark 1.5.2. This function should be thought of as a deformation of the funda- mental period of the Carlitz module πe. Indeed, we have

lim(t − θ)ω(t) = −π. t→θ e

In Anderson’s theory of t-motives (again see [1]), the function ω(t) plays the role of the rigid analytic trivialization of the Carlitz motive where its importance arises from the following proposition.

Theorem 1.5.3 (Anderson-Thakur, [1]). The function ω(t) generates the rank one

Fq[t]-submodule of T of solutions to the τ-difference equation

τ(X) = (t − θ)X.

Definition 1.5.4. For a ∈ A, we denote by χt(a) the image of a in the Tate algebra obtained from the change of variables θ 7→ t.

Corollary 1.5.5 (Pellarin, [22] Lemma 29). For all a ∈ A,

ca(ω) = χt(a)ω ∈ T. 20 Pellarin used the twisting operation to give another characterization of the func-

−1 X −(i+1) i tion ω(t). Write πe(θ−t) for the element of T given by the power series πe θ t . i≥0 Theorem 1.5.6 (Pellarin, [22] (4) pg. 2058). As elements of the Tate algebra, we have i X X πq ω(t) = e (π(θ − t)−1) = e (πθ−(i+1))ti = e . C e C e D · (θqi − t) i≥0 i≥0 i While enough was known to Carlitz to establish the next result, it first appears implicitly in Anderson’s fundamental paper on log-algebraicity, and it was Pellarin who extended these ideas to the realm of operators on the Tate algebra and wrote everything as clearly and plainly as we do now.

Theorem 1.5.7 (Pellarin, [22] Lemma 31). We have the following equality of oper- ators on B1(|πe|), X X −1 log = a ca.

d≥0 a∈A+(d) Remark 1.5.8. We draw the reader’s attention to the analogy between the basic roles

n played by the polynomials z for the action of the integers Z on the multiplicative group of C and the polynomials Ca(z) for the action of A given by the Carlitz module on the additive group of C∞. Both of these actions are uniformized by analytic exponential functions. In the first case, the classical exponential ez does this, and in the second, the Carlitz exponential eC (z) plays the role. With this in mind, take note of the remarkable similarity between the expression for the operator log on the right of the equality above, and the power series expansion about the origin for the classical logarithm, that is

X xn log(1 − z) = − . n n≥0

21 The Theorems of F. Pellarin

We have now stated enough to formally see (and in fact, everything is rigorously justi-

fiable) the next beautiful theorem of Pellarin. We will refer to this result throughout this dissertation as Pellarin’s formula. Just as the classical logarithm is closely related to the special values of Dirichlet L-series at 1, so too is the Carlitz logarithm.

Theorem 1.5.9 (Pellarin, [22] Theorem 1). We have the following identity of ele- ments of the Tate algebra,

X χt(a) ω(t) = log(ω(t)) = π(θ − t)−1. a e a∈A+

We state one more result discovered by Pellarin through the use of his theory of deformations of vectorial modular forms.

ω(t) X χt(a) Theorem 1.5.10. For k ≡ 1 mod (q − 1), the series ∈ K [[t]] is πk ak ∞ e a∈A+ equal in T to a rational function in K(t).

X χt(a) The series are our first examples of special values of Pellarin’s L-series ak a∈A+ in one indeterminate. In the next chapter, we shall introduce similar series in an arbitrary, finite number of indeterminates and state a theorem of Angl`esand Pellarin demonstrating just how far rationality results like the one above go. A major result in this dissertation is the solution of Pellarin’s question to determine the explicit rational functions in K(t) that appear in the last theorem stated above. We further generalize this computation to Pellarin’s L-series in s ≤ q indeterminates which will be introduced in the next chapter.

22 1.6 The Anderson Generating Function

In order to catch a glimpse of the underlying formalism to which this dissertation gives evidence, we introduce a slight generalization of the function ω(t) of the last section. First, we make some preliminary observations.

In the last section we mentioned that ω is a generator of the rank one Fq[t]- submodule of T satisfying τ(ω) = (t − θ)ω. Also, recall that τ is multiplicative on

T.

Definition 1.6.1. For i ≥ 0, define bi(t) ∈ A[t] by the equality

i τ (ω) = bi(t)ω.

Explicitly b0(t) = 1, and for i ≥ 1 i−1 Y qj bi(t) = (t − θ ). j=0

Remark 1.6.2. Observe that both the families of polynomials Di and `i are special-

qi izations of the bi(t). Indeed, for all i ≥ 0, we have bi(θ ) = Di and `i = τ(bi(t))|t=θ.

Recall from 1.3.1 the definition of the polynomials Ei(z) ∈ K[z].

Theorem 1.6.3. Let a ∈ A, and let deg(a) denote its degree in θ. The following equality holds in A[t], deg(a) X χt(a) = bi(t)Ei(a). i=0

Proof. In Corollary 1.5.5 we observed that ca(ω) = χt(a)ω in T. Expanding out the left side we have,

deg(a) X i X χt(a)ω = ca(ω) = Ei(a)τ ω = bi(t)Ei(a)ω. i=0 i≥0 × Using the fact that ω ∈ T , we obtain the desired equality in T. Hence, both members in the equality are polynomials in A[t] which agree for t in the closed unit disc in C∞. We conclude that they are identically equal in A[t].

23 1.6.1 The Wagner Series for χt and the Anderson Generating Function

Theorem 1.6.3 shows that the formal series

X Ξt(z) := bi(t)Ei(z) (1.6.1) i≥0 stores the information contained in the function χt on A. Indeed, upon making the formal replacement z 7→ a for some a ∈ A the series on the right becomes χt(a). This formal series may be considered in two different directions. In Section 3.4.1 we consider it as a function of z on the completion Av of A at a finite place v for those t such that χt extends to a continuous function from Av to itself. Now we proceed to analyze its ∞-adic properties.

Convention 1.6.4. In working with elements of the Tate algebra we shall always reserve the variable z for the action of the operators τ i given by τ i(z) = zqi , and we shall continue with the Fq[t]-linear action of τ. As an example, see the next definition.

Definition 1.6.5. The Anderson’s Generating Function for the Carlitz module (AGF) is the function of two variables t, z ∈ C∞ such that |t| < |θ| and z is arbitrary given by the formula i X zq ω(t, z) := e (z(θ − t)−1) = . C D · (θqi − t) i≥0 i

One sees easily that for each fixed z ∈ C∞ the AGF is an element of T. This function has simple poles at t = θ, θq, θq2 ,... . Observe that the Anderson-Thakur function ω(t) is given under this formalism by ω(t, πe). For a more leisurely account of Anderson generating functions for general Drinfeld modules over A and also inter- esting explicit formulas for them, see [2].

The major theme

The author’s first discoveries were all based upon the use of the series Ξt(z), and as the reader will notice, the coefficients bi(t) appear throughout this work. There is a 24 deep connection between Pellarin’s series and the function ω(t) and more generally with the AGF ω(t, z), and it is this connection that explains why the author has obtained the explicit formulas in this dissertation. We record this connection as a moral, first in words, and then in a precise mathematical statement in the result that follows. We shall direct the reader’s attention to this underlying principle at relevant points throughout this dissertation.

The Wagner Series for the character χt is the Anderson Generating function for the Carlitz module.

Theorem 1.6.6 (Pellarin, Perkins). Let z ∈ C∞. The following equality holds in T:

X Ei(z)bi(t) = ω(t, πze )/ω(t). i≥0

Proof. By definition

−1 ω(t, πze ) = eC (πze (θ − t) ).

−1 Using Pellarin’s formula log(ω(t)) = πe(θ − t) we see that ω(t, πz) e = ω(t)−1e zlog(ω(t)). ω(t) C

We have an equality of formal twisted power series,

X i eC zlog = Ei(z)τ . i≥0 ! X i X i Thus showing Ei(z)τ (ω(t)) = Ei(z)τ (ω(t)) will finish the proof. This i≥0 i≥0 amounts to an interchange of sums that we justify now.

25 One may easily show, see [13] Lemma 3.1.8, that there exists an explicit constant c(z) depending on z and not on i such that |Ei(z)| ≤ c(z)/|Li|. Using this we have

 i+1  − q −q j(q−1)−1 X X j+1 qi j X X q−1 −qi( ) ||Ei(z)eC (π/θe ) t || ≤ c(z) q q q−1 i≥0 j≥0 i≥0 j≥0 q X −qi X −qi j = c(z)q q−1 q (q ) i≥0 j≥0 −qi q X q = c(z)q q−1 1 − q−qi i≥0 q q X −qi ≤ c(z)q q−1 q < ∞. q − 1 i≥0

Similarly, using Goss’ Lemma [13] Lemma 3.1.8 again, one sees that Ξt(z) is a uniform limit of polynomial functions on the disc O and hence resides in T. The argument above shows that ω(t, πze )/ω(t) and Ξt(z) agree on O and are thus equal in T.

The proof of the previous theorem was inspired by the methods of Pellarin in

[22]. The statement was implicit in the work of the author [23], and we gratefully acknowledge F. Pellarin for bringing the statement to light and allowing us to include it here.

We will see in Section 4.2.2 that Lemma 1.6.6 gives the simplest explanation of the explicit formulas obtained in this dissertation for the rational special values of

Pellarin’s L-series at the positive integers.

26 CHAPTER 2

PELLARIN’S L-SERIES

Throughout this chapter s will denote a non-negative integer and t1, . . . , ts indepen- dent indeterminates over C∞. By convention, any empty product or empty sum shall be taken to be 1 or 0, respectively.

2.1 First Definitions and Properties

Definition 2.1.1. For t an indeterminate (or an element of C∞), and for a ∈ A we shall denote by χt(a) the image of a in Fq[t] (or C∞, respectively) under the Fq-algebra morphism of evaluation at t, i.e. a 7→ a(t).

Special values at the positive integers

Definition 2.1.2. Let k be a positive integer. The k-th special value of Pellarin’s

L-series in s indeterminates is defined by the formal series / Euler product,

s X −k L(Π χti , k) := a χt1 (a) ··· χts (a)

a∈A+ Y −k
−1 = 1 − $ χt1 ($) ··· χts ($) .

$∈A+ irreducible

s Remark 2.1.3. Clearly, L(Π χti , k) converges (in C∞) upon specializing the ti ∈ C∞ such that |ti| ≤ 1 for all i. For any given s and k, the closed unit polydisc |ti| ≤ 1 is

s not the largest domain on which L(Π χti , k) converges. We have merely emphasized

27 s this set as it gives a uniform domain on which L(Π χti , k) converges for all positive integers s and k. We shall discuss the rigid analyticity of these functions in the ti in Theorem 2.1.8.

Special values at the negative integers

To define the special values of Pellarin’s L-series at the negative integers we shall need a lemma on the vanishing of certain power sums, just as we did for Goss’ special zeta values at the negative integers.

Lemma 2.1.4. Let k be a non-negative integer. Then the power sums

s X k Sd(Π χti , k) := a χt1 (a) ··· χts (a)

a∈A+(d) vanish for d >> 0 (depending on s and k).

Proof. See propositions 3.3.1 and 3.3.3.

Definition 2.1.5. Let k be a non-negative integer. We define the k-th special poly- nomial associated to Pellarin’s L-series in s indeterminates via

s X −d s z(Π χti , x, −k) := x Sd(Π χti , k). d≥0 Remark 2.1.6. This definition follows in the footsteps of Goss’ original extension of the Carlitz zeta values to a much larger analytic domain. Lemma 2.1.4 guarantees

−1 that the series given above is in fact a polynomial in A[t1, . . . , ts][x ]. We will show

−1 s later that for fixed s, the degree in x of the special polynomial z(Π χti , x, −k) grows logarithmically in k. This will be useful information for many applications to follow.

