CHAPTER 2 Preliminaries Since such results will be used on many occasions throughout this memoir, in this chapter we collect some preliminary results about Iwahori-spherical represen- tations as well as crystalline periods of Galois representations. 2.1. Notation Fix throughout an algebraic closure Q of Q and Qp of Qp for each prime p. We will denote by vp(−) the valuation on Q p such that vp(p) = 1 and let | · |p denote the corresponding absolute value such that |p|p = 1/p. Throughout this work, by a p-adic field we will mean a finite extension of Qp for some rational prime p. For a p-adic field K let | · |K denote the absolute value normalized such that for a uniformizer one has | |K = 1/q where q denotes the size of the residue field of K. If the context is clear then |·|K will sometimes simply be denoted by |·|. For a p-adic field K normalize local class field theory so that uniformizers correspond to lifts of geometric Frobenius. Let GK := Gal(K/K) and let WK ⊂ GK denote the Weil group and for g ∈ WK let ν(g) ∈ Z be such that g is a lift of the ν(g)’th power of geometric Frobenius. For a character χ of K× we will denote by ˜ the character of WK corresponding to it by local class field theory. Let the maximal absolutely unramified subfield of K be denoted by K0 and let K0 ur denote its maximal unramified extension. Suppose F is a number field and ρ : Gal(F/F ) → GLn(Q ) is a continu- ous representation. Consider the isomorphism class of the semi-simplification of the residual representation of the representation on a Galois stable lattice coming from ρ. Let ρ denote the scalar extension of this representation to the algebraic closure. For an extension B/A of number fields let SplB/A denote the set of finite places of A which split completely in B. 2.2. Weil-Deligne representations We briefly recall some standard results on Weil-Deligne representations and refer to [BH, Chapter 7] for more details. Let K be a p-adic field and fix an algebraic closure K. Let WK be the Weil group which is a topological group such that the inertia group IK is open and the topology of IK agrees with the topology on IK viewed as a subset of Gal(K/K) equipped with the Krull topology. With this topology the Weil group is a locally profinite topological group. As for any locally profinite topological group, one defines a smooth representation of WK as follows: Let E be a field of characteristic zero and V an E-vector space. Then a smooth representation is defined to be a 9

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