--- date: 2023-03-16 12:10 created: 2023-03-27T15:52 updated: 2023-03-27T15:52 --- Last modified: `=this.file.mday` # Background: zeta functions and special values ## Main Points - $L\dash$functions are the easiest formulations of Langlands correspondences, e.g. $L(f, s) = L(E, s)$ where $f$ is a modular (automorphic) form and $E$ is an elliptic curve (Galois rep.) - Conjecture (Langlands) For any continuous representation $$\rho: \operatorname{Gal}(\bar{F} / F) \longrightarrow G L_n(\mathrm{C})$$of the Galois group there is an automorphic representation $\pi \in \Pi(G)$, necessarily unique, such that $\sigma_v(\pi)=\sigma_v(r)$ for all $v$ outside some finite set $S \supseteq S_{\infty}$. In particular $$L_S(s, \pi)=L_S(s, \rho)$$ - Special values are often special numbers, e.g. Bernoulli numbers - Special values have analytic use, e.g. zeta regularization - Residues are often interesting, e.g. BSD and class number formula. - Can define $L\dash$functions for function fields, hope hard conjectures concerning them admit easier solutions on this side. - The values of $\zeta_{A}(s)$ at the positive integers satisfy a form of Euler's **Theorem**: $\zeta(1-2m) \approx B_{2m}$. - 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. - Establishing functional equations uses Gauss sums ## Classical Theory of L functions - Defining $L\dash$functions: introduced by Dirichlet to study primes in APs - Here are some nice properties that we would like $L$-functions to have (but which are often not known to hold). - **Local-to-global formulas**: multiply together local functions to get global function, $L(s) = \prod_{p} L_p(s)$. - **Euler product**. If $a_{m n}=a_m a_n$ whenever $m$ and $n$ are coprime, and moreover the $a_{p^r}$ for $p$ prime and $r \geq 1$ satisfy a suitable recurrence relation, then we can write $$L(X, s)=\prod_{p \text { prime }} \frac{1}{F_p\left(p^{-s}\right)},$$where $F_p \in \mathbf{C}[t]$ is a polynomial of the form $1-a_p t+\cdots$. - **Analytic continuation**. We say that an $L$-series of the form (1.7) has an analytic continuation if there exists a meromorphic function on the whole complex plane that coincides with the given series in its domain of convergence. - **Functional equation**. In many cases, one can define a dual object $X^*$, a completed $L$-function $L(X, s)$ which is a "simple" modification of the original $L(X, s)$ (the product of $L(X, s)$ with certain exponentials and $\Gamma$-functions), an integer $k$ and a complex number $\epsilon(X)$ such that (assuming analytic continuation) $$L(X, s)=\epsilon(X) L\left(X^*, k-s\right)$$ - In the case where $L(X, s)$ admits (or is expected to admit) such a functional equation, the vertical line $\{s \in \mathbf{C} \mid \Re s=k / 2\}$ is called the critical line. - **Riemann hypothesis**. One expects that all the zeroes of the meromorphic function $L(X, s)$, except the well-understood trivial zeroes, lie on the critical line. - $\zeta$ functions for algebraic varieties over $\FF_q$: proved, Weil conjectures, by Deligne. - $\zeta$ functions for number fields: no proof in sight. - **Special values**. E.g. Euler's formula $\zeta(2m)\approx B_{2m}$. - Classical zeta: $$\zeta(s)=\sum_{n \geq 1} \frac{1}{n^s}=\prod_p\left(1-p^{-s}\right)^{-1}$$ - Converges absolutely for $\Re(s) > 1$, simple pole at $s=1$, functional equation, meromorphic continuation. Expect these from general $L$ functions. - Special values are the Bernoulli boys: $$\zeta(2 m)=\frac{(-1)^{m+1}(2 \pi)^{2 m} B_{2 m}}{2(2 m) !