Is there any generalization of Weil conjecture for any non-smooth geometric-connected variety? For example, for more general curve, or at least some numerical evindence (example)?
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5$\begingroup$ The zeta function is still rational, the roots of the numerator and denominator are still concentrated on vertical lines at (integers and) half integers, but the weights are mixed, i.e. the part coming from $H^k$ is not all on the line $\operatorname{Im}(z) = \frac{k}{2}$. This is all classical; rationality of the zeta function is due to Dwork, and mixedness of the cohomology is due to Deligne. $\endgroup$– R. van Dobben de BruynCommented May 10, 2018 at 22:36
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2$\begingroup$ @R.vanDobbendeBruyn Thanks, is there any references? $\endgroup$– BonbonCommented May 10, 2018 at 22:40
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3$\begingroup$ A good place to start is any book on étale cohomology with a focus on the Weil conjecutres (e.g. Freitag–Kiehl) and look at the references from there. Deligne's result is contained in Weil II, but this is a technical paper. There should be straightforward ways to deduce the general case from the smooth projective case using hypercovers (and de Jong's alterations), similar to the hypercover construction in Deligne's Hodge III. $\endgroup$– R. van Dobben de BruynCommented May 10, 2018 at 22:41
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