We study birational invariants and their link to conjectures in smooth projective varieties over finite fields, indicating deeper connections in algebraic geometry.
We study a natural birational invariant for varieties over finite fields and show that its vanishing on projective space is equivalent to the Tate conjecture, the Beilinson conjecture, and the Grothendieck–Serre semi-simplicity conjecture for all smooth projective varieties over finite fields. We further show that the Tate, Beilinson, and 1-semi-simplicity conjecture in half of the degrees implies those conjectures in all degrees.
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Balkan et al. (2026) studied this question.
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