Sum-product phenomena: $$\mathfrak{p}$$-adic case

2020 ◽  
Vol 142 (2) ◽  
pp. 349-419
Author(s):  
Alireza Salehi Golsefidy
Keyword(s):  
Adic Case

scholarly journals Analytic Continuation of Equivariant Distributions

2018 ◽  
Vol 2019 (23) ◽  
pp. 7160-7192 ◽  
Author(s):  
Dmitry Gourevitch ◽  
Siddhartha Sahi ◽  
Eitan Sayag
Keyword(s):  
Analytic Continuation ◽  
Reductive Group ◽  
Principal Series ◽  
Real Algebraic Varieties ◽  
Series Representations ◽  
Degenerate Principal Series ◽  
Special Case ◽  
Adic Case ◽  
Principal Series Representations ◽  
Whittaker Models

Abstract We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun.

scholarly journals A note on p-adic Carlitz's q-Bernoulli numbers

2000 ◽  
Vol 62 (2) ◽  
pp. 227-234 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim
Keyword(s):  
Invariant Measure ◽  
Generating Function ◽  
Bernoulli Number ◽  
Bernoulli Numbers ◽  
Adic Case

In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.

The p-adic case

2012 ◽  
pp. 46-49
Author(s):  
J. W. S. Cassels
Keyword(s):  
Adic Case

scholarly journals ON ERGODIC BEHAVIOR OF p-ADIC DYNAMICAL SYSTEMS

2001 ◽  
Vol 04 (04) ◽  
pp. 569-577 ◽  
Author(s):  
MATTHIAS GUNDLACH ◽  
ANDREI KHRENNIKOV ◽  
KARL-OLOF LINDAHL
Keyword(s):  
Dynamical Systems ◽  
Natural Number ◽  
Unit Circle ◽  
Complex Plane ◽  
Haar Measure ◽  
Analogous Result ◽  
Topologically Transitive ◽  
Ergodic Behavior ◽  
Adic Case

Monomial mappings, x ↦ xn, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an analogous result for monomial dynamical systems over p-adic numbers. The process is, however, not straightforward. The result will depend on the natural number n. Moreover, in the p-adic case we will not have ergodicity on the unit circle, but on the circles around the point 1.

Brauer Group Analogues of Results Relating the Witt Ring to Valuations and Galois Theory

1995 ◽  
Vol 47 (3) ◽  
pp. 527-543 ◽  
Author(s):  
Yoon Sung Hwangk ◽  
Bill Jacob
Keyword(s):  
Galois Theory ◽  
Sufficient Conditions ◽  
Structure Theory ◽  
Valuation Theory ◽  
Cup Product ◽  
The Past ◽  
Relative Rigidity ◽  
Form Theory ◽  
Special Case ◽  
Adic Case

AbstractLet F be a field of characteristic different from p containing a primitive p-th root of unity. This paper studies the cup product pairing Hl(F, p) x Hl(F, p) → H2(F, p) and its relationship to valuation theory and Galois theory. Sufficient conditions on the pairing which guarantee the existence of a valuation on the field are described. In the non p-adic case these results provide a converse to the well-known structure theory in this situation. In the p-adic case, the pairing is described using the notion of "relative rigidity". These results are analogues of results in quadratic form theory developed in the past decade, which cover the special case p = 2. Applications to the maximal pro-p Galois group of F are also described.

Quadratic forms ‘à la’ local theory

1967 ◽  
Vol 63 (3) ◽  
pp. 579-586 ◽  
Author(s):  
A. Fröhlich
Keyword(s):  
Quadratic Forms ◽  
The Other ◽  
Local Theory ◽  
Real Numbers ◽  
The Real ◽  
Other Hand ◽  
Non Local ◽  
Characteristic 2 ◽  
The One ◽  
Adic Case

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

FRACTAL SETS IN THE FIELD OF p-ADIC ANALOGUE OF THE COMPLEX NUMBERS

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950053 ◽  
Author(s):  
YIN LI ◽  
HUA QIU
Keyword(s):  
Number Field ◽  
Complex Number ◽  
Geometric Structure ◽  
Classical Case ◽  
Complex Numbers ◽  
The Real ◽  
Fractal Sets ◽  
Fractal Measures ◽  
Complex Number Field ◽  
Adic Case

The [Formula: see text]-adic number field [Formula: see text] and the [Formula: see text]-adic analogue of the complex number field [Formula: see text] have a rich algebraic and geometric structure that in some ways rivals that of the corresponding objects for the real or complex fields. In this paper, we attempt to find and understand geometric structures of general sets in a [Formula: see text]-adic setting. Several kinds of fractal measures and dimensions of sets in [Formula: see text] are studied. Some typical fractal sets are constructed. It is worthwhile to note that there exist some essential differences between [Formula: see text]-adic case and classical case.

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