Throughout this paper Z, Q, C, Zp, Qp and Cp will respectively denote the ring of rational integers, the field of rational numbers, the filed of complex numbers, the ring p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp such that |p|p = p −vp(p) = p. If q ∈ Cp, we normally assu...

We introduce the notion of a prism, which may be regarded as "deperfection" perfectoid ring. Using prisms, we attach ringed site --- prismatic to $p$-adic formal scheme. The resulting cohomology theory specializes (and often refines) most known integral theories. As applications, prove an improved version almost purity theorem allowing ramification along arbitrary closed subsets (without using ...

In this paper, we introduce q-analogues of degenerate Catalan numbers and polynomials with the help a fermionic p-adic q-integrals on Zp establish some new connections Stirling first second kinds. Furthermore, also find few identities results type numbers.

In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite étale extension of R[p−1] is “almost” finite étale over R. Here, we use almost purity to lift the finite étale extension of R[p−1] to a finite étale extension of rin...

Abstract We prove that if two semi-algebraic subsets of ℚ p n {\mathbb{Q}_{p}^{n}} have the same p -adic measure, then this equality can already be deduced using only some basic integral transformation rules. On one hand, considered as a positive answer to analogue que...

Let p be an odd prime number. For n ∈ Z N ∪ {0}, let Cpn {ζ | ζpn 1} be the cyclic group of order p, and let Tp limn→∞Cpn ⋃ n≥0 Cpn Cp∞ be the space of locally constant functions in the p-adic number fieldCp. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ C, one normally assumes that |q| < 1. If q ∈ Cp, one n...

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...