Is there any set of axioms that characterize completely the zeta function of a scheme over a finite field of characteristic $p$?
$\begingroup$
$\endgroup$
5
-
1$\begingroup$ Analytic axioms or algebraic axioms? For sure, it must be a rational function in $q^{s}$ to start, and its zeros must have $\Re s =1/2$. Do you want to know what rational functions can occur? $\endgroup$– Marc PalmCommented Jul 20, 2012 at 15:55
-
$\begingroup$ Anysort of axioms.However it is preferable to have axioms uniform in $q$,hesize of the finite field. $\endgroup$– user16974Commented Jul 20, 2012 at 18:47
-
$\begingroup$ A related question is to ask for characterization （even conjectural） of Frobenius char. polynomials on cohomology of a fixed degree. For curves and abelian varieties one can use Honda-Tate，and this related question is equivalent to your original one. I believe that very little common feature is known for general schemes （not nece. proper or smooth）. For projective smooth varieties，by weak Lefschetz，the most interesting ones are the middle two， about which again very little is known. $\endgroup$– shenghaoCommented Jul 21, 2012 at 10:05
-
$\begingroup$ @shenghao: thanks, can you elaborate on the equivalence between the two related questions? $\endgroup$– user16974Commented Jul 21, 2012 at 12:49
-
$\begingroup$ With pleasure: For an abelian variety, $H^1$ determines the others: $H^i$ is the $i$-fold wedge product of $H^1$, so knowing what $H^1$ is gives the full zeta function. For a curve, $H^0$ and $H^2$ are always known, so again $H^1$ determines the zeta function. But for other varieties we are less lucky. I don't know how to tell the cup-product structure (for $X$ a proper variety) from the zeta function either. $\endgroup$– shenghaoCommented Jul 21, 2012 at 23:35
Add a comment
|