}, m \in \mathbb{Z}_{>0}, \quad\implies \zeta(1-2 m)=\frac{B_{2 m}}{2 m},$$ - Can define partial zeta functions: for $S\subseteq \ZZ$, set $$\zeta_S(s) \da \sum_{n\in S} n^{-s}$$ - Useful for **zeta regularization**: $$\sum_{n\geq 1} n \da \lim_{s\to -1}(s+1)^k\zeta(s), \qquad \sum_{a\in S} a \da \lim_{s\to -1} (s+1)^k\zeta_S(s), \qquad k\in \ZZ$$ - Can define "densities" $d(S) \da {\zeta_S(s) \over \zeta(s)} \in \CC(s)$, etc. - Dirichlet $L\dash$functions: - For $\chi$ a mod $N$ Dirichlet character, define $$L(s, \chi)=\sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}=\prod_{p \in \mathcal{P}}\left(1-\chi(p) p^{-s}\right)^{-1}, \quad \operatorname{Re}(s)>1 .$$ - Special values are often transcendental: $$\begin{aligned} & \zeta(2) = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots=\frac{\pi^2}{6} \\ & \zeta(4) = 1+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\cdots=\frac{\pi^4}{90} \\ & 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{4} \\ & 1-\frac{1}{3}-\frac{1}{5}+\frac{1}{7}+\frac{1}{9}-\frac{1}{11}-\frac{1}{13}+\cdots=\frac{\log (1+\sqrt{2})}{\sqrt{2}} \end{aligned}$$ - Dedekind zeta functions: - For $F\in \NF$, define $$\zeta_F(s)=\sum_{\mathfrak{a}\in \Id(\OO_F)}\left|\mathcal{O}_F / \mathfrak{a}\right|^{-s}=\prod_{\mathfrak{p}\in \spec \OO_F\smz}\left(1-\left|\mathcal{O}_F / \mathfrak{p}\right|^{-s}\right)^{-1},$$ - Known meromorphic continuation, simple pole at $s=1$, functional equation. ## Important applications of special values - Important application of special values: the class number formula $$\Res_{s=1} \zeta_F(s) = \frac{2^{r_1}(2 \pi)^{r_2}}{\left|d_K\right|^{1 / 2} w_K } h_F \Reg(F)$$ where - $h_F = \size \Pic(\OO_F)$ is the class number of $F$ - $r_1, r_2$ are the number of real resp. complex places of $F$ - $d_K = \disc(K/\QQ)$ - $w_K = \size \Union_n \mu_n(F)$ is the number of roots of unity in $F$. - $\Reg(F)$ is the **regulator** of $F$, - Generally: $\Reg_{n, F}: \K_n(X) \to \bigoplus_{p\geq 0} H^{2p-n}_D(X, \QQ(p))$ landing in Deligne cohomology - Think of this as a generalized Chern character. - For quadratic fields: $\K_1(F) = \OO_F\units \to \RR^{r}$ where $d\da r_1 + r_2-1$, pick fundamental units $u_1,\cdots, u_r$ and consider $$M_F \da\left(\begin{array}{ccc} N_1 \log \left|\sigma_1\left(u_1\right)\right| & \cdots & N_{r_1+r_2} \log \left|\sigma_{r_1+r_2}\left(u_1\right)\right| \\ \vdots & \ddots & \vdots \\ N_1 \log \left|\sigma_1\left(u_r\right)\right| & \cdots & N_{r_1+r_2} \log \left|\sigma_{r_1+r_2}\left(u_r\right)\right| \end{array}\right) \in \Mat_{r, r+1}(\RR), N_i = 1 \iff \RR, 2 \iff \CC$$ Then $\Reg_K = \abs{\det{M_F^r}}$, any $r\times r$ submatrix. - Volume of a "logarithmic" lattice - Small regulator means $F\units$ is large - Generalized the Dirichlet class number formula for quadratic fields, explicit equations for $L(\chi, 1)$, class number of $K$ can be extracted. - In general, expect regulator maps $r_m: \K_{2m-1}(\OO_F)\to \RR^{d_m}$ whose covolumes are related to special values of $\zeta_F(s)$ (Lichtenbaum, Borel, somewhat worked out in 70s) - $\K(\OO_F)$ carries a lot of interesting data: - $\K_0(\OO_F) = \Pic(F) \oplus \ZZ$ - $\K_1(\OO_F) = \OO_F\units$ - Can rewrite the class number formula as $$\Ord_{s=0}\zeta_F(s) = \lim_{s\to 0} s^{-r}\zeta_F(s) = - {\size \K_0(\OO_F)_\tors \over \size \K_1(\OO_F)_\tors} \cdot \Reg_F$$ - Higher regulators: Lichtenbaum conjecture $$\Ord_{s=-n} \zeta_F(s) \approx \pm {\size \K_{2n}(\OO_F)_\tors \over \size \K_{2n+1}(\OO_F)_\tors} \cdot \Reg_{n, F} \mod 2^{n}, n\gg 0$$ where $\Reg_{n, F}$ are higher regulator maps. - Again useful for zeta regularization: for $F = \QQ$, $\Reg_{n, \QQ} = 1$ and $$\zeta(-1) = \pm {\size \K_2(\ZZ) \over \size \K_3(\ZZ)_\tors} = -{\size C_2\over \size C_{48}} = -{1\over 12}$$ - - BSD: - Conjecture 3.1 (Birch and Swinnerton-Dyer [2], Tate [8]). Let $E$ be an elliptic curve over a number field $K$. Then the following holds: - The function $L(E, s)$ can be continued to an entire function on the complex plane and satisfies the functional equation $$ L(E, s)= \pm L(E, 2-s) . $$ where the sign is +1 or -1 depending on whether $\operatorname{rk} E(K)$ is even or odd, respectively. - The Tate-Shafarevich group $Ш_{E}$ is finite. - We have $$\operatorname{ord}_{s=1} L(E, s)= r \da \operatorname{rk} E(K)$$and the leading term of the power series expansion of $L(E, s)$ at $s=1$ is given by $$L(E, s) = c_0 (s-1)^r + \bigo((s-1)^{r+1}), \qquad c_0 \da \lim _{s \rightarrow 1} {(s-1)^{-\mathrm{rk} E(K)}} L(E, s) =\frac{\# Ш_{E} } {\# E(K)_{\mathrm{tor}}^2}\operatorname{Reg}_{1, E}$$ - Remark: A more standard formulation of the above conjecture also involves the product of the Tamagawa numbers at the finite places, and the product of the periods at the infinite places. Following Tate [8], we have absorbed these into the completed $L$-function $L(E, s)$, ### Special values for curves recover class numbers - We begin by recalling the classical zeta function of $X$ itself. Let - $X_{0}$ be the set of closed points of $X$. - The reader should note that $\infty$ plays no special role in this definition. - For $x \in X_{0}$, let $\mathcal{O}_{x}$ be the associated local ring with maximal ideal $M_{x}$. - Set $$ \mathbf{F}_{x}:=\mathcal{O}_{x} / M_{x},\qquad N x:=\#\left(\mathcal{O}_{x} / M_{x}\right) $$ - **Definition** 7.8.1. Let $s \in \mathbb{C}$. We set $$ \zeta_{X}(s):=\zeta_{k}(s):=\prod_{x \in X_{0}}\left(1-N x^{-s}\right)^{-1} $$ - Let $D$ be a divisor of $X$ and set $$ N D:=r^{\operatorname{deg} D} \implies \zeta_{\mathbf{k}}(s)=\sum_{D\in \Div(X) \text{ positive} } N D^{-s} $$ - Set $u:=r^{-s}$. Then it is very well-known that $$\zeta_{\mathbf{k}}(s)=Z_{\mathbf{k}}(u),\qquad Z_{\mathbf{k}}(u)=\frac{P_{\mathbf{k}}(u)}{(1-u)(1-r u)}$$ - Moreover, if $g=$ genus $X=$ genus of $\mathbf{k}$, then $P_{\mathbf{k}}(u)$ is a polynomial of degree $2 g$ in $u$ with the following (incomplete) list of properties: 1. $P_{\mathbf{k}}(0)=1$. 2. $P_{\mathbf{k}}(1)=h(\mathbf{k})$, where $h(\mathbf{k})$ is the class number of the field $\mathbf{k}$. 3. $P_{k}(u)$ satisfies the functional equation $$ P_{\mathbf{k}}(u)=r^{g} u^{2 g} P_{\mathbf{k}}(1 / r u) $$ - We set $$ \zeta_{\mathrm{A}}(s):=\left(1-N \infty^{-s}\right) \zeta_{k}(s) \qquad Z_{\mathrm{A}}(u):=\left(1-u^{d_{\infty}}\right) Z_{\mathbf{k}}(u) $$ - It is thus easy to see that $$ Z_{\mathbf{A}}(u)={ P_{\mathbf{A}}(u) \over (1-r u)}, \qquad P_{\mathrm{A}}(u):=\left(1+u+\cdots+u^{d_{\infty}-1}\right) P_{k}(u) $$ - Thus $$P_{\boldsymbol{A}}(1)=d_{\infty} h(\mathbf{k})=h(\mathbf{A})$$where $h(\mathbf{A})$ is the class number of $\mathbf{A}$ as a Dedekind domain. # Some progress on the Drinfeld side ## Pellarin L-series For $\theta, t$ independent variables over $\FF_q$, setting $$\omega_C \da \omega_C(t)= (-\theta)^{1\over q-1} \prod_{i\geq 0} \qty{1 - t\over \theta^{q^i}}\inv$$ yields a meromorphic function which converges in the Tate algebra $\TT$ of rigid analytic functions of $t$ on the closed unit disc $\bar{\DD}_\infty \subseteq \CC_\infty$. This came up in the theory of the Carlitz module. **Theorem (Pellarin, 12)**: define $\chi: \FF_q[\theta] \to \FF_q[t]$ by $a\mapsto a(t)$, the evaluation at $t$ map for polynomials $a(\theta)$, and define $$L(\chi, s) \da \sum_{a\in \FF_q[\theta]_+} \chi(a) a^{-s}.