Question the first
Let be the integer ring of a mixed characteristic local field . Fix an algebraic closure with absolute Galois group , and let be a Galois-stable -lattice in a crystalline representation of all of whose Hodge-Tate weights are and . By a remarkable theorem of Breuil and Kisin, the association is an equivalence of categories between -divisible groups over and lattices in crystalline representations of the above type.
Let us say as above is –analytic if the following two conditions hold:
1. There is an -module structure on for which the Galois action is -linear.
2. For all except for our initial embedding , the map
is an isomorphism.
Is is true that is -analytic if and only if the associated -divisible group admits a strict -action? (Recall that a strict -action is a ring map such that the induced -action on agrees with the one given by the structure map .)
Question the second
Let be a mixed characteristic local field with uniformizer , , the lower triangular Borel, the diagonal maximal torus. Fix another local field , and let be a continuous character, identified with a tuple of characters in the obvious way. Suppose for all we have the inequality , with equality for . Does the locally analytic induction
admit a -invariant norm?