Geometric monodromy - semisimplicity and maximality

• 成果类型: Article
• 署名作者: Cadoret, Anna; Hui, Chun-Yin; Tamagawa, Akio
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2017.186.1.5
• 发表日期: 2017
• 页码: 205-236
• 关键词: l-adic representations; compatible systems; abelian-varieties; l-independence; image theorem; torsion; FIELDS

摘要

Let X be a connected scheme, smooth and separated over an algebraically closed field k of characteristic p >= 0, let f : Y -> X be a smooth proper morphism and x a geometric point on X. We prove that the tensor invariants of bounded length <= d of pi(1)(X, x) acting on the etale cohomology groups H* (Y-x, F-l) are the reduction modulo-l of those of pi(1)(X, x) acting on H* (Y-x, Z(l)) for l greater than a constant depending only on f : Y -> X, d. We apply this result to show that the geometric variant with F-l-coefficients of the Grothendieck-Serre semisimplicity conjecture namely, that pi(1) (X, x) acts semisimply on H*(Y-x, F-l) for l >> 0 is equivalent to the condition that the image of pi(1) (X, x) acting on H* (Y-x, Q(l)) is `almost maximal' (in a precise sense; what we call 'almost hyperspecial') with respect to the group of Q(l)-points of its Zariski closure. Ultimately, we prove the geometric variant with F-l-coefficients of the Grothendieck-Serre semisimplicity conjecture.