Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms

2013 ◽  
Keyword(s):  
Quadratic Forms ◽  
Arithmetic Theory

scholarly journals The Arithmetic Theory of Quadratic Forms.

1951 ◽  
Vol 58 (9) ◽  
pp. 644
Author(s):  
J. D. Elder ◽  
B. W. Jones
Keyword(s):  
Quadratic Forms ◽  
Arithmetic Theory

The Arithmetic Theory of Quadratic Forms (Carus Mathematical Monograph Number Ten)

1951 ◽  
Vol 35 (313) ◽  
pp. 214
Author(s):  
J. W. S. Cassels ◽  
Burton W. Jones
Keyword(s):  
Quadratic Forms ◽  
Arithmetic Theory

scholarly journals SUMS OF FOUR SQUARES WITH A CERTAIN RESTRICTION

2021 ◽  
pp. 1-10
Author(s):  
YUE-FENG SHE ◽  
HAI-LIANG WU
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Quadratic Forms ◽  
Ternary Quadratic Forms ◽  
Arithmetic Theory ◽  
Four Square

Abstract Z.-W. Sun [‘Refining Lagrange’s four-square theorem’, J. Number Theory175 (2017), 169–190] conjectured that every positive integer n can be written as $ x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$ with $x+3y$ a square and also as $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in \mathbb {Z})$ with $x+3y\in \{4^k:k\in \mathbb {N}\}$ . In this paper, we confirm these conjectures via the arithmetic theory of ternary quadratic forms.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

Totally Wild Quadratic Forms of Type E7

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Trace Map ◽  
Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

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