Search

We begin with the observation that the p -adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts ϕ that are "Lie invariant mod p " in the sense that the "normalized" action of ϕ on 1-forms preserves mod p the space of invariant 1-forms. Our main result is that, after removing the 2-torsion sections, the above situation can be "infinitesimally deformed" in the sense that the above mod p result has a mod p 2 analogue. We end by showing that, in contrast with the case of elliptic curves, the following holds: if G is a linear algebraic group over a number field and if G is not a torus then for all but finitely many primes p the p -adic completion of G does not possess a Frobenius lift that is Lie invariant mod p. [ABSTRACT FROM AUTHOR]

We study the prime-to- p Hecke action on the projective limit of the sets of connected components of Shimura varieties with fixed parahoric or Bruhat–Tits level at p. In particular, we construct infinitely many Shimura varieties for CM unitary groups in odd variables for which the considered actions are not transitive. We prove this result by giving negative examples on the question of Bruhat–Colliot-Thélène–Sansuc–Tits and its variant, which are related to the weak approximation on tori over Q. [ABSTRACT FROM AUTHOR]

We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as well as in some other situations. This work relies in an essential way on several finiteness results for unramified cohomology. [ABSTRACT FROM AUTHOR]