---
date: 2023-03-16 00:05
aliases: ["Untitled"]
---

Last modified: `=this.file.mday`

# Drinfeld Talk 2 Draft 2

- What is $d_\infty$ and $v_\infty$?
- What is $\gens{\alpha}$ and $\gens{\alpha}^y$?
- RH for Drinfeld modules: 4.12.8.5?
- L for EC with CM factors via Hecke characters?

# Setup


- Basic comparison: $\zeta_A(s) \da \sum_{a\in A^{\mathrm{mon}} } a^{-s}$ compared to $\zeta_k(s) \da \sum_{D\in \Div^+(X)} \Norm(D)^{-s}$.


## Zeta functions (7.8)



## Partial zetas

- We can now define the partial zeta functions of interest to us for describing periods. If $I \subseteq \mathbf{A}$ is an ideal and $j \geq 0$, then we set
$$
I_{j}:=\{i \in I \mid \operatorname{deg}(i) \leq j\}
$$
Clearly $I_{j}$ is a finite dimensional $\mathbf{F}_{r}$-vector space. For $j \gg 0$, this dimension is given exactly by the Riemann-Roch Theorem.

- **Definition** 7.8.2: Partial zeta functions
	1. Let $\alpha$ be an ideal class of $\mathbf{A}$. We set
$$
\zeta_{\alpha}(s):=Z_{\alpha}(u):=\sum_{\substack{I \in \alpha \\ I \subseteq A}} N I^{-s} .
$$
2. Let $I \subseteq \mathbf{A}$ be an ideal and $a \in \mathbf{A}$. We then set
$$
\zeta_{a, I}(s):=Z_{a, I}(u):=\sum_{\substack{b \in \mathbb{A} \\ b \equiv a(I)}} N b^{-s}=\sum u^{\operatorname{deg} b},\qquad N b:=N(b \AA)
$$

- Proposition 7.8.3: Decomposition of Zeta functions by ideals
	1. $$a \equiv b \mod I \implies Z_{a, I}(u)=Z_{b, I}(u)$$
	2. $$\sum_{\substack{a \mod IJ \\ a \equiv b \mod I}} Z_{a, I J}(u)=Z_{b, I}(u)$$
	3. $Z_{b a, b I}(u)=u^{\operatorname{deg} b} Z_{a, I}(u)$ for $0 \neq b \in \mathbf{A}$.
	4. Let $\zeta \in \mathbf{F}_{r}^{*}$. Then
$$
Z_{\zeta a, I}(u)=Z_{a, I}(u)
$$


- Motivating these definitions:
	
	- The basic definition, 8.1.2.2, is motivated by the following simple observations: first of all, if $\alpha$ is positive, then 8.1 .1 simply becomes $$\alpha=\pi^{j}\langle\alpha\rangle .$$
	- Now the subgroup of $\mathrm{K}^{*}$ given by the powers of $\pi$ is obviously infinite cyclic; thus any homomorphism of it into $\mathrm{C}_{\infty}^{*}$ is determined by what $\pi$ maps to, and this we call " $x$." 
	- On the other hand, $\langle\alpha\rangle$ is a 1-unit in $K\units$; as is well known the group of 1-units is isomorphic to the countable product of $\ZZpadic$ with itself. Thus, this group has a huge group of endomorphisms and an even larger group of homomorphisms into $\mathrm{C}_{\infty}^{*}$. The simplest and most natural of these endomorphisms is raising an element to the $y$-th power for $y \in \mathbf{Z}_{p}$, and these are the ones used in the definition (i.e., $\left.\langle\alpha\rangle \mapsto\langle\alpha\rangle^{\nu}\right)$. As of this writing,

- Proposition 8.1.3. Behaves like exponentials.
	- 1. Let $\alpha$ and $\beta$ be positive elements and let $s=(x, y) \in$ $S_{\infty}$. Then $$(\alpha \beta)^{s}=\alpha^{s} \beta^{s} .$$
	- 2. Let $s_{0}, s_{1} \in S_{\infty}$ and $\alpha$ positive. Then
$$
\alpha^{s_{0}+s_{1}}=\alpha^{s_{0}} \alpha^{s_{1}}
$$
- Prototypical L-function: We are now already able to define the zeta function of $\mathbf{A}=\FF_{r}[T]$. Indeed, this function is simply given as
$$
\zeta_{\mathbf{A}}(s):=\sum_{a \in \mathbf{A} \text { monic }} a^{-o},
$$
	where the convergence, etc., will be discussed in later subsections.


## Functional analysis (8.4)

![](attachments/2023-03-16cts.png)

- Let $L$ be any field of characteristic $p$ which is complete with respect to a nontrivial absolute value $\abs{\wait}$. 
- Let $\mathcal{O} \subset L$ be the ring of integers with maximal ideal $M$. 
- We shall give a description of the space of continuous functions from $\ZZpadic$ to $L$ which is a variant of the classical theorem of Mahler. 
  
- **Definition** 8.4.2. Set $\left(\begin{array}{l}x \\ 0\end{array}\right) \equiv 1$. For $k$ a positive integer set $$\left(\begin{array}{l}x \\k\end{array}\right):=\frac{x(x-1) \cdots(x-k+1)}{k !} \in \mathbf{Q}[x] \text {. }$$
	- Thus $\left(\begin{array}{l}x \\ k\end{array}\right)$ gives rise to a continuous function from $Q_{p}$ to itself. If $x$ is a non-negative integer, then $\left(\begin{array}{l}x \\ k\end{array}\right) \in \mathbb{Z} \subset \mathbf{Z}_{p}$. As such integers are dense, we see that $$\left(\begin{array}{l} x \\k\end{array}\right): \ZZpadic \rightarrow \mathbf{Z}_{p}$$
	- By reducing these modulo $(p)$ we can consider them as continuous functions with values in $\mathbf{F}_{p} \subset \mathcal{O} \subset L$; we will also denote these functions by " $\left(\begin{array}{l}x \\ k\end{array}\right)$ ".

- Let $\left\{a_{k}\right\} \subset L$ be a sequence of elements in $L$ with $a_{k} \rightarrow 0$ as $k \rightarrow \infty$. Set
$$
\varphi(x):=\sum_{k=0}^{\infty} a_{k}\left(\begin{array}{l}
x \\
k
\end{array}\right) \in C\left(\ZZpadic, L\right)
$$



- **Definition** 8.4.3. Coefficient extraction:.
	- We set $\Delta^{0} \varphi(x):=\varphi(x)$ and $$\Delta \varphi(x):=\varphi(x+1)-\varphi(x) .$$
- Lemma 8.4.5 (Periodicity). 
	- Let $0 \leq k<p^{N}$. 
	- Then the function $\left(\begin{array}{l}x \\ k\end{array}\right): \ZZpadic \rightarrow \FF_p$ is periodic modulo $\left(p^{N}\right)$.

- **Theorem 8.4.6 (Basis of continuous functions)**
	- Every $\varphi(x) \in C\left(\ZZpadic, L\right)$ can be expanded uniquely as $$\varphi(x)=\sum a_{k}\left(\begin{array}{l}x \\k\end{array}\right)$$where $\left\{a_{k}\right\} \subset L$ and $a_{k} \rightarrow 0$ as $k \rightarrow \infty$. 
	- Moreover, the $a_{k}$ are given as:
	- Lemma 8.4.4. We have for $j \geq 0$ $$\begin{aligned}a_{j} & =\Delta^{j} \varphi(0) \\& =\sum_{i=0}^{j}(-1)^{j-i}\left(\begin{array}{l}j \\i\end{array}\right) \varphi(i) .\end{aligned}$$


- Let $f \in C\left(\ZZpadic, L\right)$ be written as $$f(x)=\sum a_{k}\left(\begin{array}{l}x \\k\end{array}\right) .$$
- We set
$$
\|f\|_{1}:=\max _{k}\left\{\left|a_{k}\right|\right\} .
$$
- Proposition 8.4.8. We have
$$
\|f\|=\|f\|_{1} .
$$



- Thus, in down to earth terms, to give an entire function for $S_{\infty}$ means that for each $y \in \ZZpadic$ we are given a power series $g_{y}(u)$ which converges for all $u$ and
$$
f(x, y)=g_{y}(1 / x) \text {. }
$$
- Furthermore, let $B \subset \mathbf{C}_{\infty}$ be any bounded subset and $\varepsilon>0$. Then we require the existence of $\delta:=\delta(B)>0$ so that if $y_{0}, y_{1} \in \ZZpadic$ and $\left|y_{0}-y_{1}\right|_{p}<\delta$ then $\left|g_{y_{0}}(u)-g_{y_{1}}(u)\right|<\varepsilon$ for all $u \in B$.

