L-functions and zeta functions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 March 2012

L-functions and zeta functions


  1. X0 be an algebraic variety over 𝔽q
  2. 0 a l sheaf on X0
  3. (X,) obtained from (X0,0) by extension of scalars to 𝔽_q F:XX and F*: F* the associated Frobenius maps.
  1. |X0| be the set of closed points in X0
  2. k(x) the residue field of X0 in X, and
  3. deg(x)=[k(x):𝔽q]

The L-function of (X0,0) is Z(X0,0,t) l[[t]] given by Z(X0,0,t) = x|X0| 1 det( 1-Fx* tdeg(x), 0 ) .

The Hasse-Weil zeta function of X0 is the L-function of (X0,l) where l is the constant sheaf on X0, 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 ζX0(s) for X0 = Spec(), ζ(s) = p>0 p   prime 1 1-p-s .

The generalised Grothendieck-Lefschetz formula is xXFn Tr(Fx*n,x) = i(-1)i Tr(F*n,Hci(X,)) (GL) which, when n=1 and =l, the constant sheaf, is the classical Grothendieck-Lefschetz formula Card(XF) = i(-1)i Tr(F*,Hci(X,l)).

Taking the logarithmic derivative of Z(X0,0,t), t d dt logZ(X0,0,t) =dfn t d dt Z(X0,0,t) Z(X0,0,t) = n>0 xXFn = X0(𝔽qn) Tr(Fx*n,0)tn. (A)

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

Notes and References

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



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