Definition 2.1.7. Let k be a non-negative integer, then the special value of Pellarin’s

L-series at −k is defined by

s s L(Π χti , −k) := z(Π χti , 1, −k) ∈ A[t1, . . . , ts]. 28 2.1.1 The Analyticity Theorem of Angl`esand Pellarin

We include here a theorem of Angl`esand Pellarin on the rigid analyticity in the ti

s of the functions L(Π χti , k). Their result generalizes the constructions and results of Goss [18] to the case of arbitrarily many indeterminates. Further, their result allows us to make the precise connection with Goss’ zeta values at the negative integers.

d For a ∈ A+(d), we let hai := a/θ . Then hai is a 1-unit in K∞. Hence, via the binomial series, we may exponentiate hai by elements y in the p-adic integers Zp.

× Theorem 2.1.8 (Angl`esand Pellarin, [3] Prop. 32). For all t1, . . . , ts ∈ C∞, x ∈ C∞ and y ∈ Zp, the series

X −d X y x χt1 (a) ··· χts (a)hai (2.1.1)

d≥0 a∈A+(d) converges to a function that is rigid-analytic in t1, . . . , ts, x and continuous in y.

As a corollary, for each y ∈ Zp the function in (2.1.1) may be represented by an −1 entire power series in x and the ti. Hence the special values of Pellarin’s series at the positive integers are rigid analytic entire functions in the ti. Also, observe that when y is a positive integer, making the change of variables x 7→ xθ−y returns our

s special polynomial z(Π χti , x, −y).

2.1.2 The Rationality Theorem of Angl`esand Pellarin

The following theorem is due to Angl`esand Pellarin [3]. The statement was inspired by the author’s work [24]. It gives the general picture, in arbitrarily many indeter- minates, of when Pellarin’s L-series is a rational function.

Theorem 2.1.9 (Angl`es,Pellarin [3], Theorem 4). Let α, s be positive integers such

29 that α ≡ s (mod q−1). Let δ be the smallest positive integer such that, simultaneously,

δ δ q − α ≥ 0 and s + lq(q − α) ≥ 2. Then the formal series

s δ−1   Y Y ti π−αL(Πsχ , α)ω(t ) ··· ω(t ) 1 − ∈ K [[t , . . . , t ]] e ti 1 s θqj ∞ 1 s i=1 j=0 is in fact a symmetric polynomial in K[t1, . . . , ts] whose total degree φ(a, s) satisfies

φ(a, s) s + l(qδ − α) ≤ − 1. s q − 1

Remarks on the proof

The proof of this theorem follows by first noting that for α and s as in the statement the formal series above is actually a symmetric polynomial in the ti with coefficients in

K∞. Proving that the coefficients are in fact in K is achieved by specializing the ti at roots of unity in the algebraic closure of Fq. When the conditions on α and s appearing in the theorem above are satisfied, these specialized L-series are the Dirichlet L-series defined by Goss times a Gauss-Thakur sum divided by a power of πe. It is shown that this specialization lies in an extension of K obtained by adjoining roots of unity.

This combined with the fact that these series are rigid analytic functions is enough to ensure the rationality in K[t1, . . . , ts] of the function under consideration in the theorem.

It is worth highlighting that Angl`esand Pellarin’s method is completely distinct from the methods used in this dissertation.

2.1.3 Deformation of Carlitz-Goss zeta values and Goss L-values

The purpose of defining these special L-values is to gain a new perspective on Dirichlet type Goss L-values and especially the Carlitz-Goss zeta values at the positive integers.

We shall also see that these series are “inductively related” by the parameter s. Let us give some examples of how this works. 30 Example 2.1.10. Let s = 1, and let t = λ be an element of the algebraic closure in

C∞ of Fq with minimal polynomial $. Then χt becomes a Dirichlet character from

A into A/$A, and L(χt, k) becomes a Dirichlet type Goss L-value. Specializations of the ti at roots of unity is a major tool in [3].

Example 2.1.11. Let 1 ≤ l ≤ s and m1, . . . , ml non-negative integers such that k −

mj X mj q q > 0. Then setting tj = θ for j = s−l+1, . . . , s returns the Pellarin special

s−l X mj L-value in s − l indeterminates L(Π χti , k − q ). Similarly, now that the analytic continuation of Pellarin’s L-series in s - indeter- minates has been shown in the Theorem 2.1.8 of Angl`esand Pellarin, we may allow

X mj the positive integers mj to vary freely, with out the constraint k − q > 0, and one obtains all special values (at both positive and negative integers) of Pellarin’s

L-series in any integer less than s indeterminates. In particular, when l = s, one X obtains the Goss zeta special values ζ(k − qmj ). We will see later in this disser- tation that these L-values of Pellarin really do allow us to gain new insight into the behavior of Carlitz-Goss zeta special values.

31 CHAPTER 3

PELLARIN’S SPECIAL L-VALUES AT THE NEGATIVE

INTEGERS

The results of this chapter are generalizations of the author’s work in [23]. In [3],

Section 3 similar results are considered as a prerequisite for the proof of analyticity of Pellarin’s series in s indeterminates. Outside of the method of specialization to

Goss zeta values, which is a major theme in the life of Pellarin’s series anyway, our methods of proof are different. Further, our results concern the special polynomials associated to Goss’ extension of Pellarin’s series and are slightly finer in general than those of [3].

3.1 Recursive Nature

In the section and the next we focus on the special polynomials

s X −d X z(Π χti , x, 0) = x χt1 (a) ··· χts (a).

d≥0 a∈A+(d)

s We remind the reader that by specializing some of the variables ti of z(Π χti , x, 0) at θqji we may deduce results about the whole class of special values of Pellarin’s series at the negative integers.

32 0 Proposition 3.1.1. Let s be a non-negative integer. Then z(Π χti , x, 0) = 1, and for all s ≥ 1,

s −1 X 1−i1 1−is s 1−ij z(Π χti , x, 0) = 1 − x t1 ··· ts z(Π χtj , x, 0). (3.1.1) s (i1,...,is)∈{0,1} P 06= j ij ≡0 mod (q−1) Proof. The proof follows in the same elementary way as Goss’ original recursive for- mula for the special polynomials associated to his zeta function; that is, for a ∈ A+(d) one writes χti (a) as tiχti (b) + λ for some b ∈ A+(d − 1) and λ ∈ Fq and expands as follows.

s Let s ≥ 1. Then z(Π χti , x, 0) equals

s 1 X −d X X Y X 1−ji ji 1 + x (tiχti (b)) λ = d≥1 b∈A+(d−1) λ∈Fq i=1 ji=0 s X −d X Y = 1 + x tiχti (b) +

d≥1 b∈A+(d−1) i=1 s 1 X −d X X Y X 1−ji ji + x (tiχti (b)) λ d≥1 × i=1 j =0 b∈A+(d−1) λ∈Fq i s X −d X Y (∗) = 1 + x tiχti (b) +

d≥1 b∈A+(d−1) i=1 s ! s ! Ps X Y 1−ji X −d X Y 1−ji X i=1 ji (∗∗) + ti x χti (b) λ . s i=1 d≥1 i=1 × (ji)∈{0,1} b∈A+(d−1) λ∈Fq

P X ji X Using the fact that λ = −1 if ji ≡ 0 mod (q − 1) and equals zero s X −d X Y otherwise, and taking account of the cancellation of x tiχti (b) in

d≥1 b∈A+(d−1) i=1 (∗) and the term arising from the s-tuple of all zeros in (∗∗) we deduce the claim.

−1 s Corollary 3.1.2. For all non-negative integers s, the degree in x of z(Π χti , x, 0) is at most bs/(q − 1)c.

Proof. The proof proceeds by strong induction on s. By the recursive formula we see immediately that the claim is true for 0 ≤ s < q − 1. Now let s ≥ q − 1, and 33 suppose the claim holds for all 0 ≤ j < s. By the recursive formula and induction,

−1 s s 1−il the highest power of x appearing in z(Π χti , x, 0) comes from any z(Π χti , x, 0) such that exactly q − 1 of the il are equal to 1, and by induction the degree of this s − (q − 1) latter special polynomial is then = bs/(q − 1)c − 1. The extra x−1 q − 1 s appearing in the recursive formula shows that the degree of z(Π χti , x, 0) is at most bs/(q − 1)c.

−1 s Remark 3.1.3. We will see in Theorem 3.3.5 that the degree in x of z(Π χti , x, 0) actually equals bs/(q − 1)c.

3.2 Trivial zeros of Pellarin’s series

Proposition 3.2.1. Suppose s is a positive integer divisible by q − 1. Then

s −1 z(Π χti , x, 0) vanishes at x = 1.

Proof. The proof proceeds by induction on s divisible by q − 1 using the recursive formula from 3.1.1.

First suppose s = q − 1. Then as the recursive formula above shows we have

s −1 z(Π χti , x, 0) = 1 − x .

Thus we have a simple zero at x = 1.

Now suppose s > q − 1 is divisible by q − 1. As the recursive formula shows, apart

0 from z(Π χti , x, 0) = 1, only special polynomials occur whose number of indetermi- nates is positive, divisible by q − 1 and less than s. By induction, these all vanish at x−1 = 1. Hence

s −1 0 z(Π χti , x, 0) = 1 − x z(Π χti , x, 0) − g(x),

−1 −1 with g(x) ∈ A[t1, . . . , ts][x ] vanishing at x = 1.

34 s Proposition 3.2.2. Suppose s is divisible by q − 1. Then the zero of z(Π χti , x, 0) at x−1 = 1 is simple.

Proof. We argue by contradiction. Let s be as in the statement, and suppose the claim is false. Since evaluation is a ring homomorphism, our assumptions imply that we have a non-simple zero of

s 0 6= z(x, −s) = z(Π χti , x, 0)|t1=θ,...,ts=θ at x−1 = 1, a contradiction.

3.3 Logarithmic growth

In the classical setting (over the complex numbers), for a primitive character χ of modulus m one sets N(T, χ) to be the number of zeros of L(σ + it, χ) in the critical strip 0 < σ < 1 such that |t| < T . It is well known that the “average number of zeros” N(T, χ)/T grows like a constant multiple of log(T ) + log(m).

One of the main results of this section will be the proof that the degree in x−1 of the special polynomials

X −d X β1 βs z(β1, . . . , βs, x) := x χt1 (a) ··· χts (a)

d≥0 a∈A+(d) equals  i i  lq(p β1) + ··· + lq(p βs) φ(β1, ··· , βs) := min . i≥0 q − 1

Specializing ts at θ and the remaining ti at roots of unity in the algebraic closure of Fq we obtain the special value at −βs for Goss’ Dirichlet L-function. The growth rates for sum of digit functions lq and logq may be easily related, and the formula obtained above for the degree of z(β1, . . . , βs, x) is strikingly similar to the classical formula for N(T, χ)/T . The analogy can be made even tighter by observing that a

35 primitive character χ with square-free modulus a = P1 ··· Ps−1 (Here the Pi are the distinct monic irreducible factors of a.) may be realized as a map on A+ with values

β1 βs−1 in the algebraic closure of Fq via χ : a 7→ a(λ1) ··· a(λs−1) for positive integers

βi and roots λi of Pi. The result stated above gives precise meaning to the statement the degree in

−1 x of the special polynomials z(β1, . . . , βs, x) grows logarithmically in the βi. Such logarithmic growth of the special polynomials associated to L-series of τ-sheaves was proven in great generality by B¨ockle, see [5]. Goss used this growth to give analytic continuation to these series using non-Archimedean measure theory. We do not pursue this direction here, but direct the interested reader to [15].