- **Definition** 8.5.3. Let $g(u)=\sum_{i=0}^{\infty} a_{i} u^{i}$ be a power series with $\left\{a_{i}\right\} \subset \mathbf{C}_{\infty}$. Let $r \in \RR$. We define
$$
\|g\|_{r}:=\max _{i}\left\{\left|a_{i}\right| r^{i}\right\} \in \mathbf{R} \cup\{\infty\}
$$
- For instance, $\|g\|_{1}=\max _{i}\left\{\left|a_{i}\right|\right\}$. If $\|g\|_{1}<\infty$, then $g(u)$ converges for all $u$ with $|u|<1$, etc. Note also that if $g(u)$ is entire, then $\|g\|_{r}<\infty$ for all $r$.
- Let $g(u)$ now be a power series that converges on the closed unit ball $\{u|| u \mid \leq 1\}$. This is clearly equivalent to having $a_{i} \rightarrow 0$ as $i \rightarrow \infty$.
- Claim 8.5.4. $\|g\|_{1}=\max _{|u| \leq 1}\{|g(u)|\}$.

- Let $f(s): S_{\infty} \rightarrow \mathbf{C}_{\infty}$ be an entire function and let $y \in \ZZpadic$. By definition
$$
f(s)=f(x, y)=\sum_{i=0}^{\infty} f_{i}(y) x^{-i}
$$
where $f_{i}(y) \in \mathbf{C}_{\infty}$. By above we see that $f_{i}(y): \mathbf{Z}_{p} \rightarrow \mathbf{C}_{\infty}$ is continuous.

- Set
$$
m_{i}:=\left\{\max _{y \in \mathbf{Z}_{\mathbf{p}}}\left|f_{i}(y)\right|\right\} .
$$
By compactness, $m_{i}$ is finite.
- **Theorem** 8.5.6. Let $r \in \mathbf{R}_{+}$. Then
$$
m_{i} r^{i} \rightarrow 0
$$
as $i \rightarrow \infty$.

- Conversely, let $f_{i}(y): \ZZpadic \rightarrow \mathbf{C}_{\infty}, i=0, \ldots, \infty$, be a collection of continuous functions. Let
$$
m_{i}:=\max _{y}\left\{\left|f_{i}(y)\right|\right\},
$$
and suppose $m_{i} r^{i} \rightarrow 0$ as $i \rightarrow \infty$ for all $r \in \RR_{+}$. We then have the following converse to Theorem 8.5.6.

- **Theorem** 8.5.7. Set $f(x, y):=\sum_{j=0}^{\infty} f_{j}(y) x^{-j}$. Then $f(x, y)$ is an entire function on $S_{\infty}$.

![](attachments/2023-03-16-24.png)

- By Theorem 8.4.6 for each $j$ we can write
$$
f_{j}(y)=\sum_{k=0}^{\infty} f_{j, k}\left(\begin{array}{l}
y \\
k
\end{array}\right)
$$
for $\left\{f_{j, k}\right\} \subset \mathbf{C}_{\infty}$ and $f_{j, k} \rightarrow 0$ as $k \rightarrow \infty$. Thus
$$
\begin{aligned}
f(x, y) & =\sum_{j=0}^{\infty}\left(\sum_{k=0}^{\infty} f_{j, k}\left(\begin{array}{l}
y \\
k
\end{array}\right)\right) x^{-j} \\
& =\sum_{k=0}^{\infty}\left(\sum_{j=0}^{\infty} f_{j, k} x^{-j}\right)\left(\begin{array}{l}
y \\
k
\end{array}\right)
\end{aligned}
$$
the change of order being justified by uniform convergence. Let us write this as
$$
f(x, y)=\sum_{k=0}^{\infty} \widehat{f}_{k}\left(x^{-1}\right)\left(\begin{array}{l}
y \\
k
\end{array}\right)
$$
- By Lemma 8.4.4,
$$
\widehat{f}_{j}\left(x^{-1}\right)=\sum_{i=0}^{j}(-1)^{j-i}\left(\begin{array}{l}
j \\
i
\end{array}\right) f(x, i) \text {; }
$$
in any case $\widehat{f}_{j}(u)$ is certainly entire in $u$.

- Let $r \in \mathbf{R}_{+}$. Then the set
$$
\left\{\left\|\widehat{f}_{j}\right\|_{r}\right\} \subset \mathbf{R}_{+}
$$
is bounded (use Theorem 8.5.6). Indeed we can use this property to characterize entire functions as follows. Let $r \in \mathbf{R}_{+}$. 

- We thus define $\|f(x, y)\|_{r}$ to be the common value of $\|f(x, y)\|_{r}^{(1)}$ and $\|f(x, y)\|_{r}^{(2)}$

- **Theorem** 8.5.11. Let $\hat{f}_{j}$ be as in 8.5.8. Then for $r \in \RR_{+},\left\|\widehat{f}_{j}\right\|_{r} \rightarrow 0$ as $j \rightarrow \infty$. Conversely, let $\left\{\widehat{f}_{j}(u)\right\}$ be a collection of entire functions such that $\left\|\hat{f}_{j}\right\|_{r} \rightarrow 0$ as $j \rightarrow \infty$ for $r \in \RR_{+}$. Set
$$
f(x, y):=\sum_{j=0}^{\infty} \widehat{f}_{j}\left(x^{-1}\right)\left(\begin{array}{l}
y \\
j
\end{array}\right) .
$$
Then $f(x, y)$ is an entire function on $S_{\infty}$.

### Algebraicity

%%- The $L$-series that we will define will always turn out to be entire functions on $S_{\infty}$. However, they will also possess a very strong algebraicity component. This algebraicity mirrors that of exponentiation itself, and so we begin again by examining the entire function $$f(s):=a^{-s} \text{ where $a\in \AA$ positive }\implies f(s)=x^{-d}\langle a\rangle^{-y}$$where $s=(x, y), d=\operatorname{deg}(a)$ and $\langle a\rangle=\pi^{d / d_{\infty}} \cdot a$. 
- Recall that we choose $\pi_{*}$ to be a fixed $d_{\infty}$-th root of $\pi$ in $\mathbf{C}_{\infty}$. Suppose now that $y=-j$ for $j \geq 0$. Set
$$
\begin{aligned}
h_{f}(x,-j):=f\left(x \pi_{*}^{j},-j\right) & =x^{-d} \pi_{*}^{-d j}(a)^{j} \\
& =x^{-d} \pi_{*}^{-d j} \pi^{j d / d_{\infty}} \cdot a^{j} \\
& =x^{-d} a^{j} .
\end{aligned}
$$
Thus, by using the substitution, $x \mapsto x \pi_{*}^{j}$ at $y=-j$, we have succeeded in removing the 1-unit part of $a$. This leads to our next definition.%%

%%- **Definition** 8.5.12. Let $f(s)=f(x, y)$ be an entire function on $S_{\infty}$. We say that $f(s)$ is **essentially algebraic** if and only if $h_{f}(x,-j):=f\left(x \pi_{*}^{j},-j\right)$ is a polynomial in $x^{-1}$ with algebraic coefficients for all $j \geq 0$. 
	- We further require that all such coefficients (for all $j$ ) generate a finite extension of $\mathbf{k}$.%%
	  
- In practice the coefficients of $h_{f}(x,-j)$ will be elements of $\mathbf{A}$. Or there will exist an entire function $g$ with $h_{g}(x,-j)$ having $\mathbf{A}$ coefficients and $f$ sits in a factorization of $g$ (e.g., $f$ is some sort of $L$-function). Thus in practice the following results will be sufficient for detecting essentially algebraic entire functions.

