The field $Q_{p}$ of $p$-adic numbers is defined as the completion of the field of the rational numbers $Q$ with respect to the  $p$-adic norm $|.|_{p}$. In this paper, we study the continuous and discrete $p-$adic shearlet systems on $L^{2}(Q_{p}^{2})$. We also suggest discrete $p-$adic shearlet frames. Several examples are provided.

Introduction 2 1. Harmonic analysis and unitary duals 2 2. Non-discrete locally compact fields, classical groups, reductive groups 4 3. K0-finite vectors 7 4. Smooth representations 8 5. Parabolically induced representations 9 6. Jacquet modules 12 7. Filtrations of Jacquet modules 14 8. Square integrable and tempered representations 15 9. Langlands classification 16 10. Geometric lemma and alg...

We formulate a version of the o-minimal group conjectures from [11], which is appropriate for groups G definable in a (saturated) p-adically closed field K, We discuss the conjectures in two cases, when G is defined over Qp and when G is of the form E(K) for E an elliptic curve over K.

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...

1.1. The p-adic numbers. A rational number x ∈ Q× may be uniquely written as x = ab p n with a, b and n nonzero integers such that p ab. We define ordp(x) = n, |x|p = p−n, |0|p = 0. |·|p defines an absolute value on Q, satisfying the stronger ultrametric triangle equality |x+ y|p ≤ max(|x|p, |y|p). We define Qp to be the completion Q with respect to this metric and we use the same notation | · ...