For the remainder of this section we only use the sum of base q digits function lq and omit the subscript q.

3.3.1 Some preliminary bounds

s In this section we study the power sums Sd(Π χti , k) and related special polynomials introduced and studied in the last section. We give a lower bound on d, depending on s and k, above which these sums vanish. Further, we determine the exact degree in

−1 s x of the special polynomials z(Π χti , x, 0). In Section 3.3.3 we draw corollaries for the action of Goss’ group S(q) on the degrees of certain special polynomials obtained by specialization from those of the last several sections.

In this section alone we will make use of the following notation: For s, k non- negative integers, define l(k) + s φ(s, k) := . q − 1

36 Method One

From Proposition 3.1.1 we acquire a new proof that the special polynomials Sd(k) vanish for d > bl(k)/(q − 1)c. We will deduce this from a slightly more general

s result about the special polynomials z(Π χti , x, −k) associated to Pellarin’s L-series. We will give a second proof of this result in 3.3.3 by different means, assuming the vanishing of Carlitz power sums Sd(k). Here, no assumption on the Sd(k) will be used.

Proposition 3.3.1. For all non-negative integers s and k, the degree in x−1 of the

s special polynomial z(Π χti , x, −k) is at most φ(s, k).

i1 il(k) Proof. Write k = q + ··· + q with 0 ≤ ij ≤ ij+1 and ij < ij+q, so that collecting the powers of q according to their exponents we have the base q expansion of k.

r s Letting r = s + l(k) we have z(Π χ , x, 0)| i = z(Π χ , x, −k), and ti qi1 q l(k) ti ts+1=θ ,...,tr=θ the claim follows from Corollary 3.1.2.

Remark 3.3.2. Taking advantage of the Fq[t1, . . . , ts]-linear extension of the p-th s power map, we can slightly improve the upper bound on the degree of z(Π χti , x, −k) s + l(pik) to min . i≥0 q − 1

Method Two

We thank G. B¨ockle for suggesting this simple proof, a version of which appears in the author’s paper [24].

Proposition 3.3.3. Let s, k be non-negative integers. Then the power sums

s X k Sd(Π χti , k) := a χt1 (a) ··· χts (a)

a∈A+(d) vanish for d > φ(s, k).

37 Proof. This is proved by induction on s. When s = 0, this follows from a result of

Carlitz. See B¨ockle [6].

Now assume the result holds for some s ≥ 0. Let d > φ(s + 1, k). We view

s+1 Sd(Π χti , k) as a polynomial in ts+1 over A[t1, . . . , ts]. We shall show that this

qm polynomial vanishes for ts+1 = θ for all m sufficiently large, as we now describe.

s+1 qm Denote by Sd(Π χt , k) qm the evaluation at ts+1 = θ . Then i ts+1=θ

s+1 s m Sd(Π χt , k) qm = Sd(Π χt , k + q ). i ts+1=θ i

By induction the right side above vanishes for d > φ(s, k + qm). Now, for all

m m m > logq(k) + 1 we have l(k + q ) = l(k) + 1, and hence, φ(s, k + q ) = φ(s + 1, k).

Thus we have shown for d > φ(s + 1, k) and m > logq(k) + 1 the polynomial s+1 Sd(Π χt , k) qm vanishes. Thus it must be identically zero. This finishes the i ts+1=θ proof.

3.3.2 Exact Degree

In this section and the next we shall use the following obvious lemma without com- ment.

Lemma 3.3.4. If j, k are two positive integers such that there is no base q carry-over in the sum j + k, then l(j + k) = l(j) + l(k).

−1 Theorem 3.3.5. For any s positive integers β1, . . . , βs, the exact degree in x of the special polynomial

X −d X β1 βs z(β1, . . . , βs, x) := x χt1 (a) ··· χts (a)

d≥0 a∈A+(d) equals  i i  l(p β1) + ··· + l(p βs) φ(β1, ··· , βs) := min . i≥0 q − 1

38 Proof. The upper bound φ(β1, . . . , βs) already follows from similar specializations to those in the proof of Proposition 3.3.1 and the additivity of the p-th power mapping.

It remains to argue that the degree cannot be lower, and this is done by specializing to the one variable case.

We argue by contradiction. Suppose the degree in x−1 is strictly less than

φ(β1, . . . , βs). Let m0 = 0, and recursively choose a sequence of positive integers

mj mj−1 e−1 m1, . . . , ms−1 such that q > q p βj. Then for i = 0,..., logp(q) − 1 there is no

i i m1 i ms−1 base q carry-over in the sum p β1 + p q β2 + ··· + p q βs. Hence for these i,

i i i i m1 ms−1 l(p β1) + l(p β2) + ··· + l(p βs) = l(p (β1 + q β2 + ··· + q βs)).

Thus  i m1 ms−1  l(p (β1 + q β2 + ··· + q βs)) φ(β1, . . . , βs) = min . i≥0 q − 1 But, by B¨ockle [6, Theorem 1.2 (a)], the exact degree in x−1 of the non-zero polyno- mial

m m 1 s−1 m z(x, −(β + q β + ··· + q β )) = z(β , . . . , β , x)| qm1 q s−1 1 2 s 1 s t1=θ,t2=θ ,...,ts=θ is φ(β1, . . . , βs), a contradiction.

3.3.3 The action of Goss’ group

In his ζ-phenomenology paper [17], Goss has introduced a group of digit permutations that appear to have much to say about the symmetries of special zeta values in positive characteristic.

Definition 3.3.6. Let ρ be a permutation of the non-negative integers, and let k =

X i kiq be a p-adic integer written in base q, i.e. 0 ≤ ki < q for all i ≥ 0. Define

∗ X ρ(i) ρ (k) := kiq .

We let S(q) be the collection of all such digit-permutations. 39 Remark 3.3.7. It is easy to see that S(q) stabilizes the non-negative integers, and

Goss has shown that the group S(q) acts as continuous automorphisms of the p-adic integers that also stabilizes the non-positive integers.

Now we wish to show now that the degrees in x−1 of the special polynomials z(β1, . . . , βs, x) are invariant under a natural action of the group S(q).

∗ Lemma 3.3.8. Let ρ ∈ S(q) and k a non-negative integer. Then for all i ≥ 0,

l(pik) = l(piρ∗(k)).

Proof. Let e = logp(q). It suffices to prove this for i = 0, 1, . . . , e − 1, so let i be one

X l such number in this range. Write k = klq in base q.

i i Clearly for each l ≥ 0 we may write p kl = al + blq with p ≤ al < q or al = 0, and

i 0 ≤ bl < p . Hence for all j, k ≥ 0 there is no base q carry over in the sums aj + bk.

i X l X l+1 Thus p k = alq + blq is the sum of two positive integers written in base q, l≥0 l≥0 i X and there is no carry over in this decomposition. Hence l(p k) = (al+bl). Similarly, l≥0 i ∗ X ρ(l) X ρ(l)+1 p ρ (k) = alq + blq is the sum of two positive integers written in base l≥0 l≥0 i ∗ X q, and there is no carry-over in this decomposition. Hence l(p ρ (k)) = (al + bl) = l≥0 i lq(p k).

−1 Theorem 3.3.9. The degree in x of z(β1, . . . , βs, x) is invariant under the action

∗ s of ρ ∈ (S(q)) defined by

∗ ∗ ∗ ρ (z(β1, . . . , βs, x)) := z(ρ1β1, . . . , ρsβs, x)

∗ ∗ ∗ s for all ρ := (ρ1, . . . , ρs) ∈ (S(q)) .

Proof. This is an immediate consequence of the previous result and Theorem 3.3.5.

40 3.4 Measure coefficients for Pellarin’s series

Here we describe the calculation of the non-Archimedean measure whose moments are the special polynomials z(χt, x, −k). We begin by analyzing the Wagner series for the character χt as a function on the completions of A at finite places.

3.4.1 The Wagner Series for χt on Av

We recall that we have chosen θ as the Fq-algebra generator of A, and χt depends on this choice by our definition χt : θ 7→ t. As we saw in the introduction, the formal series X Ξt(z) := bi(t)Ei(z) i≥0 stores the information contained in the function χt on A in the sense that making the formal replacement z 7→ a ∈ A we have Ξt(a) = χt(a). We have already studied

Ξt(z) as a function of z on C∞. While reading this section keep in mind Theorem 1.6.6, i.e. ∞-adically,

The function Ξt(z) is the Anderson Generating Function for the Carlitz module.

In this section we consider Ξt(z) as a function of z on a v-adic space, for v a finite place of A. We shall abuse notation by also writing v for the monic irreducible polynomial from which the place v arises. We shall write vA for the ideal generated by v, and Av for the completion of A at v. Finally, we write Kv for the completion of K at v and | · |v for the absolute value making Kv complete, normalized so that

|v|v = 1/q. Our main references for this background material are Goss’ book [16] Chapter 8.22 and Thakur’s paper [29].

The following sets will serve as convenient places to choose t from that make χt

41 qi continuous on A with respect to | · |v: let Θ := {θ : i ≥ 0} and Θ + vA := {x + $ : x ∈ Θ and $ ∈ vA}.

Remark 3.4.1. The set Θ + vA is by no means the largest choice that provides t that make χt continuous. For example, call a polynomial affine if it is of the form

X qi γ = γc + γiθ for some γc, γ0, γ1, · · · ∈ Fq. i≥0 If v is an affine irreducible element of A, and α is another affine element of A satisfying (v(1) − v(0) + 1)α(0) + v(0) = α(v(0)), then one easily shows that v|v(α), where here we are treating θ as a variable and using parenthesis to denote composition as usual.

s Lemma 3.4.2 (v-adic continuity of Π χti ). Let ti ∈ Θ + vA for i = 1, . . . , s. Then

s the map Π χti : A → A given by a 7→ χt1 (a) ··· χts (a) is continuous with respect to the absolute value on Av.

qj Proof. Let ti be as above. Then we have χti (v) = v(θ + $) for some j ≥ 0 and

qj $ ∈ vA. Reducing modulo v we see plainly that χti (v) ≡ v(θ) ≡ 0 mod v. Thus

n n since evaluation is an Fq-algebra morphism, if v divides a, then v divides χti (a).

Thus χti is continuous at zero, and continuity in general now follows using the linearity

of χti . Finally, the proposition follows by observing that a product of continuous functions is continuous.

Thus Wagner’s Theorem (Theorem 1.3.4 above) tells us that the continuous ex-

s tension of Π χti (here we have fixed ti ∈ Θ + vA for all i) to Av has a Wagner series representation. We now move toward proving that Ξt(z) as in (1.6.1) is the Wagner series for χt.

Lemma 3.4.3. For t ∈ Θ + vA, the coefficients bi(t) lie in Av and tend to zero as i → ∞.

42 Proof. Let t ∈ Θ + vA. Then t = θqj + $. We will prove the lemma for j = 0, and the modifications necessary for j > 0 will be completely evident. i−1 Y qk qk We have bi(t) = ($ + θ − θ ). Now v divides θ − θ depending on whether k=0 −b(i−1)/dc d = deg v divides k or not. Hence |bi(t)|v ≤ q → 0 as i → ∞.