%%- Proposition 8.5.13. 
	- Let $f(s)$ be an entire function such that the power series $$h_{f}(x,-j)=f\left(x \pi_{*}^{j},-j\right)$$has $\AA\dash$-coefficients of all $j \geq 0$. Then $f$ is essentially algebraic.

- Corollary 8.5.14. 
	- Let $f(s)$ be as in 8.5 .13 and suppose $f(s)=g(s) h(s)$ where both $g(s)$ and $h(s)$ are entire. Then $h_{g}(x,-j)$ and $h_{h}(x,-j), j \geq 0$, (defined as in 8.5.13) are both polynomials.
	- Note that no such result as 8.5 .14 can be established complex analytically. Indeed, using $e^{x}$, counterexamples are easily found.%%

- Essential algebraicity follows very easily from a function being entire. Essential algebraicity is extremely important precisely because it is the "glue" holding together the theories at the different places of $\mathbf{k}$. Indeed, one is able to pass between the theories at the different primes only by using the polynomials of 8.5.12. These polynomials interpolate to the functions defined at all the places of $\mathbf{k}$.

- For instance in our prototypical case of the zeta function of $\mathbf{A}=\mathbf{F}_{r}[T]$, we will see that $\zeta_{\mathrm{A}}(s)$ is entire on $S_{\infty}$. We immediately deduce from 8.5.13 that it is essentially algebraic. Thus the power series $$h_{\zeta}(x,-j)=\sum_{t=0}^{\infty} x^{-t}\left(\sum_{\substack{a \text { monic } \\ \text { deg } a=t}} a^{j}\right)$$are, in fact, polynomials. 
	- The $v$-adic theory, for $v \in \operatorname{Spec}(\mathbf{A})$, (also given in Subsection 8.9) and continuity now imply that these polynomials also determine the $v$-adic functions $\zeta_{v}\left(\mathbf{A}, e_{v}\right)$. The general theory always works the same way!

## L-series (8.6)


- Example of cmpatible families:
	- Let $\psi$ be a Drinfeld module of rank $d$ over $L$. Let $v \in \operatorname{Spec}(\mathrm{A})$ and let $T_{v}:=T_{v}(\psi)$ be the $v$-adic Tate module of $\psi$. Let $H_{v}^{1}\left(\psi, \mathbf{k}_{v}\right):=\operatorname{Hom}_{\mathbf{A}_{v}}\left(T_{v}, \mathbf{k}_{v}\right)$ with the natural dual action of $G$. 
	- Let $\mfp$ be a prime of $\mathcal{O}_{L}$ not lying over $v$ such that $\psi$ has good reduction at $\mfp$. Then $T_{v}(\psi)$ and $H_{v}^{L}\left(\psi, \mathbf{k}_{v}\right)$ are both unramified at $\mfp$. Moreover the action of the geometric Frobenius on $H_{v}^{1}\left(\psi, \mathbf{k}_{v}\right)$ may be computed (Subsection 4.12) over the finite field $F_{\varphi}=\mathcal{O}_{L} / \rho$ via the action of the Frobenius endomorphism of of the reduction of $\psi$ at $p$. 
	- Let $\rho_{v}$ be the action of $G$ on $H_{v}^{1}\left(\psi, \mathbf{k}_{v}\right)$. Then the discussion of Subsection 4.12 assures us that $\left(\rho_{v}\right)$ is a strictly compatible integral family. We can define $B$ to be the set of finite primes where $\psi$ has bad reduction. This construction is basic to the theory of Drinfeld modules (and is totally analogous to its classical counterpart for elliptic curves and abelian varieties).
- Example:
	- Let $E$ be a $T$-module over $L$ where A acts via "complex multiplication." Although we have not worked out explicitly any of the details, it is clear that $E$ - which only involves finitely many equations - has "good" reduction at almost all primes of $\mathcal{O}_{L}$, etc. Thus one can also obtain strictly compatible integral families of representations here also. As an example, one has the tensor product of two Drinfeld modules. As we will not pursue this further, the details are left to the reader.








## Dirichlet series (8.7)


- Let $\mathbf{K}_{1} \subset \mathbf{C}_{\infty}$ be a finite extension of $\mathbf{K}$. For each ideal $I$ of $\mathbf{A}$ let $c(I) \in \mathbf{K}_{1}$. Let $s \in S_{\infty}$. We call a formal series of the form
$$
L(s):=\sum_{I \subseteq \mathbf{A}} c(I) I^{-*}
$$
a Dirichlet series.

- **Theorem** 8.7.1. (Determined by special values) 
	- Let $L(s)$ be a Dirichlet series with non-trivial half-plane of convergence $H$ as above. Then $L(s)$ is uniquely determined by $\{L(j)\}$ such that $j \geq N$ ( $N$ as above) and
$$
j \equiv 0\left(h(\mathbf{A})\left(r^{d_{\infty}}-1\right)\right) .
$$


- Corollary 8.7.2. The elements $\left\{I^{-s}\right\}$ are linearly independent.
Corollary 8.7.3. Suppose $L(s)$ has an Euler product over the primes of $\mathbf{A}$. Then this Euler product is unique.

8.8 proves that $L(\hat\rho, s)$ is entire for certain families of compatible representations $\hat\rho$.
Example:
![](attachments/2023-03-16-48.png)

- Let $W(d) \subset \boldsymbol{F}_{r}[u]$ be the subspace of polynomials of degree $\leq d$ with trivial constant term. Let $v$ measure the vanishing order at $u=0$, and let $j \geq 0$.
Proposition 8.8.2.
$$
v\left(\sum_{f \in W(d)} f(u)^{j}\right) \geq(r-1) \frac{d(d+1)}{2}
$$
![](attachments/2023-03-16-49.png)

- As a basic example of how these estimates are used, we examine the $x$-adic behavior of the function
$$
L(x, j)=\sum_{j=0}^{\infty} x^{d}\left(\sum_{f \in W(d)} f^{j}\right)
$$
where $j \geq 0$. By 8.8 .2 we see immediately that the $u$-adic valuation of the coefficient of $x^{d}$ grows quadratically in $d$. Thus $L(x, j)$ is entire in $x$. Our method of handling arbitrary abelian L-series is simply a generalization of this.



![](attachments/2023-03-16-51.png)

- In this subsection we will prove that the $L$-series associated to finite characters are entire. Let $\mathbf{A}, \mathbf{k}, \mathbf{K}, \mathbf{C}_{\infty}$, etc., be as usual. Let $\overline{\mathbf{k}}$ be an algebraic closure of $\mathbf{k}$ and let $L \subset \overline{\mathbf{k}}$ be a finite extension which may have non-trivial inseparability degree. (In [Go4] we assumed that $L$ was indeed separable. However, the remarks at the beginning of Subsection 8.6 assure us that this assumption is not necessary.) Let $\mathcal{O}:=\mathcal{O}_{L} \subset L$ be its ring of $\mathrm{A}$-integers.
Let $G=\operatorname{Gal}\left(L^{\text {sep }} / L\right)$, where $L^{\text {sep }} \subset \overline{\mathbf{k}}$ is the separable closure. Let $\chi: G \rightarrow$ $\mathbf{C}_{\infty}^{*}$ be a homomorphism of Galois type i.e., there exists a finite abelian extension $L_{1} / L$ such that $L_{1} \subset L^{\text {sep }}$ and $\chi$ factors through $G_{1}:=\operatorname{Gal}\left(L_{1} / L\right)$. Let $\mathcal{B}$ be the conductor of $L_{1}$ which we write as $\mathcal{B}=\mathcal{B}_{\infty} \mathcal{B}_{f}$; here $\mathcal{B}_{f}$ is made up of finite primes of $L$ (and so can be considered as an ideal of $\mathcal{O}_{L}$ ) and $\mathcal{B}_{\infty}$ is made up of infinite primes.
![](attachments/2023-03-16-52.png)

- Let $\mfp$ be a finite prime of $L$. If $\mfp$ ramifies in $L_{1}$, then we set $\chi(\mfp):=0$. If $\mfp$ is unramified, then we set $\chi(\mfp):=\chi\left(\left(\mfp, L_{1} / L\right)^{-1}\right)$ where $\left(\mfp, L_{1} / L\right)$ is the Artin symbol at $\mfp$. (The reader should note again that we are using the inverse of the Artin symbol in keeping with previous conventions.)

