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date: 2023-03-13 00:19
aliases: ["Untitled"]
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Last modified: `=this.file.mday`

<www-personal.umich.edu/~asnowden/seminar/2017/drinfeld/>

# 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 $\zeta_A(s) \da \qty{1 - \Norm(\infty)^{-s} }\zeta_k(s)$
![](attachments/2023-03-13class.png)

![](attachments/2023-03-13ff.png)![](attachments/2023-03-13analog.png)

# 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)

![](attachments/2023-03-13lfn.png)

Exponentiating ideals:

![](attachments/2023-03-13ideal.png)
![](attachments/2023-03-13dicih.png)

Converges on some half-plane of $S_\infty$.

![](attachments/2023-03-13entire.png)

![](attachments/2023-03-13cms.png)

![](attachments/2023-03-13divisib.png)

![](attachments/2023-03-13mainconj.png)

![](attachments/2023-03-13zeta.png)

![](attachments/2023-03-13functional.png)

![](attachments/2023-03-13asdasd.png)
![](attachments/2023-03-13entire-1.png)

![](attachments/2023-03-13gemoburn.png)

Real:
![](attachments/2023-03-13real.png)