Theorem 3.4.4. For t ∈ Θ + vA, the series Ξt(z) defined in (1.6.1) gives the con- tinuous extension of χt to Av.

Proof. Let t ∈ Θ + vA. Then the previous lemma shows that the coefficients bi(t) of

Ξt(z) tend to zero as i → ∞. Thus by Wagner’s Theorem, Ξt(z) defines a continuous function on Av. Hence uniformly continuous by the compactness of Av. This function agrees with χt on A which is dense in Av. This finishes the proof.

On the differentiability of χt on Av and connection with v-adic L-values

Throughout this section, we fix t in Θ + vA. Given the Wagner series Ξt for χt on

Av, one may be tempted to think that Ξt(z) is differentiable in z. If Ξt(z) were differentiable in z on Av, with non-zero derivative, then a simple argument using the multiplicativity of Ξt shows it must be the identity function which it is not, see [23] for more details.

Wagner gives the following criterion in terms of a function’s Wagner coefficients for differentiability.

Theorem 3.4.5 (Wagner [33], Theorem 5.1). Suppose f is a continuous Fq-linear X function on Av with Wagner series AiEi(z). If Ai/`i → 0 as i → ∞, then f is X differentiable everywhere on Av with derivative equal to Ai/`i. i≥0 The theorem of Wagner just stated plus a result from the next chapter allow us to obtain as a corollary the vanishing of the Goss (θ − λ)-adic special L-value

X X χλ(a) L (χ , 1) := . θ−λ λ a d≥0 a∈A+(d) 43 × Here λ ∈ Fq .

Remark 3.4.6. For arbitrary irreducibles v, working in the extension of Av contain-

0 ing the root λ of v, it appears that we may obtain the vanishing of Lv(χλ0 , 1) via the same method. We will not use this remark in the sequel and leave the verification to the interested reader.

Corollary 3.4.7 (Vanishing of Goss’ Dirichlet L-values for certain odd characters).

× Let λ ∈ Fq , and let v = θ − λ. Then the Goss special L-value Lv(χλ, 1) vanishes.

Proof. Later in Theorem 4.1.2 we shall show that we have an equality of polynomials

X χt(a) bd(t) = ∈ K[t]. a `d a∈A+(d)

X χλ(a) Goss has shown that for t = λ ∈ × the sums tend to zero v-adically Fq a a∈A+(d) as d → ∞. Thus by Wagner’s theorem above Ξλ is differentiable on Av with derivative

X bd(λ) X X χλ(a) = = Lv(χλ, 1). `d a d≥0 d≥0 a∈A+(d)

Thus since Ξλ is not the identity function, we conclude that Lv(χλ, 1) must vanish. This agrees with the well known fact that the v-adic special Dirichlet L-values at 1 vanish for odd characters χ. Our character χλ is an example of this phenomenon.

Remark 3.4.8. Let λ ∈ Fq and v = θ − λ. As F. Pellarin points out, the sums X χλ(a) = b (λ)`−1 tending to zero v-adically as d → ∞ suggests the v-adic a d d a∈A+(d) analytic continuation in the variable t of the formal sum

X X χt(a) L (χ , 1) := . v t a d≥0 a∈A+(d) (a,θ−λ)=1 In Corollary 4.1.9 we show that

X χt1 (a) ··· χtq (a) = `−1 ([d]b (t ) ··· b (t ) + b (t ) ··· b (t )) a d d−1 1 d−1 q d 1 d q a∈A+(d) 44 for all d ≥ 1, and from this it is easy to show that

d X X χt1 (a) ··· χtq (a) = `−1b (t ) ··· b (t ). a d d 1 d q e=0 a∈A+(e)

Thus if we let ti = λ for i = 1, . . . , q − 1 and t = tq ∈ Θ + vA we learn that

−1 q−1 Lv(χt, 1) = lim `d bd(λ) bd(t) d→∞ exists in Av and is identically equal to zero.

3.4.2 Calculation of the Measure coefficients for z(χt, x, −k)

In this section we show, for select x, t, the existence of the non-Archimedean measure whose k-th moment is the special polynomial z(χt, x, −k). We then go on to calculate the coefficients of the divided power series associated to µx,t in two ways. The first technique, while just a sketch, is intended to generalize to any number of characters.

The second technique will be a calculation straight from the definitions, following

Thakur’s original calculation. We stick to the particular example of the evaluation character χt, but it will be clear to the reader that the techniques below extend inductively (but not easily!) to any product of continuous Fq-linear functions on Av. The next theorem appeared as Theorem 1.3.14. We state it again here for the convenience of the reader.

Theorem 3.4.9. For each x ∈ Kv such that |x| ≥ 1 there is a unique Av-valued Z k measure µx whose k-th moment z dµx(z) is the value at x of the k-th special polynomial associated to the Goss zeta function z(x, −k). For all j ≥ 0 the measure

45 coefficients µx(j) of µx are given by µx(0) = 1, and   if j = cql + ql − 1  −1 l  (−x )  for some l ≥ 0 and 0 < c < q − 1,   l l µx(j) = if j = cq + q − 1 −1 l −1 l+1  (−x ) + (−x )   for some l ≥ 0 and c = q − 1,    0 otherwise .

Definition 3.4.10. Given a continuous function h : Av → Av and measure µ we Z write h dµ for the measure which takes the value gh dµ for all continuous functions g : Av → Av.

Following a tip from Anderson, Goss observed in [13] for continuous additive maps h : Av → Av that what he calls the abstract Fourier transform on measures µ 7→ h dµ is a derivation on the algebra of measures. The next result is an instance of this.

Proposition 3.4.11 (Equality of Measures). Let µx be as defined in the previous theorem. Then for each x ∈ Kv such that |x| ≥ 1 and t ∈ Θ+vA, there exists a unique

Av-valued measure µx,t whose k-th moment is the special polynomial z(χt, x, −k). It is given by

dµx,t = χtdµx. Z k Proof. It suffices to prove that the moments y χt(y) dµx(y) all equal z(χt, x, −k). Av We will unfold the definition of z(χt, x, −k) in terms of µx. By definition, we have,

X −d X k −1 z(χt, x, −k) := x a χt(a) ∈ A[t, x ],

d≥0 a∈A+(d) where the degree in x−1 is at most D(k) := b(l(k) + 1)/(q − 1)c by Prop. 2.1.4.

46 Z k Hence, we rewrite the integral y χt(y) dµx(y) in two parts as Av Z D(k) Z k X k X y bi(t)Ei(y) dµx(y) + y bi(t)Ei(y) dµx(y), (3.4.1) Av i=0 Av i>D(k) Z k and we will show first that y Ei(y)dµx(y) vanishes for i > D(k). Av Let i > D(k), and define the coefficients αi,j ∈ A through the equality,

i X qi DiEi(y) = αi,jy . j=0

Then (we drop the subscript Av) Z i Z k X k+qj y DiEi(y)dµx(y) = αi,j y dµx(y) j=0 i X X −d X k+qj = αi,j x a

j=0 d≥0 a∈A+(d) X −d X k (∗) = x a DiEi(a).

d≥0 a∈A+(d) Again by 2.1.4 the degree in x−1 of the last polynomial (∗) above is at most D(k).

Hence, Ei(a) vanishes for all a ∈ A+(d) for 0 ≤ d ≤ D(k). Thus the polynomial in (∗) vanishes. This finishes the proof of the vanishing of the second integral in (3.4.1).

Now one easily sees by the same method as above, i.e. using the representation for the DiEi(y) in powers of y and the definition of µx, that the first member of (3.4.1) is exactly z(χt, x, −k). We do not repeat the argument.

In order to calculate the measure coefficients for µx,t we shall need to understand how the Carlitz basis elements behave under multiplication. In theory, the following lemma should also be useful for inductively calculating the Wagner series for products of continuous Fq-linear functions.

X i Lemma 3.4.12 (The multiplication lemma). Write l = liq in base q. We have two cases: 47 1. Suppose li < q − 1 for some i ≥ 0. Then GlEi = Gl+qi .

2. Suppose li = q − 1 for some i, and let be k ≥ 0 be such that li+j = q − 1 for

0 ≤ j ≤ k and li+k+1 < q − 1. Then

k j ! X Y GlEi = [i + m] G i+j Pj i+n l+q −(q−1) n=0 q j=0 m=1 k+1 ! Y + [i + m] G i+k+1 Pk i+n . l+q −(q−1) n=0 q m=1

Proof. The first claim is immediate from the definitions.

For the second, we shall use the recursive formula

q Ei(z) = [i + 1]Ei+1(z) + Ei(z), (3.4.2) which holds for all i ≥ 0 and can be proven by observing that both sides are poly- nomials in z of the same degree that have the same set of zeros and the same value at z = θi+1, see e.g. Goss [16] the second proof of Theorem 3.1.5. This also follows from the functional equation for the Carlitz exponential by using that Ei(z) is the

qi coefficient of x in eC (z logC (x)).

Assume li = q − 1 for some i ≥ 0 and let k be as in case two. We shall prove the desired equality by induction on k. The case where k = 0 is immediate from (3.4.2).

Assume now that k ≥ 1, and that the desired equality holds for all 0 ≤ j < k. Then

q i i i i (∗) GjEi = Gj−jiq Ei = [i + 1]Gj−jiq Ei+1 + Gj−jiq +q .

i Now, j − jiq has ji+1 = ji+1 = ··· = ji+1+k−1 = q − 1. Hence, by induction on k we

i may apply the formula to Gj−jiq Ei+1, and (∗) becomes

k−1 j ! X Y Gj−(q−1)qi+qi + [i + 1] [i + 1 + m] G i i+1+j Pj i+1+n j−(q−1)q +q −(q−1) n=0 q j=0 m=1 k ! Y +[i + 1] [i + 1 + m] G i i+1+k Pk−1 i+1+n . j−(q−1)q +q −(q−1) n=0 q m=1 48 Now combining the two terms in the first line and re-indexing the first and second lines finishes the proof.

Theorem 3.4.13 (Measure coefficients of µx,t). We have µx,t(0) = 1, and   (i) if j = cql + ql − 1 − qe     for some l ≥ 0, 1 < c < q − 1,   −1 l  (−x ) be(t) and 0 ≤ e ≤ l;    l l e  or if j = q + q − 1 − q    for some l ≥ 1 and 0 ≤ e < l; µx,t(j) =  (ii) if j = ql+1 − 1 − qe  −1 l −1 l+1
 (−x ) + (−x ) be(t)   for some l ≥ 0 and 0 ≤ e ≤ l;    (iii) if j = ql+1 − 1  −1 l −1 l+1  (−x ) bl(t) + (−x ) bl+1(t)   for some l ≥ 0,    0 (iv) otherwise .

Remark 3.4.14. As Goss points out, this theorem highlights the importance of the coefficients bi(t) of Ξt, and we draw the attention of the reader again to the relationship between Ξt and the Anderson Generating Function.

First method of calculation

Here we indicate how the proof of Theorem 3.4.13 will go if one wishes to use Thakur’s theorem.

First proof (Sketch). We have the following string of equalities:

Z Z X Z µx,t(j) := Gj dµx,t = Gj χt dµζ = bi(t) GjEi dµ. Av Av i≥0 Av

Here the first equality is by definition, the second by Prop. 3.4.11, and the third replaces χt with its Wagner series Ξt and uses the continuity of µ. The remainder

49 of the proof is an elementary calculation of the rightmost member of the string of equalities above following from Thakur’s theorem and Lemma 3.4.12. The full details are left to the interested reader.