![](attachments/2023-03-16-53.png)

- Let $\psi$ be a Drinfeld module of rank $d$ over $L$ with lattice $M$. We say that $\psi$ has sufficiently many complex multiplications if and only if the set of $\alpha \in \mathbf{C}_{\infty}$ with $\alpha M \subseteq M$ is an order of rank $d$ over A. In this case the $L$-series $L(\psi, s)$ factors into a product of $L$-series associated to Hecke characters exactly as for elliptic curves. In turn, these $L$-series can be shown to be entire by the same technique as used here, for more see [G04]. See also Subsection 10.4.

## Teich. character

![](attachments/2023-03-16-54.png)

- Recall that in Subsection 8.1 we defined the elements $s_{j} \in S_{\infty}$ with the property that if $I=(i), i$ positive, then
$$
I^{s_{j}}=i^{j}
$$
We then set $\mathbf{V}=\mathbf{k}\left(\left\{I^{s_{1}}\right\}\right) \subset \mathbf{C}_{\infty}$; so $[\mathbf{V}: \mathbf{k}]<\infty$ as we have seen.
Let $\mathcal{O}_{\mathbf{V}}$ be the ring of $\mathbf{A}$-integers in $\mathbf{V}$ and let $\sigma: \mathbf{V} \rightarrow \overline{\mathbf{k}}_{\boldsymbol{p}}$ be an embedding over $\mathbf{k}$ where $\mfp \in \operatorname{Spec}(\mathbf{A})$ is fixed and $\overline{\mathbf{k}}_{\boldsymbol{p}}$ is an algebraic closure equipped with its canonical metric. This is completely similar to what was done in

![](attachments/2023-03-16-55.png)

- Subsection 8.3. Let $\bar{\mfp}$ be the finite prime of $\mathbf{V}$ over $\emptyset$ which is associated to $\sigma$

As before let $\mathcal{I}$ be the group of $\mathbf{A}$-fractional ideals of $\mathbf{k}$ with $\mathcal{I}(\mfp)$ the subgroup of $\mathcal{I}$ generated by those primes $\neq \boldsymbol{p}$. If $I \in I(p)$ then $I^{s_{1}}=I^{1} \in \mathbf{V}$ is prime to $\mfp$ by Proposition 8.2.9. Let
$$
\rho: I(\mfp) \rightarrow\left(\mathcal{O}_{\mathbf{V}} / \bar{\emptyset}\right)^{*}
$$
be the mapping
$$
I \mapsto I^{1}+\bar{\mfp}=I^{s_{1}}+\bar{\mfp} .
$$
Lemma 8.11.1. Let $\alpha \in \mathbf{k}^{*}$ with $\alpha \equiv 1(\bmod \boldsymbol{p}), \alpha$ positive. Then
$$
\rho((\alpha))=1 \in \mathcal{O}_{\mathbf{V}} / \bar{\mfp}
$$

![](attachments/2023-03-16-56.png)

- Recall that we set $\mathcal{P}_{p}^{+}$to be group of principal ideals $(\alpha)$ such that $\alpha$ is positive and $\equiv 1 \quad(\bmod p)$. From 8.11.1 we immediately deduce a homomorphism
$$
\hat{\rho}: \mathcal{I}(p) / \mathcal{P}_{p}^{+} \rightarrow\left(\mathcal{O}_{\mathbf{V}} / \bar{p}\right)^{*} \text {. }
$$
Let $W=W_{\sigma}$ be the Witt ring of finite field $\mathcal{O}_{\mathbf{V}} / \bar{\varphi}$ which we consider as lying in $\overline{\mathbf{Q}}_{p}$ for our fixed algebraic closure $\overline{\mathbf{Q}}_{p}$ of $\mathbf{Q}_{p}$.

Let Teich: $\left(\mathcal{O}_{\mathbf{V}} / \bar{\emptyset}\right)^{*} \rightarrow W^{*}$ be the usual map taking an element to its Teichmüller representative.
Definition 8.11.2. We set
$$
\omega_{\sigma}:=\omega_{\bar{p}}:=\text { Teich } \circ \widehat{\rho}: \mathcal{I}(\boldsymbol{p}) / \mathcal{P}_{p}^{+} \rightarrow W^{*} .
$$
The homomorphism $\omega_{\sigma}$ is the generalized Teichmüller character.

![](attachments/2023-03-16-57.png)

- Let $\psi$ be a sgn-normalized Hayes-module for a fixed sign function sgn, and to $\mathbf{k}$ the p-division points of $\psi$ in $\mathbf{C}_{\infty}$. By 7.55 we know that there is an isomorphism
via the Artin map. Thus we may consider $\omega_{\sigma}$ as a $W$-valued character of $G$.

## Special values (8.12)

![](attachments/2023-03-16-58.png)

- 8.12. Special Values at Negative Integers
Let $L \subset \overline{\mathbf{k}} \subset \mathbf{C}_{\infty}$ be a fixed finite extension of $\mathbf{k}$ as before. Let $\mathcal{O}:=\mathcal{O}_{L}$ be the ring of $\mathbf{A}$-integers of $L$. We are interested here in the function $\zeta_{O_{L}}(s)$ which we recall (see Subsection 8.6) is defined as
$$
\begin{aligned}
\zeta \mathcal{O}_{L}(s) & =\prod_{\substack{p \subset \mathcal{O}_{L} \\
p \text { prime }}}\left(1-n \mfp^{-s}\right)^{-1} \\
& =\sum_{I \subseteq \mathcal{O}_{L}} n I^{-s} .
\end{aligned}
$$
![](attachments/2023-03-16-59.png)

- From the main result of Subsection 8.9 we know that $\zeta_{O_{L}}(s)$ continues to an entire function on $S_{\infty}$. Thus it is essentially algebraic, either by the argument at the end of Subsection 8.9 (which uses 8.5.14) or directly via 8.8.1.1. Therefore if $\pi_{*}$ is our fixed root in $\mathbf{C}_{\infty}$ of
$$
z^{d \infty}-\pi
$$
and $j$ is a non-negative integer, then
$$
z(x,-j):=z_{\zeta}(x,-j):=h_{\zeta}(x,-j)=\zeta \mathcal{O}_{L}\left(x \pi_{*}^{j},-j\right)
$$
is a polynomial in $x^{-1}$ with coefficients in the ring $\mathcal{O}_{\mathbf{V}}$ of $\mathrm{A}$-integers of the value field $\mathbf{V}$. This subsection is devoted to the arithmetic interpretation of these polynomials.

![](attachments/2023-03-16-60.png)

- We now want to relate $z(x,-j)$ to the generalized Teichmüller character $\omega_{\sigma}$ of Subsection 8.11. As mentioned in that subsection, we view $\omega_{\sigma}$ as a $W$-valued character of $\operatorname{Gal}(\mathbf{k}(\boldsymbol{\rho}) / \mathbf{k})$ where $W$ is the Witt ring of $\mathcal{O}_{\mathbf{v}} / \bar{\mfp}$. Let
$$
L(\mfp):=L \cdot \mathbf{k}(\mfp) \text {; }
$$
of course $L(\mfp) / L$ is abelian with Galois group canonically isomorphic to a subgroup of $\operatorname{Gal}(\mathbf{k}(\mfp) / \mathbf{k})$. Therefore, $\omega_{\sigma}$ also gives rise to a character $\omega_{\sigma, L}$ of $\operatorname{Gal}(L(\mfp) / L)$. Let $\mathcal{B}$ be a finite prime of $\mathcal{O}_{L}$ not dividing $\mfp$; thus $L(\mfp)$ is unramified at $\mathcal{B}$. Class field theory implies that
$$
\omega_{\sigma, L}\left((\mathcal{B}, L(\mfp) / L)^{-1}\right)=\omega_{\sigma}\left((n \mathcal{B}, \mathbf{k}(\mfp) / \mathbf{k})^{-1}\right) .
$$
Let $u$ be a variable.
Definition 8.12.3. Let $L\left(\omega_{\sigma, L}^{j}, u\right)$ be the classical (characteristic zero valued) $L$-series of $\omega_{\sigma, L}^{j}$.
Thus let $\mathcal{B}$ be as above. The local factor at $\mathcal{B}$ in $L\left(\omega_{\sigma, L}^{j}, u\right)$ is
$$
\left(1-\omega_{\sigma, L}^{j}\left((\mathcal{B}, L(\mfp) / L)^{-1}\right) u^{\operatorname{deg}_{1} B}\right)
$$
where $\operatorname{deg}_{1} \mathcal{B}$ is the degree of $\mathcal{B}$ with respect to the full field $F$ of constants of $L$ (and where we have continued to use the geometric Frobenius). We note that $L\left(\omega_{\sigma, L}^{j}, u\right)$ is formed as a product over all places, finite or infinite, of $L$.

