On the Number of Representations of a Natural Number by Certain Quaternary Quadratic Forms

2020 ◽  
pp. 173-198
Author(s):  
B. Ramakrishnan ◽  
Brundaban Sahu ◽  
Anup Kumar Singh
Keyword(s):  
Natural Number ◽  
Quadratic Forms

scholarly journals Pairs of quadratic forms modulo one

1993 ◽  
Vol 35 (1) ◽  
pp. 51-61
Author(s):  
R. C. Baker ◽  
J. Brüdern
Keyword(s):  
Natural Number ◽  
Quadratic Forms ◽  
Image Position ◽  
Real Quadratic

Let s be a natural number, s ≥ 2. We seek a positive number λ(s) with the following property:Let ε > 0. Let Q1(x1, …, xs), Q2(x1, …, xs) be real quadratic forms, then for N > C1(s, ε) we havefor some integers n1, …, ns,

scholarly journals On the Eisenstein Series Corresponding to Quadratic Forms of Certain Type

2006 ◽  
Vol 13 (4) ◽  
pp. 737-740
Author(s):  
Nikoloz Kachakhidze
Keyword(s):  
Natural Number ◽  
Eisenstein Series ◽  
Quadratic Forms ◽  
J 13

Abstract The Eisenstein series corresponding to quadratic forms of type (𝑓/2, 4𝑁, χ) (𝑁 is a square-free natural number) are constructed using the bases of the spaces of Eisenstein series given in [Kachakhidze, Georgian Math. J. 13: 55–78, 2006].

On the Representation of Numbers by Positive Diagonal Quadratic Forms with Five Variables of Level 16

1998 ◽  
Vol 5 (1) ◽  
pp. 91-100
Author(s):  
D. Khosroshvili
Keyword(s):  
Natural Number ◽  
General Formula ◽  
Theta Function ◽  
Quadratic Forms ◽  
Additional Term ◽  
Singular Series

Abstract A general formula is derived for the number of representations r(n; f) of a natural number n by diagonal quadratic forms f with five variables of level 16. For f belonging to one-class series, r(n; f) coincides with the sum of a singular series, while in the case of a many-class series an additional term is required, for which the generalized theta-function introduced by T. V. Vepkhvadze [Vepkhvadze, Acta Arithmetica 53: 433–990] is used.

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.

Linked Tori, II

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

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

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