Second method of calculation

Now we give a full proof of Theorem 3.4.13 straight from the definitions with no reference to Thakur’s theorem. We shall need two small lemmas. The first highlights the immense cancellation that occurs when calculating the measure coefficients in terms of the moments. We use the logarithmic growth in the degrees of the special polynomials associated to Pellarin’s series in the proof.

Lemma 3.4.15. Let d ≥ 0 be an integer. Then for integers k in Id,1 := I1 := [qd, qd+1 − qd − 1),

−d X µt,x(k) = x Gk(a)χt(a).

a∈A+(d) d+1 d d+1 For integers k in Id,2 := I2 := [q − q − 1, q − 1],

−d X −(d+1) X µt,x(k) = x Gk(a)χt(a) + x Gk(a)χt(a).

a∈A+(d) a∈A+(d+1)

k k−1 Proof. Write Π(k)Gk(z) = z + ak−1z + ··· + a0 ∈ A[z].

For k in I1 Z Π(k)µt,x(k) = Π(k)Gk(z) dµt,x Ap k X = aiz(χt, x, −i) i=0 k d X X −e X i = ai x a χt(a).

i=0 e=0 a∈A+(e)

It is worth making a special note here that the second sum in the last line only goes up to d when k ∈ I1 precisely because of the logarithmic growth of the degrees of the

50 special polynomials obtained in the last section. Interchanging sums and observing that Gk(z) vanishes on polynomials of degree less than d gives

−d X µt,x(k) = x Gk(a)χt(a).

a∈A+(d)

For k in I2, we have the same calculation, except that now we must include the degree d + 1 monic polynomials in the double sum for z(χt, x, −i), and the last line becomes k d+1 X X −e X i ai x a χt(a).

i=1 e=0 a∈A+(e)

Again Gk(z) vanishes on all polynomials of degree less than d, and we obtain

−d X −(d+1) X µt,x(k) = x Gk(a)χt(a) + x Gk(a)χt(a).

a∈A+(d) a∈A+(d+1)

The second result, just below, is a corollary to Jeong’s Theorem 8 in [20].

Lemma 3.4.16 (Orthogonality relations). For j < qm and l ≥ 0,   if j + ql = qm − 1  m  (−1) X m Gj(a)El(a) = or if l = m and j = q − 1,  a∈A+(m)   0 if j + ql 6= qm − 1.

A Second Proof

Second proof of Theorem 3.4.13. Let d ≥ 0, and suppose k ∈ [qd, qd+1 − qd − 1). Let

d kd be the base q digit of q in k. Observe that by our restriction on k, it necessarily follows that 1 ≤ kd ≤ q − 2. By our first lemma and the definition of the polynomials

Gj we have

−d X d µt,x(k) = x Gk−kdq (a)χt(a). a∈A+(d)

51 Now replacing χt by its Wagner series we see from the orthogonality relations that

−1 d d d e d d e µt,x(k) = (−x ) be(t) if k = q +q −1−q for some 0 ≤ e < d or k = kdq +q −1−q for some 0 ≤ e ≤ d and 2 ≤ kd ≤ q − 1, and µx,t(k) = 0 otherwise. This completes the proof of part (i) of the theorem.

Say k = qd+1 − qd − 1. Then, using our lemmas,

−d X −(d+1) X d µx,t(k) = x Gk−kdq (a)χt(a) + x Gk(a)χt(a) a∈A+(d) a∈A+(d+1) −1 d −1 d+1 = ((−x ) + (−x ) )bd(t).

Now suppose k ∈ (qd+1 − qd − 1, qd+1), and observe that the digit of qd in k when written in base q necessarily equals q − 1. By our first lemma

−d X −(d+1) X µt,x(k) = x Gk−(q−1)qd (a)χt(a) + x Gk(a)χt(a).

a∈A+(d) a∈A+(d+1)

By our orthogonality lemma, the first summand is non-zero precisely when k =

(q − 1)qd + qd − 1 − qe for some 0 ≤ e < d or when k = qd+1 − 1. In the first case the

−1 d d d e first summand equals (−x ) be(t). For those k = (q − 1)q + q − 1 − q for some 0 ≤ e < d, the second summand is also non-zero since then k + qe = qd+1 − 1, and

−1 d+1 hence equals (−x ) be(t). Thus for such k,

−1 d −1 d+1 µx,t(k) = ((−x ) + (−x ) )be(t).

This completes the proof of part (ii) of the theorem.

d+1 −1 d In the second case, when k = q −1, the first summand equals (−x ) bd(t) and

−1 d+1 the second summand equals (−x ) bd+1(t). This completes the proof of part (iii) of the theorem.

It remains to remark that for k ∈ (qd+1 − qd − 1, qd+1) the second summand is non-zero precisely when the first summand is non-zero. This completes the proof.

52 CHAPTER 4

EXPLICIT FORMULAE FOR PELLARIN’S SPECIAL

L-VALUES AT THE POSITIVE INTEGERS

In this chapter we calculate various explicit formulae for certain power sums arising in connection with the special values of Pellarin’s L-series at 1 as well as the explicit rational special values of Pellarin’s L-series when the number of indeterminates is at most q.

Our premier result, Theorem 4.2.1, will follow most easily from the observation that the rational special values of Pellarin’s L-series in less than q indeterminates arise from generating series in a way that is totally analogous to Bernoulli polynomials classically. The following equality, which is due to the author and which holds for s < q, will be made precise in Section 4.2.2,

Qs −1 i=1 ω(ti) eC (zL(χti , 1)ω(ti)) X χt1 (a) ··· χts (a) −1 = . π eC (πz) z − a e e a∈A

Recall that eC is the Fq[t1, . . . , ts]-linear operator arising from the Carlitz exponential, and that by Pellarin’s formula (Theorem 1.5.9) eC (zL(χti , 1)ω(ti)) is the AGF for the

Carlitz module. Also observe that because χti (0) = 0, both sides of the equality above have power series expansions in z in a non-empty open disc about the origin. It is exactly by comparing coefficients of these power series expansions that one obtains the explicit formulae for the rational values of Pellarin’s series. We expect that there are deep connections with Taelman’s unit module [27] present in the above equality 53 some of which are already under investigation by various authors. However it is beyond the scope of this dissertation to introduce them here.

4.1 Explicit formulae for power sums when s < q, k = 1

In this section, we shall employ the method of polynomial interpolation, as first exploited by Carlitz over A, to calculate the value at k = 1 of Pellarin’s L-series when the number of indeterminates s is strictly less than q. More precisely, we shall calculate the sums

X χt (a) ··· χt (a) S (s, −1) := 1 s ∈ K[t , . . . , t ]. d a 1 s a∈A+(d) Extending our method of proof to s ≥ q appears possible using the Multiplication

Lemma (i.e. Lemma 3.4.12), but things become complicated quite fast. We do not pursue this here, and we leave the search for a better method to future works.

4.1.1 The general principle of interpolation and twisted power sums

Recall that the normalized linear Carlitz functions are defined for d ≥ 0 by the equality Ed(z) := ed(z)/Dd ∈ K[z].

Lagrange interpolation shows that given any function function f from A+(d) to an integral domain B that is a K-algebra we may interpolate f with the polynomial

X Ed(z − a) M f(z) := ` f(a) ∈ B[z]. d d z − a a∈A+(d)

Hence, if we know the coefficients of Mdf then, following an idea of Carlitz, we may X re-express the twisted power sums a−1f(a) in terms of these coefficients by

a∈A+(d) −1 merely evaluating `d Mdf(z) at z = 0.

Similarly, since we know the coefficients of Ed explicitly for all d ≥ 0 and since

d Ed(z − a) = Ed(z − θ ) for all a ∈ A+(d), we may explicitly calculate the sums 54 X −k −1 d a f(a) by determining the expansion of `d Mdf(z)/Ed(z − θ ) about z = a∈A+(d) 0. While this gives explicit formulas for these power sums they are usually too complicated to use outside of special cases.

4.1.2 Calculation of Sd(s, −1)

Recall the formal series X Ξt(z) := bi(t)Ei(z) i≥0 defined in (1.6.1) and that by Theorem 1.6.3 this series agrees with χt(a) for all a ∈ A upon the formal replacement z 7→ a. It follows that

d−1 X (Mdχt)(z) = bd(t) + bi(t)Ei(z). i=0

We now use the general principle of interpolation to calculate the sums Sd(s, −1). We will be dealing with products of polynomials expressed in the linear Carlitz basis Ei with constant terms. Therefore, it is convenient to introduce some notation to ease writing. For d ≥ 1 define   Ej(z) (0 ≤ j ≤ d − 1)  d  Ej (z) = 1 (j = d)    0 otherwise.

Example 4.1.1. For d ≥ 1 we have the following equality of functions on A+(d),

d X d (Mdχt)(z) = bi(t)Ei (z). i=0 By abuse of notation, we shall often refer to the right side above as a polynomial.

Recall the Fq[t1, . . . , ts]-linear extension of the Carlitz logarithm

X −1 j log = `j τ , j≥0 55 j and that τ ω(t) = bj(t)ω(t) for all j ≥ 0. One of the theoretical benefits of the following result is the connection it suggests between the special values of Pellarin’s series at k = 1 and the action of log on ω(t1) ··· ω(ts). Such a connection is currently being made precise by various authors. We shall discuss some immediate benefits after the statement and proof.

Theorem 4.1.2. Let d ≥ 1 be an integer, and let 0 ≤ s < q. Then

bd(t1) ··· bd(ts) Sd(s, −1) = . `d

Proof. Theorem 1.6.3 shows that χt1 (z) ··· χts (z) agrees with the polynomial s d ! Y X b (t )E d (z) for z ∈ A (d). The degree of this polynomial in z is sqd−1, ji i ji + i=1 ji=0 which by our assumption on s is strictly less than |A+(d)|. Hence,   Qs Pd b (t )E d (z) i=1 j =0 ji i ji i X χt1 (a) ··· χts (a) d = . (4.1.1) `dEd(z − θ ) z − a a∈A+(d)

Evaluating at z = 0 proves the claim.

Remark 4.1.3. Again, as we described earlier, the equality (4.1.1) contains enough

X χt (a) ··· χt (a) information to calculate S (s, −k) := 1 s for all k ≥ 1. One d ak a∈A+(d) expands both sides of the equality about the origin and compares coefficients. We do not pursue this here because the formulas we obtain in this way, while pretty, are difficult to use. However, see the second proof of Theorem 4.2.2 where we bypass these complications with a closely related idea.

Application: An elementary proof of Pellarin’s formula for L(χt, 1)

We thank F. Pellarin for allowing us to include here his elementary proof for the π formula L(χ , 1) = e that follows from the explicit formula for S (1, −1). t (θ − t)ω(t) d q In the next section, we give an explicit formula for L(Π χti , 1) via the same route.

56 Theorem 4.1.4 (Pellarin’s Formula). The following equality holds in T:

π L(χ , 1) = e . t (θ − t)ω(t)

Proof. It suffices to prove this for t ∈ C∞ such that |t| ≤ 1. Merely observe that d X bd+1(t) by Theorem 4.1.2 the partial sum S (1, −1) telescopes to . Now an e (t − θ)` e=0 d easy algebraic manipulation makes plain that these partial sums do indeed limit to π e as d → ∞. (θ − t)ω(t)

4.1.3 Example: Calculation of Sd(q, −1)

As an example of extending the results of the last subsection, we indicate how to

calculate (Mdχt1 ··· χtq )(z) for all d. A clean expression for Sd(q, −1) will follow from

this line of thought as will the rational function to which L(χt1 ··· , χtq , 1) gives rise.