## Trivial zeros

![](attachments/2023-03-16-61.png)

- **Definition** 8.13.2. We set $\tilde{z}(x,-j)$ equal to
$$
z(x,-j)\left(1-\omega_{\sigma, L}^{-j}\left(\infty_{i_{1}}\right) x^{-\operatorname{deg}\left(\infty_{i_{1}}\right)}\right)^{-1} \cdots\left(1-\omega_{\sigma, L}^{-j}\left(\infty_{i_{\ell}}\right) x^{-\operatorname{deg}\left(\infty_{i_{t}}\right)}\right)^{-1},
$$
where, as usual, deg means "degree with respect to $F_{r}$. "
![](attachments/2023-03-16-62.png)

- **Definition** 8.13.5. The zeroes of $z(x,-j) / \tilde{z}(x,-j)$ are the trivial zeroes of $z(x,-j)$

Of course the trivial zeroes of $z(x,-j)$ immediately translate to trivial zeroes of $\zeta_{O_{L}}(s)$, as, by definition, the value of $z(x,-j)$ at $x$ equals $\zeta_{O_{L}}\left(x \pi_{*}^{j},-j\right)$. If $x=1$ is a trivial zero for $z(x,-j)$ then we say $-j \in S_{\infty}$ is a trivial zero for $\zeta_{O_{1}}(s)$, etc. Moreover, the same techniques can be used to give trivial zeroes of abelian $L$-series, etc. See Subsection 8.17.

![](attachments/2023-03-16-63.png)

- Example 8.19.6. Let $\infty$ be a rational point over $\mathbf{F}_{r}$ and let $L=\mathbf{k}$; thus $\mathcal{O}_{L}=\mathbf{A}$. Let $\mfp$ be a prime of $\mathbf{A}$. We know that the decomposition and
278
8. $L$-series .
inertia groups at $\infty$ inside $\operatorname{Gal}(\mathbf{k}(\emptyset) / \mathbf{k})$ correspond to $\FF_{r}^{*} \subseteq \operatorname{Gal}(\mathbf{k}(\emptyset) / \mathbf{k})$. Let $j$ be a positive integer; thus $\omega_{\sigma}^{-j}$ is ramified at $\infty$ if and only if $j$ is not divisible by $r-1$. We then have two cases:
1. $(r-1) \nmid j$. In this case $\infty$ does not contribute to $\omega_{\sigma}^{-j}$; in consequence $-j$ is not a trivial zero for $\zeta_{\mathbf{A}}(s)$. (N.B.: This does not imply that $\zeta_{\mathbf{A}}(-j) \neq 0$; only that such a zero is not "trivial.")
2. $(r-1) \mid j$. Then $\infty$ contributes $1-x^{-1}$, as above, and we find a trivial zero at $-j \in S_{\infty}$.


## Class groups

![](attachments/2023-03-16-64.png)

- As mentioned in the introduction to this section, the reader may profit by just assuming that $\mathbf{A}$ is a principal ideal domain (or even $\mathbf{F}_{r}[T]$ ) during a first reading. This may be especially true of this subsection where the use of general $\mathbf{A}$ will cause us to introduce a fair amount of notation. In any case, in this subsection we will use our double congruence (Theorem 8.13.3) to establish results on the divisibility of certain class numbers by $p$, where $p$ is the characteristic of $\mathbf{C}_{\infty}$. While there may be some technicalities involved, the basic idea used here is very simple: a p-primary Galois component of the class group is non-trivial if and only if some classical $L$-value is not divisible
280
8. $L$-series
by $p$. On the other hand, this $L$-value modulo $p$ turns out to be the same (up to isomorphism) as some special zeta-value (in finite characteristic!) modulo the prime $\mfp$ of A. Thus the divisibility of this characteristic $p$ value by $\mfp$ determines the non-triviality of the $p$-primary Galois component we want to study.


![](attachments/2023-03-16-65.png)

- **Definition** 8.14.1. Let $G_{p}$ be as above and $G_{\sigma} \subset G_{p}$ the kernel of $\omega_{\sigma}$. Let $\mathbf{k}(\mfp)_{\sigma}$ be the subfield fixed by $G_{\sigma}$; we call $\mathbf{k}(\mfp)_{\sigma}$ the subfield cut out by $\omega_{\sigma}$.
Thus, if $\mathbf{A}$ is a principal ideal domain, then $\mathbf{k}(\mfp)_{\sigma}=\mathbf{k}(\mfp)$; in general, however, it will be a much smaller subfield.

Proposition 8.14.2. The extension $\mathbf{k}(\mfp) / \mathbf{k}(\mfp)_{\sigma}$ is everywhere unramified.

![](attachments/2023-03-16-66.png)

- Now $L \subset \mathbf{C}_{\infty}$ be a finite extension of $\mathbf{k}$. Let $\mfp \in \operatorname{Spec}(\mathbf{A})$ be fixed. To simplify matters, we assume that $\mfp$ is unramified in $L$; so, in particular, $L / \mathbf{k}$ is separable. Let $z(x,-j)$ be as in Subsection 8.12 and let $\tilde{z}(x,-j)$ be as in Definition 8.13.2. Set $L_{1}:=\mathrm{H}^{+} \cdot L$. Thus $L_{1} \subseteq L(\mfp)=L \cdot \mathbf{k}(\mfp)$, and is also unramified at the primes above $\mfp$. Finally, $L(\mfp) / L_{1}$ is totally ramified at the primes above $\mfp$. Set $G_{\mathfrak{p}, L}:=\operatorname{Gal}(L(\mfp) / L)$. Let $\omega_{\sigma, L}$ be as in Subsection 8.12 and let $o(\sigma, L)$ be its order. Thus $\omega_{\sigma, L}: G_{p, L} \rightarrow W_{\sigma}$; we denote the kernel by $G_{\sigma, L}$. One can now form, in the obvious fashion, $L(\mfp)_{\sigma}$, etc. At the primes above $\mfp$ there is also a version of the above proposition which we leave to the reader.

Lemma 8.14.3. The p-primary component $G_{p, L}^{(p)}$ of $G_{p, L}$ is contained in the kernel $G_{\sigma, L}$.

