Let p be a prime number. Let w2 and w ∗ 2 denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies w∗ 2(ξ) ≤ w2(ξ) ≤ w∗ 2(ξ) + 1, with w∗ 2(ξ) = w2(ξ) = 2 for almost all ξ. By means of Schneider’s continued fractions, we give explicit examples of p-adic numbers ξ for which ...

Exercise 1 (Maximal compact subgroups of G). A lattice in Qp is a finitelygenerated Zp-submodule of Qp that generates Qp as vector space. In particular, it’s free of rank n. Note that G acts transitively on the set of lattices in Qp . (i) Show that K = StabG(Zp ). (ii) Suppose that K ′ is a compact subgroup of G. Show that K ′ stabilises a lattice. (Hint: show that the K ′-orbit of Zp is finite...

For a fairly general reductive group G/Qp , we explicitly compute the space of locally algebraic vectors in the Breuil-Herzig construction Π(ρ)ord, for a potentially semistable Borel-valued representation ρ of Gal(Q̄p/Qp). The point being we deal with the whole representation, not just its socle – and we go beyond GLn(Qp). In the case of GL2(Qp), this relation is one of the key properties of the...

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Background Cardiovascular magnetic resonance (CMR) flow quantification for determining the ratio of flow between the pulmonary and systemic circulation (Qp/Qs) is used to quantify cardiac shunts. However, both the accuracy and precision of Qp/Qs can be influenced by measurement errors in flow quantification. Post-processing methods for stationary tissue background correction have been proposed ...

1.1. The case ` 6= p. In this case, we are in the setting of the classical Local Langlands Correspondence, which can be stated (roughly) as follows: Let n ≥ 1. We then have an injective map (1)  continuous representations of Gal(Qp/Qp) on n-dimensional Q`-vector spaces, up to isomorphism  ↪−→  irreducible, admissible representations of GLn(Qp) on Q`-vector spaces, up to isomor...