Lemma 4.1.5. We have the following identity of functions on A+(d) for all j ≥ 0:

d q d d (Ej ) = [d + 1]Ej+1 + Ej . (4.1.2)

Proof. We have seen that the equality claimed above holds for the functions Ej, i.e.

q the polynomial identity Ej = [j + 1]Ej+1 + Ej holds for all j. Thus what remains to be checked is only the equality when j = d − 1. In this case we have

d q q d d (Ed−1) = Ed−1 = [d]Ed + Ed−1 = [d] + Ed−1 = [d + 1]Ed + Ed−1

as functions on A+(d). This proves the claim.

As a polynomial in z over K[t1, . . . , ts], the polynomial (Mdχt1 ··· χtq )(z) is de- termined by its values on A+(d). Hence we could use the previous lemma and the interpolation polynomial for the product of characters in one less indeterminate to compute its Wagner expansion. We will not pursue this, but we will use this idea to

compute the constant coefficient of (Mdχt1 ··· χtq )(z). 57 s s S Let B = {(j1, . . . , js) : 0 ≤ ji ≤ d for i = 1, . . . , s}. Let D = {(j1, . . . , js) ∈ B :

s s s j1 = j2 = ··· = js} and C = B \D .

Proposition 4.1.6. For all d ≥ 1 we have the following equality of functions on

A+(d),

d q q X Y X Y M χ ··· χ = [j + 1]E d + E d
b (t ) + b (t )E d . d t1 tq j+1 j j i ji i ji q j=0 i=1 (ji)∈C i=1

d d Remark 4.1.7. Observe that Ed+1 = 0, and Ed = 1 so that the expression on the

d right side is equal as a function on A+(d) to a polynomial of degree at most q − 1

and thus is the unique polynomial interpolating the product χt1 ··· χtq on A+(d).

q d Y X Proof. Simply multiply out b (t )E d and use Lemma 4.1.5. ji i ji i=1 ji=0 Corollary 4.1.8. For all d ≥ 1 we have

q q ! −1 Y Y Sd(q, −1) = `d [d] bd−1(ti) + bd(ti) . i=1 i=1 Proof. Recall that

(Mdχt1 ··· χtq )(z) Sd(q, −1) = − d . `dEd(z − θ ) z=0 d The constant term of (Mdχt1 ··· χtq )(z) comes from the coefficients of the functions Ed in Prop. 4.1.6. From the way we have split the sum above, our claim is evident.

d X The next corollary follows by observing that the partial sums Se(q, −1) tele- e=0 scope to a quantity whose limit we can understand. That is, the proof is exactly similar to Pellarin’s elementary proof of his formula for L(χt, 1) given earlier in this chapter.

Corollary 4.1.9. The following equality holds in Tq,

q Y L(Πqχ , 1) = π/ ω(t ). ti e i i=0 58 Proof. It suffices to check this for ti ∈ C∞ such that |ti| ≤ 1 for i = 1, . . . , q. For such ti we have,

d q X L(Π χt , 1) = lim Se(q, −1) i d→∞ e=0 q −1 Y = lim `d bd(ti) d→∞ i=1 −1 Qd  qj  j=1 1 − θ/θ = lim d→∞ Qq Qd−1 qj
−1 i=1 j=0 1 − ti/θ q Y = π/e ω(ti) i=1

Remark 4.1.10. The author first obtained the corollary above in [24] via a strict analysis of the remainder term appearing in the proof of Theorem 4.2.2. For this approach see Remark 4.2.7.

A different point of view for the calculation 4.1.6

We have seen that the interpolation polynomial

d−1 X Md(χt)(z) = bd(t) + bj(t)Ej(z) j=0

q Y agrees as a function of z with χt on A+(d). Thus the product Md(χti )(z) agrees i=1 with the product character χt1 ··· χtq on A+(d). However, the degree in z of this

d polynomial is too large (it is exactly q ) for it to be Md(χt1 ··· χtq ). The main point

q is that powers of z which are too large are coming from the function Ed−1(z) ob-

q tained after multiplying the above product out. Happily, via the relation Ed−1(z) =

[d]Ed(z)+Ed−1(z) and the fact that Ed(z) is identically equal to the constant function

59 1 on A+(d), we see exactly how to remove the powers of z that are too large, while still interpolating the product of characters. Indeed,

q q Y Y Md(χti )(z) − (Ed(z) − 1)[d] bd−1(ti) i=1 i=1 clearly agrees with the product character on A+(d), and one easily checks that this

d difference is a polynomial in z of degree strictly less than |A+(d)| = q . The E d notation is a slick way of disguising all of this, but we thought it would be helpful to the reader to include these remarks. The constant term is also clear from the representation given above.

4.2 Explicit formulae when s ≤ q and k ≡ s mod (q − 1)

The main theorem of this section is the explicit calculation of the rational special values of Pellarin’s L-series when the number of indeterminates s is at most q. Again we use the Wagner series for χt, and we give two distinct calculations - one by “unfolding” Pellarin’s series just from its definition and another using the Anderson generating function. We believe that our explicit results helped inspire the major work of Angl`esand Pellarin [3] mentioned above.

Theorem 4.2.1. Let k, s be positive integers such that 1 ≤ s ≤ q, such that k ≥ s, and such that k ≡ s mod (q − 1). The following equality holds in Ts, s " s # Ps ki −k s Y X BC(k − i=1 q ) Y 1 π L(Π χt , k) ω(ti) = s k . e i P ki q i Π(k − i=1 q ) Dki · (θ − ti) i=1 k1,...,ks i=1

Here the sum is over all s-tuples of non-negative integers (k1, . . . , ks) such that s X k − qki ≥ 0. i=1 We shall prove the following theorem from which the previous theorem imme- diately follows using the Carlitz-Euler relations (1.1.5) and Pellarin’s formula for

L(χt, 1), Theorem 4.1.4. 60 Theorem 4.2.2. Let s, k be positive integers such that 1 ≤ s ≤ q, such that k ≥ s and such that k ≡ s mod (q − 1). The following formula holds in Ts,

" s ! s # b (t )L(χ , qki ) X X ki Y ki i ti L(χt1 ··· χts , k) = ζ k − q , (4.2.1) Dki k1,...,ks i=1 i=1 where the sum is over all combinations of s non-negative integers k1, . . . , ks such that s X k − qki ≥ 0. i=1

4.2.1 A first proof of Theorem 4.2.2 by “unfolding”

We copy here word for word the proof given in [24]. While not as easy as the proof involving generating series appearing in the next section, this proof has interest in its own right as it uses such classical results as the vanishing of the Goss zeta function at the negative “even” (divisible by q − 1) integers and the logarithmic growth of the special polynomials associated to Goss’ zeta function. This proof also gives the case of s = q indeterminates.

The proof proceeds by grouping the sum over A+ in Pellarin’s series according to degree and for each a ∈ A+(d) one replaces each instance of χt(a) by its polynomial interpolation on A+(d). One can then justify an interchange of sums which results in the desired explicit formulae.

Proof by “unfolding”. Let ti ∈ C∞ be such that |ti| ≤ 1 for i = 1, . . . , s. We break the proof into several lemmas for the convenience of the reader and to highlight how the various conditions on s and k arise. The first lemma uses in a crucial way the well known vanishing of the power sums

X ak

a∈A+(d) for positive integers k, d whenever k ≤ qd − 1, see Corollary 1.2.5. Its proof gives rise to the condition on s appearing in the main result. 61 Lemma 4.2.3 (Condition on s). Let 1 ≤ s ≤ q and k ≥ s, then L(χt1 , . . . , χts , k) equals

∞ ∞ i1 is s ! X X X X Y bij (tj)αij ,kj ··· ··· ζ(k − (qk1 + ··· + qks )). (4.2.2) Dij i1=0 is=0 k1=0 ks=0 j=1

Proof. We begin by replacing χti by its Wagner representation and interchanging sums. The equation

e X X χt (a) ··· χt (a) 1 s (4.2.3) ak d=0 a∈A+(d) becomes

e d d s ! s ! X X X Y bij (tj) X −k Y ··· a eil (a) . (4.2.4) Dij d=0 i1=0 is=0 j=1 a∈A+(d) l=1 Next we wish to interchange the sum indexed by the variable d with the sums indexed by the variables i1, . . . , is. It is convenient to observe that due to the vanishing of the polynomials ei(b) whenever b ∈ A(i) (i.e. whenever i > degθ(b)) we may first increase the upper limits of the sums indexed by i1, . . . , is from d to e without changing anything. Then equation (4.2.4) becomes

e e s ! e s ! X X Y bij (tj) X X −k Y ··· a eil (a) . (4.2.5) Dij i1=0 is=0 j=1 d=0 a∈A+(d) l=1

Next we replace the polynomials ei(z) with their series representations

ij X qkj eij (z) = αij ,kj z

kj =0 and interchange sums. Equation (4.2.5) becomes

e e i1 is s ! e X X X X Y bij (tj)αij ,kj X X k1 ks ··· ··· aq +···+q −k. (4.2.6) Dij i1=0 is=0 k1=0 ks=0 j=1 d=0 a∈A+(d) Next we insert and remove the tail of Goss zeta special values

∞ X X k1 ks aq +···+q −k

d=e+1 a∈A+(d) 62 and (4.2.6) becomes

e e i1 is s ! X X X X Y bij (tj)αij ,kj ··· ··· ζ(k − qk1 + ··· + qks ) − (4.2.7) Dij i1=0 is=0 k1=0 ks=0 j=1

e e i1 is s ! ∞ X X X X Y bij (tj)αij ,kj X X k1 ks ··· ··· aq +···+q −k. (4.2.8) Dij i1=0 is=0 k1=0 ks=0 j=1 d=e+1 a∈A+(d) Now we wish to show that the remainder term (4.2.8) vanishes as e → ∞. In order to do this, we use the well known fact (see Lemma 1.2.5) that the power sums

X 0 0 ak vanish for non-negative integers k0 such that k0 < qd − 1 and d0 ≥ 1. 0 a∈A+(d ) We observe that the maximum power of a which can occur in the sum above is sqe − k, and this is less than qe+1 − 1 since 1 ≤ s ≤ q and k ≥ s. Hence for all possible exponents d of a occurring in the sum over A+(d) in equation (4.2.8) we s X ki d have q − k < q − 1. Hence for all e > logq(k − s + 1) the sums indexed by i=1 k1, . . . , ks are over s-tuples (k1, . . . , ks) of non-negative integers such that 0 ≤ kj ≤ ij s X for j = 1, . . . , s and qki − k < 0. We indicate this with a prime in the summation i=1 over k1, ..., ks. Explicitly, we will write (4.2.8) as e e s ! ∞ 0 X X X Y bij (tj)αij ,kj X X k1 ks ··· aq +···+q −k, (4.2.9) Dij i1=0 is=0 k1,...,ks j=1 d=e+1 a∈A+(d) where the prime indicates only s-tuples (k1, . . . , ks) such that 0 ≤ kj ≤ ij for j = s X ki 1, . . . , s and q − k < 0 appear, and from now on we suppose e > logq(k + 1 − s). i=1 Now we estimate.