![](attachments/2023-03-16-67.png)

- Let $\mathrm{Cl}_{L}(\mfp)^{(p)}$ be the $p$-primary class group of the field $L(p)$, i.e., the $p$ primary component of the $\mathbf{F}_{1}$-rational points of the Jacobian of $C$ (as curve

![](attachments/2023-03-16-68.png)

- pal divisors of elements in $L(\mfp)^{*}$. As before the Galois module $\mathrm{Cl}_{L}(\mfp)^{(p)} \otimes W$ may be decomposed under powers of $\omega_{\sigma, L}$. One has
$$
\mathrm{Cl}_{L}(\mfp)^{(p)}\left(\omega_{\sigma, L}^{-i}\right) \simeq T_{p}\left(\omega_{\sigma, L}^{-i}\right) /(1-F) T_{p}\left(\omega_{\sigma, L}^{-i}\right) .
$$
Thus we dedure that
$$
\operatorname{ord}_{p} L\left(\omega_{\sigma, L}^{-i}, 1\right)=\operatorname{ord}_{p} L_{\mathrm{un}}\left(\omega_{\sigma, L}^{-i}, 1\right)
$$
is the length of $\mathrm{Cl}_{L}(\mfp)^{(p)}\left(\omega_{\sigma, L}^{-i}\right)$ as $W$-module.
We then have the following result.
Theorem 8.14.4. Under the above hypotheses on $\mfp$ and $i$ we have
$$
\mathrm{Cl}_{L}(\mfp)^{(p)}\left(\omega_{\sigma, L}^{-i}\right) \neq\{0\} \Leftrightarrow \bar{\mfp} \mid \tilde{z}(1,-i) .
$$

## Arith/geom, Kummer-Vandiver

![](attachments/2023-03-16-69.png)

- In a similar vein, concepts related to constant field extensions will be called "arithmetic" whereas those related to Drinfeld modules will be labeled "geometric." For instance, the theory of the classical $L$-series of function fields, due to Artin, Hasse, Weil, etc., is well-known to be intimately connected to constant field extensions (via Tate modules, etc.). Thus we refer these $L$-series as the arithmetic $L$-series. Those $L$-series associated to Drinfeld modules etc. are then geometric $L$-series.

The geometric/arithmetic dichotomy in the theory of function fields is found quite widely. For instance, in Theorem 8.14 .4 we proved a criterion for the existence of certain class groups which depended on the $\bar{\xi}$-divisibility of $\tilde{z}(1,-i)$. The element $\tilde{z}(1,-i) \in \mathbf{V}$ is very much like a classical Bernoullinumber. Because it is connected with the arithmetic $L$-series, it is an arithmetic Bernoulli-element (or arithmetic Bernoulli function). Similarly, in future subsections, we will encounter geometric Bernoulli-elements.

![](attachments/2023-03-16-70.png)

- In classical cyclotomic theory, one is very interested in whether certain class groups are cyclic or not. More precisely, let $p$ be prime, $\zeta_{p}$ a primitive $p$-th root of unity. Let $C^{(p)}$ be the $p$-primary class group of $\mathbb{Q}\left(\zeta_{p}\right)$ and $\omega$ the standard Teichmüller character of $(\mathbb{Z} / p)^{*} \simeq \operatorname{Gal}\left(\mathbb{Q}\left(\zeta_{p}\right) / \mathbb{Q}\right)$. One forms $C^{(p)}\left(\omega^{i}\right)$ in the usual way and one is interested in whether it is a cyclic $\mathbf{Z}_{p}$-module or not.
The most important sufficient condition for cyclicity in classical theory is the Kummer-Vandiver Conjecture, $p \nmid h(p)^{+}$, where $h(p)^{+}$is the class number of $\mathbb{Q}\left(\zeta_{p}+\zeta_{p}^{-1}\right)$ (and, of course $\left.h(p)^{-}=h(p) / h(p)^{+}\right)$. One then deduces cyclicity via the reflection theorem, see [Wa1, §10].

In the function field setting we are also interested in cyclicity, and in this section we will present a sufficient criterion for it. In a following subsection (Subsection 8.22), we present a possible analog of the Kummer-Vandiver Conjecture. Another remarkable analog of the Kummer-Vandiver Conjecture is given in the new work of Anderson discussed in Subsection 10.6.

![](attachments/2023-03-16-71.png)

- We use the set-up and decomposition conventions of Subsection 8.14. Thus $L \subset \mathbf{C}_{\infty}$ is a finite extension of $\mathbf{k}, \mfp \in \operatorname{Spec}(\mathbf{A})$ is unramified in $L$, and $\omega_{\sigma, L}$ is the Teichmüller character of $\operatorname{Gal}(L(\mfp) / L)$ associated to $L$ and an embedding $\sigma$ of the value field $\mathrm{V}$. Let $o(\sigma, L)$ be the order of $\omega_{\sigma, L}$ as before and let
$$
0<i<o(\sigma, L) \text {. }
$$
We assume that $r^{\operatorname{deg} p}-1$ does not divide $i$ (so we may use an appropriate double congruence).

![](attachments/2023-03-16-72.png)

- As in Subsection 8.14, let $L_{\mathrm{un}}\left(\omega_{\sigma, L}^{-i}, u\right)$ be the unit root $L$-series of $\omega_{\sigma, L}^{-i}$. Let $W$ now be the Witt ring of $\mathcal{O}_{\mathrm{v}} / \bar{\emptyset}$ and let $\bar{W}$ be its integral closure in some algebraic closure of the fraction field of $W$. We write
$$
L_{\mathrm{un}}\left(\omega_{\sigma, L}^{-i}, u\right)=\prod_{\alpha_{,} \text {a unit }}\left(1-\alpha_{j} u\right)
$$
in $\bar{W}[u]$
Proposition 8.16.1. Let $i$ be positive integer not divisible by $r^{\operatorname{deg} p}-1$. Let $\bar{M} \subset \bar{W}$ be the maximal ideal. Suppose that at most one of $\left\{\alpha_{j}\right\}$ is congruent to $1(\bmod \bar{M})$. Then $\mathrm{Cl}_{L}(\mfp)^{(p)}\left(\omega_{\sigma, L}^{-i}\right)$ is a cyclic $W$-module.

![](attachments/2023-03-16-73.png)

- Let $\tilde{z}(x,-i)$ be as in Subsection 8.14. Let $\mathbf{V}_{\overline{\boldsymbol{p}}}$ be the completion of $\mathbf{V}$ at $\bar{\mfp}$ and let $\overline{\mathbf{V}}_{\overline{\boldsymbol{p}}}$ be an algebraic closure of $\mathbf{V}_{\bar{p}}$ with integers $\overline{\mathcal{O}}_{\bar{p}}$ and maximal ideal $\bar{M}_{\bar{p}}$. Over $\overline{\mathcal{O}}_{\bar{p}}$ we write
$$
\tilde{z}(x,-i)=\prod\left(1-\theta_{e} x^{-t_{r}}\right) .
$$
Our double congruence, Theorem 8.13.3, then immediately gives the following result.

Corollary 8.16.2. Let $i$ be as above. Then $\mathrm{Cl}_{L}(p)^{(p)}\left(\omega_{0, L}^{-i}\right)$ is a cyclic $W$ module if at most one of $\left\{\theta_{e}\right\}$ is $\equiv 1\left(\bmod \bar{M}_{\bar{p}}\right)$.

Corollary 8.16 .2 is the arithmetic criterion for cyclicity. It is "arithmetic" precisely since it relates to the theory of $\zeta_{\mathrm{A}}(s), s=(x, y)$, when $y$ is a negative integer and these values are related to arithmetic $L$-series.

Remark. 8.16.3. It is easy to reformulate 8.16 .2 in terms of the value and derivative at $x=1$ of $\tilde{z}(x,-i)$. For details see [Go4, §4.4].

![](attachments/2023-03-16-74.png)

- Question 8.17.5. Let $G$ be as in 8.17 .1 and let $W$ now be a finite dimensional $\mathbf{C}_{\infty}$-vector space; so we are taking representations in characteristic $p$. Let $\rho: G \rightarrow G L(W)$ be a representation of Galois type. We now define the $L$-series of $\rho, L(\rho, s)$, by using the inertia and decomposition groups at all finite primes in the usual fashion. Can $L(\rho, s)$ be continued to an essentially algebraic entire function? The answer is certainly "yes" if $\operatorname{dim} W=1$ as we have seen. Moreover, Theorem 8.17.1 combined with [Se, $\S 16]$ tells us that $L(\rho, s)$ automatically has a meromorphic continuation as a quotient $f_{1}(s) / f_{2}(s)$ with $f_{i}(s)$ essentially algebraic and entire for $i=1,2$. This is exactly similar to what was concluded via Brauer induction at the beginning of this section for representations in characteristic 0 .

