## L-functions and zeta functions

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 ${ℚ}_{l}$ sheaf on ${X}_{0}$
3. $\left(X,ℱ\right)$ obtained from $\left({X}_{0},{ℱ}_{0}\right)$ by extension of scalars to ${\stackrel{_}{𝔽}}_{q}$ $F:X→X and F*: F*ℱ→ℱ$ the associated Frobenius maps.
Let
1. $|{X}_{0}|$ be the set of closed points in ${X}_{0}$
2. $k\left(x\right)$ the residue field of ${X}_{0}$ in $X,$ and
3. $\mathrm{deg}\left(x\right)=\left[k\left(x\right):{𝔽}_{q}\right]$

The L-function of $\left({X}_{0},{ℱ}_{0}\right)$ is $Z\left({X}_{0},{ℱ}_{0},t\right)\in {ℚ}_{l}\left[\left[t\right]\right]$ given by $Z(X0,ℱ0,t) = ∏x∈|X0| 1 det( 1-Fx* tdeg(x), ℱ0 ) .$

The Hasse-Weil zeta function of ${X}_{0}$ is the L-function of $\left({X}_{0},{ℚ}_{l}\right)$ where ${ℚ}_{l}$ is the constant sheaf on ${X}_{0},$ $Z(X0;t) = ∏x∈|X0| 1 1-tdeg(x) and ζX0(s) = Z(X;q-s)$ so that $ζX0(s) = ∏x∈|X0| 1 1-N(x)-s where N(x) = Card(k(x)).$

The Riemann zeta function is ${\zeta }_{{X}_{0}}\left(s\right)$ for ${X}_{0}=\mathrm{Spec}\left(ℤ\right),$

The generalised Grothendieck-Lefschetz formula is $∑x∈XFn Tr(Fx*n,ℱx) = ∑i(-1)i Tr(F*n,Hci(X,ℱ)) (GL)$ which, when $n=1$ and $ℱ={ℚ}_{l},$ the constant sheaf, is the classical Grothendieck-Lefschetz formula $Card(XF) = ∑i(-1)i Tr(F*,Hci(X,ℚl)).$

Taking the logarithmic derivative of $Z\left({X}_{0},{ℱ}_{0},t\right),$ $t d dt logZ(X0,ℱ0,t) =dfn t d dt Z(X0,ℱ0,t) Z(X0,ℱ0,t) = ∑n∈ℤ>0 ∑ x∈XFn = X0(𝔽qn) Tr(Fx*n,ℱ0)tn. (A)$

For a linear transformation $F:V\to V$ on a vector space $V,$ $t d dt log 1 det(1-Ft,V) = ∑n∈ℤ>0 Tr(Fn,V)tn. (C)$ Substituting (GL) in (A) and using (C) gives $t d dt logZ(X0,ℱ0,t) = ∑i (-1)it d dt log( 1 det(1-F*t,Hci(X,ℱ)) )$ so that $Z(X0,ℱ0;t) = ∏i ( det(1-F*t,Hci(X,ℱ)) ) (-1)i+1 .$

## Notes and References

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

References?