e e s ! ∞ 0 X X X Y bij (tj)αij ,kj X X qk1 +···+qks −k ··· a ≤ Dij i1=0 is=0 k1,...,ks j=1 d=e+1 a∈A+(d) e e s ∞ 0 X X X Y bij (tj)αij ,kj X X qk1 +···+qks −k ··· a ≤ Dij i1=0 is=0 k1,...,ks j=1 d=e+1 a∈A+(d) e e s k +1 ∞ 0 k q j X −qi1 X −qis X X −k q j + X (qk1 +···+qks −k)d q ··· q q j q−1 q =

i1=0 is=0 k1,...,ks j=1 d=e+1

e e s k +1 k1 ks 0 k q j (q +···+q −k)(e+1) X −qi1 X −qis X X −k q j + q q ··· q q j q−1 . 1 − qqk1 +···+qks −k i1=0 is=0 k1,...,ks j=1 63 Now we use the upper bound q(qk1 +···+qks −k)(e+1) ≤ q−(e+1), which holds because, s X as we have noted, qks − k < 0 (and in particular is at most −1 since this quantity i=1 is an integer) for all indices ki appearing in the primed summation. Thus the last line above is less than or equal to

kj +1 e e s −k qkj + q 0 j q−1 X i1 X is X X q q−(e+1) q−q ··· q−q . (4.2.10) 1 − qqk1 +···+qks −k i1=0 is=0 k1,...,ks j=1

Now the sum indexed by k1, . . . , ks is finite and the upper limit depends only on k and i1, . . . , is but not on e as described above. We remove the dependence on i1, . . . , is by dropping the condition that kj ≤ ij for j = 1, . . . , s. This has the effect of adding more positive terms to the sum indexed by k1, . . . , ks. Thus equation (4.2.10) is maximized by

 kj +1  e e ! s −k qkj + q 00 j q−1 −(e+1) X −qi1 X −qis X X q q q ··· q  k k  , (4.2.11) 1 − qq 1 +···+q s −k i1=0 is=0 k1,...,ks j=1 where the double prime signifies summing over all s-tuples (k1, . . . , ks) such that s X ki q − k < 0. The sums indexed by i1, . . . , is are convergent as e → ∞, the sum i=1 −(e+1) indexed by k1, . . . , ks is a constant which does not depend on e, and q → 0 as e → ∞. Thus the error term in (4.2.8) vanishes as e → ∞, and we conclude that

L(χt1 , . . . , χts , k) equals

∞ ∞ i1 is s ! X X X X Y bij (tj)αij ,kj ··· ··· ζ(k − (qk1 + ··· + qks )). (4.2.12) Dij i1=0 is=0 k1=0 ks=0 j=1

The second lemma uses the vanishing of Goss zeta special values for negative even integers (Lemma 1.2.8), i.e. those negative integers divisible by q − 1, in a crucial way that is totally analogous to how the vanishing of the power sums was used above.

This lemma gives rise to the congruence condition on k appearing in the statement of the main theorem. 64 Lemma 4.2.4 (Condition on k). Let s, k be as in the statement of Theorem 4.2.2,

then L(χt1 , . . . , χts , k) equals s ∞ ! ∞ ! X X X bi (t1)αi ,k X bi (ts)αi ,k ζ(k − qki ) 1 1 1 ··· s s s , (4.2.13) Di1 Dis k1,...,ks i=1 i1=k1 is=ks where the sum over k1, . . . , ks includes all s-tuples of non-negative integers such that s X k − qki ≥ 0. i=1 Proof. We begin with equation (4.2.12), and we observe that by assumption, k ≡ s mod (q − 1); hence k − (qk1 + ··· + qks ) ≡ 0 mod (q − 1). Thus ζ(k − (qk1 + ··· + qks )) vanishes as soon as k − (qk1 + ··· + qks ) < 0. This limits the upper bound on the sum indexed by k1, . . . , ks and allows for arguments entirely analogous to those above to show that we may interchange sums.

We conclude the proof with a little algebraic lemma.

Lemma 4.2.5. Let j be a non-negative integer. We have ∞ X bi(t) bj(t) α = L(χ , qj). D i,j D t i=j i j Proof.

∞ ∞ j i−j X bi(t) X bj(t)τ bi−j(t) (−1) Di α = D i,j D qj i=j i i=j i DjLi−j ∞ bj(t) X bi−j(t) = (−1)i−jτ j D L j i=j i−j ∞ bj(t) X bi−j(t) = τ j (−1)i−j D L j i=j i−j ∞ bj(t) X bi(t) = τ j (−1)i D L j i=0 i

bj(t) j = τ L(χt, 1) Dj In the last two lines we use Theorem 4.1.2.

Making the substitution in (4.2.13) finishes the proof of the theorem.

65 4.2.2 A second proof of Theorem 4.2.2 when s < q

In this section we give a proof of Theorem 4.2.2 in the special case where s < q which emphasizes the main theme of this dissertation: The Wagner series for χt is the Anderson Generating Function. This point of view allows us to give generating series for the rational special values of Pellarin’s series and demonstrates a similarity of these rational special values to Bernoulli polynomials classically. See also the remarks in the introduction of this chapter.

Theorem 4.2.6. Let 1 ≤ s < q. Let z ∈ C∞ be such that |z| < 1. We have the following equality in Ts,

χ (a) ··· χ (a) s ω(t , πz) −1 X t1 ts Y i e πe eC (πze ) = . z − a ω(ti) a∈A i=1

Proof. Let ti ∈ C∞ be such that |ti| ≤ 1 for i = 1, . . . , s. We begin with the following equality in K[t1, . . . , ts, z],

s d−1 Y X X `dEd(z) b (t )E (z) = χ (a) ··· χ (a) , (4.2.14) ji i ji t1 ts z − a i=1 ji=0 a∈A(d) which holds since both are polynomials in z whose degrees are strictly less than qd by

our assumption on s, and both interpolate χt1 ··· χts for z ∈ A(d). Now [16] Theorem

−1 3.2.8 shows that `dEd(z) → πe eC (πze ) as d → ∞, for all z ∈ C∞. Further, for |z| < 1 X (even z ∈ C∞ \ A) and |ti| ≤ 1 the limit lim χt1 (a) ··· χts (a)/(z − a) exists. d→∞ a∈A(d) Thus for z and ti as above, we have

X χt1 (a) ··· χts (a) −1 X χt1 (a) ··· χts (a) lim `dEd(z) = π eC (πz) . d→∞ z − a e e z − a a∈A(d) a∈A

Now by Theorem 1.6.6 letting d tend to infinity on the left side of (4.2.14) finishes the proof.

66 Proof of Theorem 4.2.2 when s < q. We mentioned in the introduction that

X 1 X π/e (πz) = = 1/z + ζ(k + 1)zk. e C e z − a a∈A k≥0 k+1≡0 mod (q−1)

In a similar way we have

X χt (a) ··· χt (a) X 1 s = L(Πsχ , k + 1)zk. z − a ti a∈A k≥0 k+1≡s mod (q−1)

−1 Now by definition ω(t, πze ) = eC (zπe(θ −t) ), and using Pellarin’s formula we may −1 write πe(θ − t) = L(χt, 1)ω(t). Thus

X −1 i X −1 i qi ω(t, πze ) = Di τ (zL(χt, 1)ω(t)) = Di L(χt, q )bi(t)ω(t)z . i≥0 i≥0

Hence Theorem 4.2.6 shows that

X s k L(Π χti , k + 1)z k≥0 k+1≡s mod (q−1) equals   s ! j  X k Y −1 X −1 ji q i 1/z + ζ(k + 1)z  ω(ti) D L(χti , q )bji (ti)ω(ti)z .   ji k≥0 i=1 ji≥0 k+1≡0 mod (q−1)

Multiplying everything out as power series in z and comparing the coefficients proves

Theorem 4.2.2.

Remark 4.2.7. In the case s = q we have

Qs χ (a) π π s ω(t , πz) X i=1 ti e e Y i e = Qs + . z − a ω(ti) eC (πz) ω(ti) a∈A i=1 e i=1 For the coefficients of the non-zero powers of z, the proof is as above working back- s Y wards from Theorem 4.2.2. The constant term π/e ω(ti) may be obtained in two i=1

67 s Y ways. We have already calculated that L(Πqχ , 1) = π/ ω(t ) in this chapter. ti e i i=1 We may also obtain this from closer examination of the error term in (4.2.8) above.

Indeed, the proof of Theorem 4.2.2 goes through in exactly the same way in the case s = q and k = 1 except that one has a single non-zero term arising from the remainder, namely q ! Y be(tj) X e+1 aq −1. De j=1 a∈A+(e+1)

X qe+1−1 Carlitz showed that a = De+1/`e+1, and we have the recursive formula

a∈A+(e+1) q qe+1 De = De+1/(θ − θ). Inputting these ingredients and taking the limit as e → ∞ s Y we obtain L(Πqχ , 1) = π/ ω(t ). ti e i i=1

4.3 Applications

4.3.1 Numerators of the Bernoulli-Carlitz numbers

s In this section we let λ ∈ Fq and s = q − 1 in Theorem 4.2.1 so that χλ becomes the trivial character with conductor θ − λ from A to Fq. Doing so allows us to obtain new recurrence relations between the Bernoulli-Carlitz numbers and from that new divisibility results for the numerators of the Bernoulli-Carlitz numbers.

A lemma on the function ω

The following lemma, discovered by the author, has its own interest and will be used in the sequel. The reader should compare with similar results that hold for products of

Carlitz torsion [25], and products of Thakur’s Gauss sums. See [3] for general results connecting the specializations of the function ω at roots of unity with Thakur’s Gauss sums.

68 Lemma 4.3.1. Let Fq ⊆ C∞ be the algebraic closure of Fq. Let λ ∈ Fq, and let f ∈ A be its minimal polynomial. Suppose f has degree d. Then d Y ω(τ iλ)q−1 = (−1)df. i=1 d Y Proof. With the notations above we have f = (θ − τ iλ). It is a basic property of i=1 limits that if an → a and bn → b, then anbn → ab. With this and the continuity of the map x 7→ xq−1 we have

d d n −1!q−1 Y Y Y  τ iλ ω(τ iλ)q−1 = lim ι 1 − n→∞ θqj i=1 i=1 j=0 d n −(q−1) Y Y  τ iλ = ι(q−1)d lim 1 − n→∞ θqj i=1 j=0 n d −(q−1) Y Y  τ iλ = (−θ)d lim 1 − n→∞ θqj j=0 i=1 n Y j = (−θ)d lim (f(θ)/θd)−q (q−1) n→∞ j=0 = (−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→∞

New recursive relations for the Bernoulli-Carlitz numbers

Now we proceed toward the new recursion relations for the Bernoulli-Carlitz numbers.

From the definition of the Bernoulli-Carlitz numbers, by comparing coefficients in the formula ∞ X BC(j) z = e (z) zj C Π(j) j=0 69 one may obtain the recurrence relations BC(0) = 1 and for k ≥ 1

X Π(k) BC(k) = − BC(k + 1 − qj). Π(qj)Π(k + 1 − qj) q≤qj ≤k+1 k Π(k) For non-negative integers k, j let us write := , and observe j A Π(k − j)Π(j) k that ∈ A for all j, k ≥ 0, see e.g. [16] Section 9.1. We now give a new j A recurrence relation satisfied by the BCs that will be used in Theorem 4.3.5 to give new divisibility results for the numerators of the Bernoulli-Carlitz numbers.

