--- date: 2023-03-13 00:19 aliases: ["Untitled"] --- Last modified: `=this.file.mday` # Ch 7 - Setup: $k$ a global field over $\FF_r$, $A$ the subring of functions regular away from $\infty$ - Drinfeld/$T\dash$module structure: a representation of $A$ as a ring of operators on $\GG_a^d$ for some $d$. - Shtuka: proper model of this action; locally free sheaves on the complex curve $X$ correspond to $k$. - Zeta functions: $\Norm(x) \da \size(\OO_{X ,x}/ \mfm_x)$ and set $\zeta_X(s) \da \prod_{x\in \abs{X}} (1 - \Norm(x)^{-s})\inv$. - In terms of divisors, $\zeta_k(s) = \sum_{D\in \Div(X), D > 0} \Norm(D)$ where $\Norm(D) \da r^{\deg D}$. - Then $Z_k(u) = { p_k(u) \over (1-u)(1-ru)}$ for $u\da r^{-s}$, where $p_k(1)$ is the class number of $k$ and there is a functional equation $p_k(u) = r^g u^{2g} p_k\qty{1\over ru}$. - Complete to obtain $\zeta_A(s) \da \qty{1 - \Norm(\infty)^{-s} }\zeta_k(s)$ - We set $$ \zeta_{A}(s):=\left(1-N \infty^{-s}\right) \zeta_{k}(s) $$ and $$ Z_{\mathrm{A}}(u):=\left(1-u^{d_{\infty}}\right) Z_{\mathbf{k}}(u) $$ It is thus easy to see that $$ Z_{\mathrm{A}}(u)=P_{\mathrm{A}}(u) /(1-r u) $$ where $P_{\mathrm{A}}(u):=\left(1+u+\cdots+u^{d_{\infty}-1}\right) P_{k}(u)$. Thus $$ P_{\mathbf{A}}(1)=d_{\infty} h(\mathbf{k})=h(\mathbf{A}) $$ where $h(\mathbf{A})$ is the class number of $\mathbf{A}$ as a Dedekind domain. - Let sgn continue to be our fixed sign function. Let $\pi$ be our fixed positive uniformizer in $\mathbf{K} \subset \mathrm{C}_{\infty}$. Let $x \in \mathbf{K}^*$, then $x$ can be written uniquely as $$x=\operatorname{sgn}(x) \pi^j u$$where $j \in \mathbb{Z}$ and $u \in \mathbf{K}^*$ is a 1 -unit. - **Definition** 7.9.1. We set $\langle x\rangle:=\langle x\rangle_{\infty}:=\langle x\rangle_\pi:=u$. - Lemma 7.9.2. $\langle x y\rangle=\langle x\rangle\langle y\rangle$. Thus $x$ is written uniquely as $$ \begin{gathered} x=\operatorname{sgn}(x) \pi^j\langle x\rangle \\ \text { where } j=v_{\infty}(x)=-\operatorname{deg}(x) / d_{\infty}, d_{\infty}=\left[\mathbb{F}_{\infty}: \mathbf{F}_r\right] \end{gathered} $$ - Remark. 7.9.3. There are somewhat analogous decompositions in classical Archimedean theory. In $\mathbf{R}^{*}$ we have $$ x= \pm|x|, $$ and in $\mathbb{C}^{*}$ we have $$ x=|x| e^{i \theta} $$ for $\theta=\arg (x)$. Note, of course, that $\left|e^{i \theta}\right|=1$. # Ch 8 $L_V(s) \da \prod_{w \in \abs X} f_w(\Norm(w)^{-s})\inv$ where $f_w(u) \da \det \qty{\id - \Frob^{\mathrm{geom}}_w u \mid V^{I_w}}$ where $I_w$ is the decomposition group in $\Gal(k^\sep/k)$ and $V$ is e.g. $H^1_\et(Z; \QQladic)$ where $Z/k$ is a abeian variety. Exponentiating: ![](attachments/2023-03-13exponen.png) - **Definition** 8.1.2. - 1. We set $$S_{\infty}:=\mathbf{C}_{\infty}^{*} \times \mathbb{Z}_{p}$$We make $S_{\infty}$ into a topological group in the obvious fashion with group action written additively. - 2. If $\alpha \in K^{*}$ is positive and $s=(x, y) \in S_{\infty}$, then we set $$ \alpha^{s}:=x^{\operatorname{deg}(\alpha)}\langle\alpha\rangle^{y}=x^{-d_{\infty} v_{\infty}(\alpha)}\langle\alpha\rangle^{y} $$ - We note that $\langle\alpha\rangle^{y}=\sum_{j=0}^{\infty}\left(\begin{array}{l}y \\ j\end{array}\right)(\langle\alpha\rangle-1)^{j}$ converges precisely because $\langle\alpha\rangle$ is a 1-unit. - Before passing on to more general theory, just to show the reader that the functions defined in this section are truly easily understood, we point out that we are now already able to define the zeta function of $\mathbf{A}=\mathbf{F}_{r}[T]$. Indeed, this function is simply given as $$ \zeta_{\mathrm{A}}(s):=\sum_{a \in \mathrm{A} \text { monic }} a^{-s}, $$ where the convergence, etc., will be discussed in later subsections. This function is the prototype for all the $L$-series we will eventually define. ## Exponentiating ideals - **Definition** 8.2.5. Let $I \subseteq$ A be a fractional ideal and $s=(x, y) \in S_{\infty}$. We then set $$ I^{*}:=x^{\operatorname{deg} I}\langle I\rangle^{y} $$ where $\langle ?