![](attachments/2023-03-16-75.png)

- **Theorem** 8.17.1. Let $L$ be a finite extension of $\mathbf{k}$. Let $G$ be the Galois group over $L$ of the separable closure of $L$, and let $V$ be a finite dimensional $\bar{Q}_{p}$ vector space. Let $\rho: G \rightarrow G L(V)$ be a representation of Galois type. Then the $L$-series $L(\rho, s)$ is an essentially algebraic entire function on $S_{\infty}$.

![](attachments/2023-03-16-76.png)


## Special values

- Let sgn be our fixed sign function. Let $I \subseteq \mathbf{A}$ be an ideal. From Hayes' theory (cf. Th. 7.4.8 and Cor. 4.9.5.2) we can always find a sgn-normalized Hayes module $\psi$ with lattice $\Lambda=\xi \cdot I$, for some $\xi \in C_{\infty}^{*}$. Thus
$$
e_{\Lambda}(z)=e_{\psi}(z)=z \prod_{0 \neq \alpha \in \Lambda}(1-z / \alpha),
$$
and is an entire $\FF_{r}$-linear mapping of $\mathbf{C}_{\infty}$ to itself. As $\psi$ can be defined over the A-integers $\mathcal{O}^{+}$of $\mathrm{H}^{+}, e_{\psi}(z)$ has coefficients in $\mathrm{H}^{+}$.

We now assume to begin with, for simplicity of exposition, that $\infty$ is rational $\left(d_{\infty}=1\right)$.


- Lemma 8.18.1. Let $j$ be a positive integer with $(r-1) \mid j$. Then
$$
\xi^{-j} \sum_{\substack{i \in I \\ i \text { monic }}} i^{-j} \in \mathrm{H}^{+},
$$
where $\mathbf{H}^{+}=\mathbf{H}$ as $d_{\infty}=1$.

- **Theorem** 8.18.6. Let $j$ be a positive integer divisible by $r^{d_{\infty}}-1$. Then
$$
\xi^{-t j} \zeta_{\OO_{L}} (j)
$$
is non-zero, contained in the maximal totally-real subfield of $\mathbf{k}(B)$, and
$$
\left(\xi^{-t j} \zeta_{O_{L}}(j)\right)^{2} \in \mathbf{H}
$$


- First of all, Yu's general result on periods [Yu1], together with Theorems 8.18.3 and 8.18.6, establishes the transcendence over $\mathbf{k}$ of zeta values at positive integers divisible by $r^{d_{\infty}}-1$. 
- Now let $\mathbf{A}=\FF_{r}[T]$ in which case much more can be said. Indeed, let $\xi$ be the period of the Carlitz module. Then using the special points on the higher tensor powers of the Carlitz module found by Anderson and Thakur [AT1], Jing Yu [Yu3] established the following remarkable result which goes far beyond classical theory.

- Theorem 8.18.8 (Yu). Let $\mathbf{A}=\FF_{r}[T]$. Let $\zeta(s)$ be the zeta function of $\mathbf{A}$ (as a function on $S_{\infty}$ ).
	1. We have $\zeta(i)$ is transcendental over $\mathbf{k}$ for all positive $i$.
	2. Let $i$ now not be divisible by $r-1$. Then $\zeta(i) / \xi^{i}$ is also transcendental over $\mathbf{k}$.
	3. ( $v$-adic version) Let $i$ be as in Part 2 and let $\zeta_{v}\left(\mathbf{A}, e_{v}\right)$ be the $v$-adic interpolation as in Subsection 8.6 (there is no non-trivial $\sigma$ in this case). Then $$\zeta_v(\mathbf{A}, 1, i, i) = =\sum_{j=0}^{\infty}\left(\sum_{\substack{\text { monic } \\ v\{n \\ \operatorname{deg} n=j}} n^{-i}\right)$$ is transcendental over $\mathbf{k}$

## Functional equation

![](attachments/2023-03-16-82.png)

- In classical number theory, Euler discovered the functional equation for the Riemann zeta function $\zeta(s):=\zeta_{0}(s)$ by computing the values of $\zeta(s)$ at both positive even and negative odd integers and then noting that both expressions can be described in terms of Bernoulli numbers. See [Ay1] for a very readable account of this.

![](attachments/2023-03-16-83.png)

- recall it. The reader should note, of course, that a very similar phenomenon takes place with arbitrary Dedekind zeta functions (as well as $L$-functions of more general motives). Thus let $\Gamma(s)$ be Euler's gamma function and set
$$
\xi(s):=\Gamma(s / 2) \pi^{-s / 2} \zeta(s) .
$$
Then, as is universally known, $\xi(s)$ has a meromorphic continuation to $\mathbf{C}$ with simple poles at $s=0$ and $s=1$, and
$$
\xi(s)=\xi(1-s) .
$$
This is the symmetric form of the functional equation. A non-symmetric form states that
$$
\zeta(s)=\pi^{s-1 / 2} \zeta(1-s) \Gamma\left(\frac{1-s}{2}\right) / \Gamma\left(\frac{s}{2}\right) ;
$$
various other forms are possible using standard $\Gamma$-identities. Finally if one wants to work with entire functions, as opposed to meromorphic ones, simply set
$$
\hat{\xi}(s):=s(1-s) \xi(s) ;
$$
so $\hat{\xi}(s)$ obviously satisfies the same functional equation as $\xi(s)$.

![](attachments/2023-03-16-84.png)

- Remark. 8.19.1. Let $\mathbf{k}$ be our base function field, as usual, and let $\zeta_{k}(s)$ be its classical (complex) arithmetic zeta function in the sense of Artin and Weil. So
$$
\zeta_{k}(s)=\frac{P_{k}\left(r^{-s}\right)}{\left(1-r^{-s}\right)\left(1-r^{1-s}\right)},
$$
where $P_{\mathbf{k}}(u) \in \mathbb{Z}[u]$ has degree $2 g, g=$ genus of $\mathbf{k}$. As is very well-known, $P_{k}(u)$ satisfies the functional equation
$$
P_{k}(u)=r^{g} u^{2 g} P_{k}\left(\frac{1}{r u}\right) \text {. }
$$
which, as the reader may readily see, endows $\zeta_{k}(s)$ with a functional equation under $s \mapsto 1-s$. Set
$$
\xi_{\mathrm{k}}(s):=r^{s(g-1)} \zeta_{\mathrm{k}}(s) \text {; }
$$
Then one verifies directly that

![](attachments/2023-03-16-85.png)

- $$
\xi_{\mathbf{k}}(s)=\xi_{\mathbf{k}}(1-s)
$$
If one wants to deal with "entire" functions, one sets
$$
\begin{aligned}
\hat{\xi}_{\mathbf{k}}(s): & =\frac{r^{s(g-1)} \zeta_{\mathbf{k}}(s)}{r^{-s} \zeta_{\mathbf{p} 1}(s)} \\
& =r^{s g} P_{\mathbf{k}}\left(r^{-s}\right) .
\end{aligned}
$$
We now return to studying the zeta function
$$
\zeta_{\mathrm{A}}(s)=\sum_{I \subseteq \mathbf{A}} I^{-\theta}
$$
as before. The special values of $\zeta_{\mathbf{A}}(s)$ at the integers are similar to those of $\zeta_{\mathbb{Q}}(s)$. For instance, as $\zeta_{\mathbf{A}}(s)$ is essentially algebraic, we see immediately that
$$
\zeta_{\mathbf{A}}(-j) \in \mathcal{O}_{\mathbf{V}}
$$
where $\mathbf{V}$ is the value field, $\mathcal{O}_{\mathbf{V}}$ its $\mathbf{A}$-integers, and $j \geq 0$. Moreover, as in our last subsection, the values of $\zeta_{\mathrm{A}}(s)$ at the positive integers satisfy a form of Euler's Theorem. However, all attempts to link these two types of special values in the classical fashion of $\zeta_{\mathbf{Q}}(s)$ have so far been unsuccessful.

![](attachments/2023-03-16-86.png)

- In this subsection we will establish a "functional equation" for the special values of $\zeta_{\mathbf{A}}(s)$ at the positive integers which are divisible by $r^{d_{\infty}}-1$. Recall that we set
$$
\mathbf{T}:=\mathbf{H}^{+} \cdot \mathbf{V}
$$
Let $\psi$ be a sgn-normalized rank one Drinfeld module defined over the Aintegers $\mathcal{O}^{+}$of $\mathbf{H}^{+}$. Without loss of generality we can and shall assume that the lattice $\Lambda$ of $\psi$ is $\xi$. A.

