28 nov 2025 -- 12:00
aula 6, Dipartimento di Matematica ed Informatica Via Machiavelli 30, Ferrara
Abstract.
Let X be smooth projective defined over a finite field, and f:X->X be a q-polarised morphism (i.e q is an integer >2 and there is an ample divisor L so that f*(L)=qL). Note: q does not need to be the cardinality of the finite field! Inspired by Weil's Riemann hypothesis, Tate proposed the following questions:
Tate 1: any eigenvalue of f* on H{etale}j(X) has absolute value = q{j/2}.
Tate 2: the action of f* on H{etale}j(X) is semisimple, i.e. diagonalisable.
Both Tate 1 and Tate 2 are consequences from the Standard conjectures of Bombieri - Grothendieck.
When f=FrX the Frobenius morphism of X, then Tate 1 is Weil's Riemann hypothesis, which is a well known theorem by Deligne.
This talk presents some new developments since the last year. In particular, I will explain 2 main results:
The proof of Tate 1, in fact a more general statement for all endomorphisms of X, which is now a theorem of Junyi Xie, following joint work with Fei Hu and Junyi Xie. The proof of the fact that Standard Conjecture D alone is enough to solve Tate 2. Some further generalisations of Tate 2, for example to rational maps of X. Also, some ideas on how this can be solved.
The talk includes joint work with Fei Hu and Junyi Xie.