L-functions and zeta functions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 March 2012

L-functions and zeta functions

Let

1. $X_0$ be an algebraic variety over ${𝔽}_q$
2. ${ℱ}_0$ a ${\mathbb{Q}}_l$ sheaf on $X_0$
3. $(X,ℱ)$ obtained from $(X_0,{ℱ}_0)$ by extension of scalars to ${\stackrel{\_}{𝔽}}_q$ $$F:X\to Xand{F}^{*}:F^{*}ℱ\to ℱ$$ the associated Frobenius maps.

Let

1. $\left|X_0\right|$ be the set of closed points in $X_0$
2. $k\left(x\right)$ the residue field of $X_0$ in $X,$ and
3. $\mathrm{deg}\left(x\right)=[k\left(x\right):{𝔽}_q]$

The L-function of $(X_0,{ℱ}_0)$ is $Z(X_0,{ℱ}_0,t)\in {\mathbb{Q}}_l\left[\left[t\right]\right]$ given by $$Z(X_0,{ℱ}_0,t)=\prod _{x\in \left|X_0\right|}\frac{1}{\det (1-F_x^{*}{t}^{\mathrm{deg}\left(x\right)},{ℱ}_0)}.$$

The Hasse-Weil zeta function of $X_0$ is the L-function of $(X_0,{\mathbb{Q}}_l)$ where ${\mathbb{Q}}_l$ is the constant sheaf on $X_0,$ $$Z(X_0;t)=\prod _{x\in \left|X_0\right|}\frac{1}{1-t^{\mathrm{deg}\left(x\right)}}and{\zeta }_{X_0}\left(s\right)=Z(X;q^{-s})$$ so that $${\zeta }_{X_0}\left(s\right)=\prod _{x\in \left|X_0\right|}\frac{1}{1-N{\left(x\right)}^{-s}}whereN\left(x\right)=\mathrm{Card}\left(k\left(x\right)\right).$$

The Riemann zeta function is ${\zeta }_{X_0}\left(s\right)$ for $X_0=\mathrm{Spec}\left(\mathbb{Z}\right),$ $$\zeta \left(s\right)=\prod _{\genfrac{}{}{0ex}{}{p\in {\mathbb{Z}}_{>0}}{p \; prime}}\frac{1}{1-p^{-s}}.$$

The generalised Grothendieck-Lefschetz formula is $$\begin{array}{ll}\sum _{x\in X^{F^n}}\mathrm{Tr}(F_x^{*n},{ℱ}_x)=\sum _i{(-1)}^i\mathrm{Tr}(F^{*n},H_c^i(X,ℱ))& (GL)\end{array}$$ which, when $n=1$ and $ℱ={\mathbb{Q}}_l,$ the constant sheaf, is the classical Grothendieck-Lefschetz formula $$\mathrm{Card}\left(X^F\right)=\sum _i{(-1)}^i\mathrm{Tr}(F^{*},H_c^i(X,{\mathbb{Q}}_l)).$$

Taking the logarithmic derivative of $Z(X_0,{ℱ}_0,t),$ $$\begin{array}{rcll}t\frac{d}{dt}\log Z(X_0,{ℱ}_0,t)& \underset{d{f}_n}{=}& \frac{t\frac{d}{dt}Z(X_0,{ℱ}_0,t)}{Z(X_0,{ℱ}_0,t)}& \\ & =& \sum _{n\in {\mathbb{Z}}_{>0}}\sum _{x\in X^{F^n}=X_0\left({𝔽}_{q^n}\right)}\mathrm{Tr}(F_x^{*n},{ℱ}_0)t^n.& (A)\end{array}$$

For a linear transformation $F:V\to V$ on a vector space $V,$ $$\begin{array}{ll}t\frac{d}{dt}\log \frac{1}{\det (1-Ft,V)}=\sum _{n\in {\mathbb{Z}}_{>0}}\mathrm{Tr}(F^n,V)t^n.& (C)\end{array}$$ Substituting (GL) in (A) and using (C) gives $$t\frac{d}{dt}\log Z(X_0,{ℱ}_0,t)=\sum _i{(-1)}^{i}t\frac{d}{dt}\log \left(\frac{1}{\det (1-F^{*}t,H_c^i(X,ℱ))}\right)$$ so that $$Z(X_0,{ℱ}_0;t)=\prod _i{\left(\det (1-F^{*}t,H_c^i(X,ℱ))\right)}^{{(-1)}^{i+1}}.$$

Notes and References

These notes are part of the processing of §1 of Deligne's paper Weil I.

References

References?

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