$$ Then $$L(\chi, 1) = {\tilde \pi \over (t-\theta)\omega_C}$$ where $\tilde \pi \da - \Res_{t=\theta}( \omega_C)$ is the period of the Carlitz module. - **Theorem (Green-Papanikolas 16)**: let - $A = \FF_q[t, y]$ be the coordinate ring of an elliptic curve - $\AA = \FF_q[\theta, \eta]$ is an isomorphic copy of $\AA$. - $\rho: \AA\to H\ts{\tau}$ be a (particular sign-normalized) rank 1 Drinfeld module over - $H$ the Hilbert class field of $K\da \FF_q(\theta, \eta)$ where - Note that $\rho$ comes a construction of Thakur, Drinfeld, Mumford on the *shtuka function* for $E$. - Define $$L(\AA, s) \da \sum_{\mfa \subseteq A} \chi(\mfa) ( \del\rho_\mfa )^{-s} \in \TT,$$ then $$L(\AA, s)\mid_{\Xi} = h(\AA)$$ where - $\Xi \da (\theta, \eta)\in E(K)$ - $\rho_\mfa\in H[\tau]$ is a certain extension to integral ideals of $\AA$ - $\bd \rho_\mfa$ is the coefficient of $\tau^0$. # The Drinfeld Theory ## The complex plane $S_\infty$ (8.1) - Decomposing $\alpha\in K\units$: let - $\sgn$ be a fixed sign function, - $\pi \in K\units$ be a fixed positive (i.e., $\operatorname{sgn}(\pi)=1$ ) uniformizer, - Then $$\alpha=\operatorname{sgn}(\alpha) \pi^{j}\langle\alpha\rangle \qquad 8.1.1$$where - $j=v_{\infty}(\alpha)$ and - $\langle\alpha\rangle$ is a 1-unit which depends on $\pi$. - So if $\alpha$ is positive, then $$ \alpha=\pi^{j}\langle\alpha\rangle $$ - Fact: $$ s = x+iy\in \CC \implies n^{s}=e^{(x+i y) \log n}=e^{x \log n} \cdot e^{i y \log n},\,\quad \left|n^{s}\right|=\left|e^{x \log n}\right|,\,\, \left|e^{i y \log n}\right|=1 $$ - In the function field case, we now use (8.1.1) to do something similar. - **Definition** 8.1.2. - 1. We set $$S_{\infty}:=\mathbf{C}_{\infty}^{*} \times \ZZpadic$$We make $S_{\infty}$ into a topological group in the obvious fashion with group action written additively. - 2. If $\alpha \in \mathbf{K}^{*}$ is positive and $s=(x, y) \in S_{\infty}$, then we set $$ \alpha^{s}:=x^{\operatorname{deg}(\alpha)}\langle\alpha\rangle^{y}=x^{-d_{\infty} v_{\infty}(\alpha)}\langle\alpha\rangle^{y} \text {. } $$ 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. ## Exponentiating ideals (8.2) - Issue: Dedekind domains are PIDs iff factorial, so for $\ZZ$ we identify $(a)$ with $a$, but we can't generally do this for $A$ and need extend exponentiation to ideals. - Let $\mathcal{I}$ be the group of fractional ideals of $\mathbf{A}$. - Let $\mathcal{P}$ be the subgroup of principal ideals - Let $\mathcal{P}^{+} \subseteq \mathcal{P}$ the subgroup generated by positive elements. - Let $U_{1} \subset \mathbf{K}^{*}$ be the 1-units. - Let $\widehat{U}_{1} \subset \mathbf{C}_{\infty}^{*}$ be the group of 1-units in $\mathbf{C}_{\infty}$. - **Definition** 8.2.5: Exponentiating ideals. - Let $I \subseteq$ A be a fractional ideal and $s=(x, y) \in S_{\infty}$. We then set $$ I^{\theta}:=x^{\operatorname{deg} I}\langle I\rangle^{y}, $$ where $\langle ?\rangle$ is the canonical extension to $\mathcal{I}$ of $\langle ?