scholarly journals Valuations and Hermitian forms on skew fields

2002 ◽  
pp. 103-115 ◽  
Author(s):  
Thomas Craven
Keyword(s):  
Hermitian Forms ◽  
Skew Fields

Skew Fields

1983 ◽  
Author(s):  
P. K. Draxl
Keyword(s):  
Skew Fields

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

Non unimodular Hermitian forms

1996 ◽  
Vol 123 (1) ◽  
pp. 233-240 ◽  
Author(s):  
Eva Bayer-Fluckiger ◽  
Laura Fainsilber
Keyword(s):  
Hermitian Forms

On embedding some G-filtered rings into skew fields

2014 ◽  
Vol 55 (6) ◽  
pp. 1017-1041
Author(s):  
A. I. Valitskas
Keyword(s):  
Skew Fields

Spherical functions and local densities on the space of p-adic quaternion Hermitian forms

2021 ◽  
pp. 1-38
Author(s):  
Yumiko Hironaka
Keyword(s):  
Laurent Polynomial ◽  
Spherical Functions ◽  
Plancherel Formula ◽  
Explicit Formulas ◽  
Hermitian Forms ◽  
Spherical Fourier Transform ◽  
Local Densities ◽  
Laurent Polynomial Ring ◽  
Injective Map ◽  
Residual Characteristic

We introduce the space [Formula: see text] of quaternion Hermitian forms of size [Formula: see text] on a [Formula: see text]-adic field with odd residual characteristic, and define typical spherical functions [Formula: see text] on [Formula: see text] and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to [Formula: see text], and define a spherical Fourier transform on the Schwartz space [Formula: see text] which is Hecke algebra [Formula: see text]-injective map into the symmetric Laurent polynomial ring of size [Formula: see text]. Then, we determine the explicit formulas of [Formula: see text] by a method of the author’s former result. In the last section, we give precise generators of [Formula: see text] and determine all the spherical functions for [Formula: see text], and give the Plancherel formula for [Formula: see text].

scholarly journals HIGHER PRODUCT PYTHAGORAS NUMBERS OF SKEW FIELDS

2010 ◽  
Vol 03 (01) ◽  
pp. 193-207
Author(s):  
Dejan Velušček
Keyword(s):  
Laurent Series ◽  
Skew Field ◽  
Skew Fields ◽  
Pythagoras Number

We introduce the n–th product Pythagoras number p n(D), the skew field analogue of the n–th Pythagoras number of a field. For a valued skew field (D, v) where v has the property of preserving sums of permuted products of n–th powers when passing to the residue skew field k v and where Newton's lemma holds for polynomials of the form Xn - a, a ∈ 1 + I v , p n(D) is bounded above by either p n( k v ) or p n( k v ) + 1. Spherical completeness of a valued skew field (D, v) implies that the Newton's lemma holds for Xn - a, a ∈ 1 + I v but the lemma does not hold for arbitrary polynomials. Using the above results we deduce that p n (D((G))) = p n(D) for skew fields of generalized Laurent series.

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