Weil's insight: A Lefschetz fixed point formula for varieties?
In this video, we explain Weil's insight into his conjectures linking the topology of a variety over the complex numbers, to the point counts of the variety over finite fields. We first look at the Frobenius morphism in this setting, and see how the set of rational points over finite fields can be viewed as fixed point sets of powers of the Frobenius morphism. We then talk about the Lefschetz fixed point formula from topology. Weil's insight was there should be some version of the formula even when you look at varieties over finite fields. This gives a handle on the zeta function and we show how for example, rationality of the zeta function would follow if there were such a formula as Weil predicted and was later found by Grothendieck.