\rangle: \mathcal{P}^{+} \rightarrow U_{1}$. - Clearly $$(I J)^{s}=I^{a} J^{s},\qquad I^{s+t}=I^{s} I^{t}, \qquad s,t\in S_\infty$$ - Moreover, if $I=(i)$ with $i$ positive, then $$ I^{s}=i^{s} . $$ - 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 $$ I^{s}=x^{-j d_{\infty}\langle i\rangle^{y}} . $$ ## Convergence - **Definition** 8.4.1. Let $C\left(\ZZpadic, \mathcal{O}\right)$ be the space of continuous $\mathcal{O}$-valued functions on $\ZZpadic$. Let $C\left(\ZZpadic, L\right)$ be the space of continuous $L\dash$valued functions. - Note that every $f \in C\left(\ZZpadic, L\right)$ has compact image as $\mathbf{Z}_{p}$ is compact. Thus there exists $\alpha \in L^{*}$ with $\alpha f \in C\left(\ZZpadic, \mathcal{O}\right)$. - Note also that $C\left(\ZZpadic, L\right)$ is an $L r$ vector space and $C\left(\ZZpadic, \mathcal{O}\right)$ is an $\mathcal{O}$-module. Finally $C\left(\ZZpadic, L\right)=C\left(\ZZpadic, \mathcal{O}\right) \otimes_\OO L$. - We can give these function spaces norm as follows: let $f \in C\left(\mathbf{Z}_{p}, L\right)$, then we set $$ \|f\|:=\max _{s \in \mathbf{Z}_{p}}\{|f(s)|\} \text {. } $$ - Both $C\left(\ZZpadic, L\right)$ and $C\left(\ZZpadic, \mathcal{O}\right)$ are complete with respect to $\|f\|$. - **Corollary** 8.4.7. Let $\varphi(x)=\sum a_{k}\left(\begin{array}{l}x \\ k\end{array}\right)$ be a continuous function from $\mathbf{Z}_{p}$ to L. Then $\varphi(x) \in C\left(\ZZpadic, \mathcal{O}\right)$ if and onty if $\left\{a_{k}\right\} \subset \mathcal{O}$ ## Entire functions (8.5) - We recall that $S_{\infty}=\mathbf{C}_{\infty}^{*} \times \ZZpadic$. Let $s=(x, y) \in S_{\infty}$ and let $f(s): S_{\infty} \rightarrow \mathbf{C}_{\infty}$ be a continuous function. we can view $f(s)$ as a continuous family of $\mathbf{C}_{\infty}$-valued functions on $\mathbf{C}_{\infty}^{*}$ or as a continuous family of continuous $\mathbf{C}_{\infty}$-valued functions on $\ZZpadic$. - The prototype of such $f(s)$, as above, is $$ f(x)=\alpha^{s}, \qquad \text{recalling }\alpha^{s}=x^{\operatorname{deg}(\alpha)}\langle\alpha\rangle^{y} \quad \text{for $\alpha \in \mathbf{K}^{*}, \alpha$ positive. } $$ - Note $\alpha^{s}$ is analytic in the $\mathrm{C}_{\infty}^{*}$-variable. - This suggests that we should restrict our attention to continuous functions on $S_{\infty}$ which are analytic in the first variable. - This is almost correct; we also want the roots of these analytic functions to "flow continuously" in the $\ZZpadic$-variable. It is well-known that this is accomplished by requiring uniform continuity on bounded subsets of $\mathbf{C}_{\infty}^{*}$. We are thus lead to the next definition. - **Definition** 8.5.1. We define an entire function on $S_{\infty}$ to be a continuous family of $\mathbf{C}_{\infty}$-valued entire power series in $x^{-1}$ which is parameterized by $\ZZpadic$. Moreover, this family is required to be uniformly convergent on bounded subsets of $\mathbf{C}_{\infty}$. ## Algebraicicity - **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 $\mathbf{k}$. Motivation: - 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. Checking: - 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. - ## Trace of Frob - **Definition** 8.6.1. A **$v$-adic representation** of $G$ is a continuous homomorphism $$ \rho: G \rightarrow \operatorname{Aut}(V) $$ where $V$ is a finite dimensional $\mathbf{k}_{v}$-vector space and $\operatorname{Aut}(V)$ inherits its topology from $\operatorname{End}_{\mathbf{k}_{v}}(V)$. - **Definition** (Trace of Frobenius): - Let $\rho$ be as in Definition 8.6.2 and unramified at $w$. Set $$P_{\rho, w}(u):=\operatorname{det}\left(1-\rho\left(F_{\bar{w}}\right) u \mid V\right)$$ - Notice that this characteristic polynomial also only depends on $w$. - Dependent definitions: - Let $w$ be any place of $L$ and let $\bar{w}$ be a place of $L^{\text {sep }}$ lying over $w$. - Let $D_{\bar{w}}$ and $I_{\bar{w}}$ be the decomposition and inertia groups at $\bar{w}$. - Thus $D_{\bar{w}} / I_{\bar{w}}$ is isomorphic to $\mathrm{Gal}\left(\overline{\FF}_{w} / \mathbf{F}_{w}\right)$ where $\mathbf{F}_{w}$ is the finite field at $w$. - Let $F_{\bar{w}} \in D_{\bar{w}} / I_{\bar{w}}$ be the geometric Frobenius at $\bar{w}$, that is $F_{\bar{w}}$ is the inverse of the automorphism $$ x \mapsto x^{N w} . $$ - Let $\widehat{w}$ be another place of $L^{\text {sep }}$ over $w$. Then $D_{\widehat{w}}, I_{\widehat{w}}$ and $F_{\widehat{w}}$ are all conjugate to $D_{\hat{w}}, I_{\hat{w}}, F_{\hat{w}}$, respectively. - **Definition** 8.6.2. Let $\rho$ he as in Definition 8.6.1. We say that $\rho$ is **unramified** at $w$ if and only if $\rho\left(I_{\bar{w}}\right)$ is the identity. - Note that this notion only depends on $w$ and not on any choice of place above it. ## Norms - Let $L$ be a finite extension of $\mathbf{k}$ with ring of $\mathbf{A}$-integers $\mathcal{O}_{L}$ - Note $\mathcal{O}_{L}$ is finite over A. - If $L$ is separable, then this is due, essentially, to Dedekind. - If $L$ is not separable, let $L^{\prime}$ be its maximal separable subfield. We thus reduce, by Dedekind, to handling totally inseparable extensions. - Without loss of generality, we need only handle the case $L / \mathbf{k}$ totally inseparable. - Let $p^{e}$ be the degree of $L / \mathbf{k}$ and let $\left\{\alpha_{1}, \ldots, \alpha_{m}\right\}$ be generators of $\mathbf{A}$ as $\FF_{r}$-algebra. Then $\mathcal{O}_{L} \subseteq \mathbf{F}_{r}\left[\alpha_{1}^{p^{-e}}, \ldots, \alpha_{m}^{p^{-e}}\right]$ and is also finite. - For each prime $w$ of $L$, let $\mathbf{F}_{w}$ be the corresponding finite field and let $\operatorname{deg} w=\left[\FF_{w}: \FF_{r}\right]$ be its degree over $\FF_{r}$. Set $$ N w:=\# \mathbf{F}_{w} . $$ - If $w$ lies over a prime $v$ of $\operatorname{Spec}(\mathbf{A})$, then we set $\mathbf{F}_{v}:=\mathbf{A} / v$, and $$\begin{aligned}n w & :=v^{\operatorname{deg} w / \operatorname{deg} v} \\& =v^{f_{w}},\end{aligned}$$where $f_{w}=\left[\FF_{w}: \FF_{v}\right]$. - We view $n w$ as an ideal of $\mathbf{A}$ and extend $n$ and $N$ to all fractional $\mathcal{O}_{L}$-ideals in the obvious multiplicative fashion. Clearly, $$N w=N n w,$$so both ways of forming norms are compatible. - Let $\alpha \in L^{*}$ and let $N_{\mathbf{k}}^{L}(\alpha)$ be the usual multiplicative norm. Let $(\alpha)$ be the A-fractional ideal generated by $\alpha$. Then we have $$\left(N_{\mathbf{k}}^{L}(\alpha)\right)=n(\alpha)$$If $L / \mathbf{k}$ is separable, this is standard. - Let $\overline{\mathbf{k}}$ be an algebraic closure of $\mathbf{k}$ which contains $L$ and let $L^{\operatorname{sep}} \subset \overline{\mathbf{k}}$ be the separable closure of $L$. Set $G:=\operatorname{Gal}\left(L^{\text {sep }} / L\right)$. Let $v \in \operatorname{Spec}(\mathbf{A})$ and $\mathbf{k}_{v}$ the completion of $\mathrm{k}$ at $v$. ## L functions of representations - **Definition** 8.6.7 (Important: $L\dash$functions). - Let $\hat{\rho}:=\left(\rho_{v}\right)$ be a *strictly compatible family* of representations with $B$ the set of bad places. - Let $w$ be a finite place of $L$ not in $B$ - Let $P_{\widehat{\rho}, w}(u)$ be the polynomial $P_{\rho_{w}, w}(u)$ for $v$ not equal to $w$. By definition, this polynomial does not depend on the choice of $v$. - Let $s \in S_{\infty}$. Recalling Frobenius traces and norms, we set $$L(\hat{\rho}, s):=\prod_{\substack{w, \text { finite } \\ w \notin B}} P_{\hat{\rho}, w}\left(n w^{-s}\right)^{-1} .