Diophantine Problems Over Local Fields: III. Decidable Fields

1966 ◽  
Vol 83 (3) ◽  
pp. 437 ◽  
Author(s):  
James Ax ◽  
Simon Kochen
Keyword(s):  
Local Fields ◽  
Diophantine Problems

Diophantine Problems Over Local Fields I

1965 ◽  
Vol 87 (3) ◽  
pp. 605 ◽  
Author(s):  
James Ax ◽  
Simon Kochen
Keyword(s):  
Local Fields ◽  
Diophantine Problems

scholarly journals Zeros of systems of 𝔭-adic quadratic forms

2010 ◽  
Vol 146 (2) ◽  
pp. 271-287 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Algebraic Geometry ◽  
Quadratic Forms ◽  
Residue Class ◽  
Residue Field ◽  
Local Fields ◽  
Class Field ◽  
Residue Class Field ◽  
Minimization Technique ◽  
Diophantine Problems

AbstractWe show that a system of r quadratic forms over a 𝔭-adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax–Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a 𝔭-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.

Local Fields

1986 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Local Fields

scholarly journals Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos A. M. André ◽  
João Dias
Keyword(s):  
Local Field ◽  
Unit Group ◽  
Basic Algebra ◽  
Local Fields ◽  
Dimensional Representation ◽  
Smooth Representation ◽  
One Dimensional ◽  
Finite Dimensional

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

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