Lemma 4.3.2 (Second Recursive Formula). We have BC(0) = 1, and for any posi- tive integer k ≡ 0 mod (q − 1) and any λ ∈ Fq,

q−1 q−1 P ki   P ki
k− i=1 q X k BC k − i=1 q · (θ − λ) BC(k) = q−1 , P ki k k − i=1 q (θ − λ)(1 − (θ − λ) ) k1,...,kq−1 A where the sum is over all (q − 1)-tuples of non-negative integers (k1, . . . , kq−1) such q−1 X that k − qki ≥ 0. i=1 Proof. This is an elementary algebraic calculation made using Theorem 4.2.1, the

q−1 fact that since χλ is the trivial character we have  1  L(χq−1, k) = 1 − ζ(k), λ (θ − λ)k and Lemma 4.3.1.

Remark 4.3.3. This same trick can be played to obtain similar recursive relations

q for L(χt, k) using our explicit formula for L(Π χti , k). The details are left to the reader.

Divisibility of the numerators by degree one primes

Definition 4.3.4. Let k ≡ 0 mod (q − 1) be a positive integer, and let X be the set q−1 X ki ∗ of all (q − 1)-tuples of non-negative integers such that k − q ≥ 0. Let (ki ) ∈ X i=1 70 q−1 q−1 X ∗ X ki ki ∗ be such that q ≥ q for all (ki) ∈ X. We call such a tuple (ki ) maximal in i=1 i=1 X.

∗ ∗ Theorem 4.3.5. Let k, X and (ki ) be as in the definition above, and let µ := q−1 X k∗ ∗ q µ∗−2 k − q i . Suppose that µ > 2, then (θ − θ) divides the numerator of BC(k). i=1 Proof. Using Lemma 4.3.2 we see that (1 − (θ − λ)k)BC(k) equals

q−1 !   q−1 X k X P ki ki k−1− i=1 q q−1 · BC k − q · (θ − λ) . P ki k − i=1 q k1,...,kq−1 A i=1

q−1 X Now µ∗ − 1 is minimal among all the differences k − 1 − qki appearing in the i=1 exponent of (θ − λ) on the right above. It follows by Carlitz von-Staudt theorem, see [16], Theorem 9.2.2, that the zeros (in C∞) of the denominators of the non-zero Bernoulli-Carlitz numbers are simple. Taking account of the possibility that (θ − λ) divides the denominator of some BC(j) on the right we conclude that (θ − λ)µ∗−2 divides the right side above and hence also divides BC(k). As λ ∈ Fq was arbitrary, this finishes the proof.

Remark 4.3.6. 1. Integers k as in the theorem above are easy to obtain. For example let d ≥ 2 and suppose k = (q − 1)qd + (q − 1)qd−1. Then µ∗ = (q − 1)qd−1 > 2, and

(θq − θ)(q−1)qd−1−2 divides the numerator of BC(k). In fact, as mentioned in the abstract, it is not difficult to show that as a subset of the positive integers such that k ≡ 0 mod (q − 1) those with µ∗(k) > 2 form a set of full natural density. This leads to the question of interpolating the Bernoulli-Carlitz numbers directly, and this question remains interesting and open for general primes. There are no known

Kummeresque congruences for these numbers. We hope that the theorem above may give some insight on a path to take toward the question of interpolation in general, and we plan to return to this problem in a future work. 71 2. Finally, it is reasonable to expect that a recursion similar to that in Lemma

4.3.2 holds for j(q −1) tuples of non-negative integers for all j ≥ 1 arising in a similar way from a closed form formula for Pellarin’s series in j(q−1) indeterminates. Perhaps

finding these formulas will allow us to obtain more information on the numerators of the BC(k) for all k ≥ 1. Such information should have interesting consequences for

L. Taelman’s new class module.

4.3.2 Recursive Relations for Pellarin’s L-Series

Now that we have obtained generating series for Pellarin’s L-series from Theorem

4.2.6, we may use them to obtain recursive relations among the rational functions occurring in Pellarin’s [22] Theorem 2. For simplicity we will stick to the case of one indeterminate.

Definition 4.3.7. Let k ≡ 1 mod (q − 1). Define

−k γk(t) := πe L(χt, k)ω(t).

Recall, we have shown that γk(t) ∈ K(t) for all k as above. Further, using the τ action one observes immediately that

k −qk k τ (γ1(t)) = πe L(χt, q )bk(t)ω(t) = bk(t)γqk (t).

−1 Hence, using Pellarin’s Formula γ1(t) = (θ − t) , we obtain the following theorem.

Theorem 4.3.8. Let k ≡ 1 mod (q − 1). If k = qj for some j, then

−1 γk(t) = . bj+1(t)

Otherwise, blogq(k)c X γk+1−qj (t) γ (t) = − . k D j=1 j

72 Proof. We handled the case when k = qj for some j above. When k 6= qj for any j we must appeal to the generating series relation. From Theorem 4.2.6 we have

X k πzωe (t, πze ) L(χt, k)z = . eC (πz)ω(t) k≥1 e k≡1 mod (q−1)

We clear denominators and change coordinates to obtain

X k eC (z) · γk(t)z = zω(t, z). k≥1 k≡1 mod (q−1)

Explicitly, blog (l−1)c q qj +1 X X γl−qj (t) X z zl = . qj Dj D · (θ − t) l≥2 j=0 j≥0 j l≡2 mod (q−1) Suppose l 6= qj + 1 for any j, then the coefficient of zl on the left vanishes, and hence

blogq(l−1)c X γl−qj (t) γ (t) = − . l−1 D j=1 j

This finishes the proof.

Example 4.3.9. As an example, on the next page we list the first ten of these γj in the case A = F3[θ]. With specialization to the Bernoulli-Carlitz numbers in mind, it is convenient to normalize by a Carlitz factorial. The reader may check that after multiplying by θ − t, evaluation at t = θ in Π(j − 1)γj(t) yields the Bernoulli-Carlitz number BC(j − 1). It is interesting to observe that for our small list below, the zeros of γj all lie in A.

73 The First Ten Non-zero γj

1 γ (t) = 1 θ − t −1 Π(2)γ (t) = 3 (θ3 − t)(θ − t) 1 Π(4)γ (t) = 5 (θ3 − t)(θ − t) −1 Π(6)γ (t) = 7 (θ3 − t)(θ − t) −θ6 − θ4 − θ2 Π(8)γ (t) = 9 −(θ9 − t)(θ3 − t)(θ − t) θ15 + θ13 + θ11 − θ9 − θ7 − θ5 − θ3 + t Π(10)γ (t) = 11 −(θ9 − t)(θ3 − t)(θ − t) θ15 + θ13 + θ11 + θ9 − θ7 − θ5 − θ3 − t Π(12)γ (t) = 13 (θ9 − t)(θ3 − t)(θ − t) θ12 − θ10 − θ4 + θ2 Π(14)γ (t) = 15 −(θ9 − t)(θ3 − t)(θ − t) −θ12 + θ10 + θ6 − θ4 Π(16)γ (t) = 17 −(θ9 − t)(θ3 − t)(θ − t) θ21 − θ19 + θ15 − θ13 + θ9 − θ7 + θ3 − t Π(18)γ (t) = 19 −(θ9 − t)(θ3 − t)(θ − t)

74 BIBLIOGRAPHY

[1] G. Anderson and D. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. of Math. (2) 132 (1990), no. 1, 159191.

[2] A. El-Guindy and M. Papanikolas, Identities for Anderson generating functions for Drinfeld modules, arXiv:1302.0238v2 (2013).

[3] B. Angles and F. Pellarin, Universal Gauss-Thakur sums and L-series, arXiv:1301.3608v2 (2013).

[4] M. Bhargava, The Factorial Function and Generalizations, Amer. Math. Monthly (9) 107 (2000), 783-799.

[5] G. B¨ockle, Global L-functions over function fields, Math. Ann. (2) 323 (2002), 737-795.

[6] G. B¨ockle, The distribution of the zeros of the Goss zeta-function for 2 3 A = F2[x, y]/(y + y + x + x + 1), Math. Z. (2013), 1-27. [7] L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J. (2) 2 (1935), 137-168.

[8] L. Carlitz, A class of polynomials, Trans. Amer. Math. Soc. 43 (1938), 167-182.

[9] L. Carlitz, A set of polynomials, Duke Math. J. (2) 6 (1940), 486 - 504.

[10] C. Y. Chang, M. Papanikolas, D. Thakur, and J. Yu, Algebraic independence of arithmetic gamma values and Carlitz zeta values, J. Number Theory (4) 223 (2010), 1137-1154.

[11] C. Y. Chang and J. Yu, Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math. (1) 216 (2006), 321-345.

[12] K. Conrad, The Digit Principle, J. Number Theory 84 (2000), 230-257.

[13] D. Goss, Fourier Series, Measures and Divided Power Series in the Theory of Function Fields, K-theory 2 (1989), no. 4, 533-555.

75 [14] D. Goss, A formal Mellin transform in the arithmetic of function fields, Trans. Amer. Math. Soc. 327 (1991), 567-582.

[15] D. Goss, Applications of non-Archimedean integration to the L-series of τ- sheaves, J. Number Theory (1) 110 (2005), 83-113.

[16] D. Goss, Basic Structures of Function Field Arithmetic, Springer-Verlag, Berlin, 1996.

[17] D. Goss, ζ-phenomenology, arXiv:0806.3463v7 (2010).

[18] D. Goss, On the L-series of F. Pellarin, J. Number Theory (3) 133 (2013), 955-962.

[19] D. Goss and W. Sinnott, Class groups of function fields, Duke Math. J. (2) 52 (1985), 507-516.

[20] S. Jeong, A Comparison of the Carlitz and Digit Derivatives Bases in Function Field Arithmetic, J. Number Theory (2) 84 (2000), 258-275.

[21] N. Katz, P -adic L-functions via moduli of elliptic curves, Proc. Symp. Pure Math. 29 (1975), 479-506.

[22] F. Pellarin, Values of certain L-series in positive characteristic, Ann. of Math. 176 (2012), 2055-2093.

[23] R. B. Perkins, On Pellarin’s L-series, Proc. Amer. Math. Soc. (2012), Ac- cepted.

[24] R. B. Perkins, Explicit formulae for L-series in finite characteristic, Submitted. arXiv:1207.1753v2, (2013).

[25] M. Rosen, Number Theory in Function Fields, Springer-Verlag, New York, 2002.

[26] J. Sheats, Riemann hypothesis for the Goss Zeta function for Fq[T ], J. Number Theory (1) 71 (1998), 121-157.

[27] L. Taelman, Dirichlet unit theorem for Drinfeld modules, Math. Ann. (4) 348 (2010), 899-907.

[28] L. Taelman, A Herbrand-Ribet theorem for function fields, Invent. Math. (2) 188 (2012), 253-275.

[29] D. Thakur, Zeta measure associated to Fq[T ], J. Number Theory 35 (1990), 1-17.

[30] D. Thakur, Function Field Arithmetic, World Scientific Publishing, River Edge, NJ, 2004.

76 [31] L.I. Wade, Certain quantities transcendental over GF (qn, x), Duke Math. J. 8 (1941), 701-720.

[32] L.I. Wade, Certain quantities transcendental over GF (qn, x), II, Duke Math. J. 10 (1943), 587-594.

[33] C. G. Wagner, Linear operators in local fields of prime characteristic, J. Reine Angew. Math. 251 (1971), 153 - 160.

[34] C. G. Wagner, Interpolation series in local fields of prime characteristic, Duke Math. J. (2) 39 (1972), 203-210.

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