$$ - The above definition is not complete in that we would also like to define Euler factors for those primes in $B$. The "obvious" way to proceed is to follow classical theory and associate to a finite prime $w \in S$ the polynomial $$P_{\rho_{v}, w}(u):=\operatorname{det}\left(1-\rho_{v}\left(F_{\bar{w}}\right) u \mid V_{v}^{I_{\bar{w}}}\right),$$where $v$ does not lie under $w$; this polynomial is independent of the choice of prime $\bar{w}$ over $w$. - It seems reasonable to expect that this polynomial has coefficients in $\mathbf{A}$ and is independent of the choice of $v$ when the representation arises from a Drinfeld module or $T$-module. - However, very little is known about these polynomials and we will not pursue them further here. - It is also easy to give a $v$-adic version of 8.6.7. Let $V$ be the value field as in Subsection 8.3. Let $\sigma: \mathrm{V} \rightarrow \overline{\mathbf{k}}_{v}$ be our fixed embedding and let $S_{\sigma, v}$ be as in Definition 8.3.4, etc. - **Definition** 8.6.8 ($v\dash$adic $L\dash$functions). - Let $e_{v}=\left(x_{v}, s_{v}\right) \in C_{v}^{*} \times S_{\sigma, v}$ (cf. Definition 8.3.5). We set $$ L_{\sigma, v}\left(\widehat{\rho}, e_{v}\right):=\prod_{\substack{w \\ w \notin B \\ w \nmid v}} P_{\widehat{\rho}, w}\left(n w^{-e_{v}}\right)^{-1} $$ ## Compatible families - **Definition** 8.6.3. Let $\rho$ be as in 8.6.2. We say that $\rho$ is **rational** (resp. **integral**) if and only if there exists a finite set $B$ of places of $L$ such that the following hold. 1. $\rho$ is unramified at all finite places of $L$ not in $B$. 2. Let $w$ be a finite place of $L$ not in $B$. Then $P_{\rho, w}(u)$ has coefficients in $k$ (resp. A). - **Definition** 8.6.4. Assume that both $\rho$ and $\rho^{\prime}$ are rational. Then $\rho$ and $\rho^{\prime}$ are said to be **compatible** if there is a finite subset $B$ of places of $L$ such that $\rho$ and $\rho^{\prime}$ are unramified at the finite places outside of $B$ and $$P_{\rho, w}(u)=P_{\rho^{\prime}, w}(u)$$ for all such $w$ not in $B$. - **Definition** 8.6.5. Compatible families - For each finite prime $v$ of $\mathbf{A}$, let $\rho_{v}$ be a rational $v$-adic representation. The family $\left(\rho_{v}\right)$ is said to be **compatible** if $\rho_{v}, \rho_{v^{\prime}}$ are compatible for any two primes $v, v_{1}^{\prime}$. - The family $\left(\rho_{v}\right)$ is said to be **strictly compatible** if there exists a finite subset $B$ of places of $L$ such that the following conditions hold:' 1. Let $B_{v}$ be the set of places of $L$ above $v$. Then for every finite $w$ not in $B \cup B_{v}$, the representation $\rho_{v}$ is unramified at $w$ and $P_{\rho_{v}, w}(u)$ has rational coefficients. 2. Let $w \notin B \cup B_{v} \cup B_{v^{\prime}}$. Then $$P_{\rho_{v}, w}(u)=P_{\rho_{v^{\prime}}, w}(u) .$$ - It is easy to see that there is a smallest set $B$ of finite places satisfying 1 and 2 above. We call $B$ the set of **bad places for the family**. - Examples 8.6.6. 1. Let $\rho_{v}=1$ be the trivial character for all $v$. Then $\left(\rho_{v}\right)$ is clearly a strictly compatible, integral, family. ## Convergence Proposition 8.6.9 (Half-planes of convergence) 1. Let $\hat{\rho}$ be the family of trivial representations. Then $L(\widehat{\rho}, s)$ converges for all $\left\{s=\left.(x, y) \in S_{\infty}|| x\right|_{\infty}>1\right\}$. 2. Let $\widehat{\rho}$ arise from a Drinfeld module $\psi$ of rank d. Then $L(\hat{\rho}, s)$ converges for all $$\left\{s=\left.