Determine the least vanishing exponent for mock Frobenius covers

Determine the least exponent e for which the pullback map induced by the mock Frobenius tower becomes zero in the relevant positive cohomological degrees, including determining the corresponding exponent for each constructible sheaf and tower under consideration.

Background

Theorem B establishes a uniform exponent e for the multiplication-by-ℓ tower of an abelian variety, with an explicit but generally nonoptimal sufficient bound. The paper notes that Esnault formulated the analogous problem for mock Frobenius covers of projective space and gave an example in which the least exponent is 2. Thus, identifying the optimal exponent remains an unresolved quantitative problem in the parallel mock Frobenius setting.

References

Esnault \S5, Remark~3 raises the question of the least such exponent e for the mock Frobenius covers and gives the example of a smooth conic tangent to the three coordinate lines for \ell=2, for which e=2.

— Artin vanishing along the $\ell$-adic tower of an abelian variety  (2609.35404 - Choudhury et al., 28 Sep 2026) in Section 1, subsection “Vanishing along the tower,” paragraph following Theorem B