For each ideal $I \subseteq A$ we have the $\mathbf{F}_{r}$-linear polynomial $\psi_{I}(\tau)$ which is monic with roots $\psi[I] \subset \mathbf{C}_{\infty}$. Let $J$ be another ideal. Then by 7.6.1 we know that
$$
\begin{aligned}
D \psi_{I J} & =D(J * \psi)_{I} D \psi_{J} \\
& =D \psi_{I}^{\sigma J} \cdot D \psi_{J},
\end{aligned}
$$
where $\sigma_{J}$ is the Artin symbol at $J$ and the action is written multiplicatively.
Lemma 8.19.2. The element $I^{s_{1}} / D \psi_{I}$ is a unit in the ring of $\mathbf{A}$-integers of T for all integers I of $\mathbf{A}$.

Measure
![](attachments/2023-03-16-87.png)

- **Definition** 8.22.11. An $\mathbf{A}_{v}$-valued measure (or $v$-adic measure) on $\mathbf{A}_{v}$ is a finitely additive $A_{v}$-valued function on the compact open subsets of $\mathbf{A}_{v}$.
Let $\mu$ be one such measure and let $f(t): \mathbf{A}_{v} \rightarrow \mathbf{A}_{v}$ be continuous. By using step-function approximations, one can form the integral $\int_{\mathbf{A}_{v}} f(t) d \mu(t)$. If $f(t)=\sum_{i=0}^{\infty} c_{i} G_{i}(t)$, then
$$
\int_{A_{v}} f(t) d \mu(t)=\sum_{i=0}^{\infty} c_{i}\left(\int_{A_{v}} G_{i}(t) d \mu(t)\right) .
$$
Thus $\mu$ is uniquely determined by the collection $\left\{\int_{A_{v}} G_{i}(t) d \mu(t)\right\} \subset \mathbf{A}_{v}$. Conversely, any such collertion $\mu=\left\{b_{i}\right\}$ can easily be seen to define a unique $\boldsymbol{v}$-adic measure.

![](attachments/2023-03-16-88.png)

- Recall now the algebra of divided power series, $\mathbf{A}_{v}\{\{z\}\}$, consisting of all formal sums $\sum_{i=0}^{\infty} c_{i} \frac{z^{i}}{i !},\left\{c_{i}\right\} \subset \mathbf{A}_{v}$, with, as usual,
$$
\frac{z^{i}}{i !} \cdot \frac{z^{j}}{j !}=\left(\begin{array}{c}
i+j \\
i
\end{array}\right) \frac{z^{i+j}}{(i+j) !}
$$
As $\left(\begin{array}{c}i+j \\ i\end{array}\right) \in \mathbb{Z}$, this definition works in any characteristic.
To the measure $\mu=\left\{b_{i}\right\}$ one assigns the divided power series $f_{\mu}(z)$ with
$$
f_{\mu}(z):=\sum_{i=0}^{\infty} b_{i} \frac{z^{i}}{i !} \text {. }
$$
Remarkably, the mapping $\mu \mapsto f_{\mu}(z)$ is readily checked to be an isomorphism between the algebra of measures with the convolution product (in the usual sense), and the algebra of divided power series.
Recall that $z(x,-j)$ is defined by
$$
z(x,-j)=\sum_{h=0}^{\infty}\left(\sum_{\substack{n \text { monic } \\ \text { deg } n=h}} n^{j}\right) x^{-h},
$$
where the sum in parentheses vanishes for $h \gg 0$; thus
$$
z(1,-j)=\zeta_{\mathrm{A}}(-j) .
$$
![](attachments/2023-03-16-89.png)

- It is very simple to give a $v$-adic measure $\mu_{x}$ on $\mathbf{A}_{v}$ such that
$$
\int_{\mathbf{A}_{v}} t^{j} d \mu_{x}(t)=z(x,-j), \quad j \geq 0 .
$$
As $G_{i}(t)$ is a polynomial in $t$, and so one can compute $\int G_{i}(t)$ in terms of $\left\{\int t^{j}\right\}$, one sees that $\mu_{x}$ is unique. Finally, we have the wonderful computation by Thakur [Th3] of $f_{\mu_{x}}(z)$ which we present in our next result.

Real zeros
![](attachments/2023-03-16-90.png)

- Of course, the classical Riemann hypothesis (Riemann's Riemann hypothe sis!) is that the zeroes of $\hat{\xi}(s)$ are of the form $\frac{1}{2}+i \beta, \beta \in R$. This may be reformulated in the following well-known fashion. First of all, set $t:=s-1 / 2$ and
$$
\omega(t):=\hat{\xi}(s)=\hat{\xi}(t+1 / 2) .
$$
As $1-s=1 / 2-t$, we see
$$
\omega(t)=\hat{\xi}(s)=\hat{\xi}(1-s)=\hat{\xi}(-t+1 / 2)=\omega(-t) .
$$
Upon expanding $\omega(t)$ about $t=0$, we find
$$
\omega(t)=\sum_{j=0}^{\infty} e_{j} t^{j}
$$
with $\left\{e_{j}\right\} \subset \mathbf{R}$ and $e_{j}=0$ for $j$ odd. Thus the Riemann hypothesis now becomes the statement that the zeroes of $\omega(t)$ are of the form $i \beta, \beta \in \mathbf{R}$. Finally we put $u:=i t$, and $\theta(u):=\omega(t)=\omega(u / i)=\omega(-i u)$. Thus
$$
\theta(u)=\sum_{j=0}^{\infty} e_{2 j}(-1)^{j} u^{2 j} .
$$
Clearly $\left\{(-1)^{j} e_{2 j}\right\} \subset \mathbf{R}=\mathbf{Q}_{\infty}$ and the Riemann hypothesis is now the statement that the zeroes of $\theta(u)$ are real. Similar statements can be made, e.g., for Dedekind zeta functions of arbitrary number fields. We shall see a little later that this formulation of the Riemann hypothesis will be echoed in the theory of $\zeta_{A}(s)$.

![](attachments/2023-03-16-92.png)

- We now present a previously unpublished conjecture of J.-P. Serre related to the simplicity of zeroes; we thank him for his kind permission to quote it. In it, we shall say that a zero $s$ of an $L$-series is "non-trivial" if and only if $\mathfrak{R}(s)>0$.

Conjecture 8.24.1 Let $G:=\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbb{Q})$ and let $\chi$ be an irreducible character of $G$.
1. All the non-trivial zeroes of $L(\chi, s)$ should be simple.
2. $s=1 / 2$ should be a zero of $L(\chi, s)$ only if $\chi$ is real and the functional equation of $L(\chi, s)$ has a minus sign.
3. If $\chi$ and $\chi^{\prime}$ are distinct, then the zeroes of $L(\chi, s)$ not equal to $s=1 / 2$ should be distinct from those of $L\left(\chi^{\prime}, s\right)$.
![](attachments/2023-03-16-91.png)

- **Theorem** 8.24.5 (Wan). Let $y \in \mathbf{Z}_{p}$ have the property that each $r$-adic digit of $-y$ is less than $p$ or equal to $r-1$. Then the zeroes of $\zeta_{\mathrm{A}}(x, y)$ are in $\mathbf{K}$ and are simple.

Corollary 8.24.6. Let $r=p$. Then all zeroes of $\zeta_{\mathrm{A}}(s)$ are in $\mathbf{K}$ and are simple.
![](attachments/2023-03-16-93.png)

- we deduce that each side of the Newton polygon has projection onto the $x$ axis of length 1. Thus for each slope there is one root of $\zeta_{\mathrm{A}}(x, 1)$ associated to it. Moreover, this root must be in $\mathbf{K}$ and be simple! As $\left\{\tilde{s}_{j}\right\} \subset \mathbf{K}=\mathbf{k}_{\infty}$, the analogy with the classical theory, and conjecture, as presented above is quite clear!