(x, y) \in S_{\infty}|| x\right|_{\infty}>r^{1 / d}\right\}$$ 3. Let $\hat{\rho}$ be as in 1 or 2 . Then $L_{\sigma, v}\left(\hat{\rho}, e_{v}\right)$ converges for all $\left\{e_{v}=\left(x_{v}, s_{v}\right) \mid\right.$ $\left.|x|_{v}>1\right\}$ - Proof. - Part 1 follows easily by definition. - Part 2 follows because of the Riemann Hypothesis for Drinfeld modules over finite fields (see 4.12.8.5). - Part 3 follows because the characteristic polynomials in question all have $\mathbf{A}$ coefficients and so are $v$-adic integers. - Can use a notion of purity to get convergence for $L\dash$functions of $T\dash$modules. - We use the notation " $\zeta_{\mathcal{O}_{L}}(s)$ " for $L(\hat{\rho}, s)$ if $\widehat{\rho}$ is the trivial family and $\mathcal{O}_{L}$ is the ring of $\mathbf{A}$-integers in $L$. - This function is completely analogous the classical Dedekind zeta functions. Indeed by definition we have $$ \zeta O_{L}(s):=\prod_{p}\left(1-n \wp^{-s}\right)^{-1} $$ where $p$ runs over all primes of $\mathcal{O}_{L}$. Thus $$ \zeta_{L}(s)=\sum_{I} n I^{-s}, $$ where now $I$ runs over all non-zero ideals of $\mathcal{O}_{L}$. The $v$-adic interpolation of this function is $$ \zeta_{\sigma, v}\left(\mathcal{O}_{L}, e_{v}\right):=\prod_{v \nmid p}\left(1-n p^{-e_{v}}\right)^{-1} $$ etc., as a function on $\CC_{v}^{*} \times S_{\sigma, v}$. These definitions clearly encompass those presented earlier for $\mathbf{A}=\mathbf{F}_{r}[T]$ and $L=\mathbf{k}$. ## Analytic continuation - **Definition** 8.17.7. - 1. We set $$ g(\rho, x):=\sum_{k=0}^{\infty} L(\rho,-k) \cdot \frac{x^{k}}{k !} . $$ 2. Let $\log (T)=\sum_{n=1}^{\infty}(-1)^{n-1} \frac{(T-1)^{n}}{n}$, as usual. We set $$f(\rho, T):=g(\rho, \log (T))$$as a power series in $T-1$. - **Theorem** 8.17.8. Let $\mathcal{O} \subset L$ be the integers of $L$.. Assume the p-adic Artin Conjecture (which follows from the Main Conjecture of Iwasawa theory) then $f(\rho, T)-f(\rho, 1)$ has $\rho$-adically bounded denominators for all finite primes $p$ of $L$. - (Under the additional hypothesis that Iwasawa's $\mu$-invariant vanishes, one actually obtains that $f(\rho, T)-f(\rho, 1) \in \mathcal{O}[[T-1]]$. - In other words, under the $p$-adic Artin Conjecture, the function $g(\rho, T)$ is $p$-adically analytic in the domain $|T-1|_{p}<1$. - Any connection, in general, between these functions, (or $\operatorname{Spec}(L[[T-1]] / f(\rho, T)$ ), etc.) and the complex analytic theory of $L(\rho, s)$ would be very interesting. - For instance, if $\operatorname{dim} \rho=1$, then $f(\rho, T)$ can be used to complex-analytically continue $L(\rho, s)$. > 8.8 proves that $L(\hat\rho, s)$ is entire for certain families of compatible representations $\hat\rho$. # 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 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** 8.7.3. - Suppose $L(s)$ has an Euler product over the primes of $\mathbf{A}$. Then this Euler product is unique. ## Entireness 8.9 shows that $L\dash$series associated to finite characters are entire: - **Definition** 8.9.1. We set $$ L(\chi, s):=\prod_{\mfp \in \spec \OO}\left(1-\chi(p) n \mfp^{-s}\right)^{-1} . $$ - As before $L(\chi, s)$ converges on a half-plane of $S_{\infty}$ and can be written $$ \sum_{I \subseteq \mathcal{O}} \chi(I) n I^{-s}, $$ where $I$ is an ideal of $\mathcal{O}$ and $\chi(I)$ is defined multiplicatively. - When $\chi \equiv 1$, then $L(\chi, s)=\zeta_{\OO}(s)$, etc. - **Theorem** 8.9.2 (Main result). - $L(\chi, s)$ is entire in the sense of Subsection 8.5.