\rangle$ is the canonical extension to $\mathcal{I}$ of $\langle ?\rangle: \mathcal{P}^{+} \rightarrow U_{1}$. - Clearly $$ (I J)^{s}=I^{s} J^{s} $$ and $$ I^{s+t}=I^{s} I^{t} $$ for $s, t \in S_{\infty}$. Moreover, if $I=(i)$ with $i$ positive, then $$ I^{s}=i^{s} \text {. } $$ More generally, let $I=(i)$ with $i$ arbitrary. Write $$ i=\operatorname{sgn}(i) \pi^{j}\langle i\rangle . $$ - **Proposition** 8.2.6. Let $s=(x, y) \in S_{\infty}$ and $I=(i)$ as above. Then $$ I^{s}=x^{-j d_{\infty}}\langle i\rangle^{y} \text {. } $$ ## 8.7. Formal Dirichlet Series - 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 following form a a **Dirichlet series**: $$L(s):=\sum_{I \subseteq \mathbf{A}} c(I) I^{-s}.$$ This converges on some half-plane of $S_\infty$. - **Definition** 8.9.1. 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.). We set $$ L(\chi, s):=\prod_{p \text { prime of } 0}\left(1-\chi(\mfp) n \mfp^{-s}\right)^{-1} $$ As before $L(\chi, s)$ converges on a half-plane of $S_{\infty}$ and can be written $$ \sum_{I \subseteq 0} \chi(I) n I^{-s} $$ where $I$ is an ideal of $\mathcal{O}$ and $\chi(I)$ is defined multiplicatively. When $\chi \equiv 1$, then $L(\chi, s)=\zeta o(s)$, etc. - **Theorem** 8.9.2 (Main result). $L(\chi, s)$ is entire in the sense of Subsection 8.5. - 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. - 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 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. - The basic flow of the proof of **Theorem** 8.17.1 is the following: We use Brauer induction to get a meromorphic continuation; then we use Weil's result for the collection $\left\{\rho \otimes \omega_{\sigma, L}^{j}\right\}$, where $j>0$ and $\mfp$ runs over a certain infinite list of primes, to conclude holomorphy at the positive integers; finally we use strong continuity to conclude the result. - Brauer induction is, of course, an essential tool of the classical theory of Artin $L$-series also. - Moreover, the fundamental "yoga" of Iwasawa theory stipulates that the Main Conjecture is the cyclotomic (in the sense of algebraic numbers) version of Weil's results. - **Theorem** 8.18.8 (Remarkable, far beyond classical theory) $(\mathbf{Y u})$. Let $\mathbf{A}=\mathbb{F}_{r}[T]$. 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 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 $$\zeta_v(\mathbf{A}, 1, i, i)=\sum_{j=0}^{\infty} \qty{\sum_{\substack{n \text { monic } \\ v \nmid n \\ \operatorname{deg} n=j}} n^{-i}}$$is transcendental over $\mathbf{k}$. - As we will need the functional equation for $\zeta(s)$ for comparison, we now 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 $\xi(s)$. - $$ P_{k}(u)=r^{9} u^{2 g} P_{k}\left(\frac{1}{r u}\right) \text {, } $$ which, as the reader may readily see, endows $\zeta_{k}(s)$ with a functional equation under $s \mapsto 1-s$. Set $$ \xi_{k}(s):=r^{s(g-1)} \zeta_{k}(s) \text {; } $$ Then one verifies directly that $$\xi_{k}(s)=\xi_{k}(1-s)s$$ - 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 $$ \zeta \mathrm{A}(s)=\sum_{I \subseteq A} I^{-s} $$ as before. The special values of $\zeta_{\mathrm{A}}(s)$ 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 $\zeta_{A}(s)$ 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 $\zeta_{\mathrm{Q}}(s)$ have so far been unsuccessful. We, therefore, see that the arithmetic/geometric dichotomy of subsection 8.15 also carries over to special values. We have called the special values at negative integers arithmetic Bernoulli-elements. Since the values at the positive integers arise from exponentials of Drinfeld modules, we call them geometric Bernoulli-elements. ## Converting to real zeros - Set $$ \hat{\xi}(s):=s(1-s) \xi(s) . $$ Thus $\hat{\xi}(s)$ is entire and satisfies the functional equation $$ \hat{\xi}(s)=\hat{\xi}(1-s) . $$ - Of course, the classical Riemann hypothesis (Riemann's Riemann hypothesis!) is that the zeroes of $\hat{\xi}(s)$ are of the form $\frac{1}{2}+i \beta, \beta \in \mathbf{R}$. 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.