Representations of indefinite quadratic forms

Hsia, J. S.; Shao, Yi; Xu, Fei

doi:10.1515/crll.1998.002

Cited by 28 publications

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“…Some effort has been made in [EH82] and [HSX98]. In this paper, we make some progress on this subject.…”

Section: Introductionmentioning

confidence: 99%

“…The notation and terminology are standard if not explained, or adopted from [Xu01], [Ome73] and [HSX98]. Let V be a quadratic space over a number field F with a non-degenerate symmetric bilinear form B (x, y).…”

Section: Introductionmentioning

confidence: 99%

“…Put i(M, K) = p<∞ i p ∈ I F . The ideal associated to i(M, K) is p<∞ p rp which is a generalization of the intersection ideal defined in [HSX98].…”

Section: Introductionmentioning

confidence: 99%

“…The method we develop here can be used to check if one form can be represented by the spinor genus of another form directly, without the presence of a third form with a certain property in the genus (see [HSX98]). We explain this by the following example taken from [Hsi99].…”

Section: Introductionmentioning

confidence: 99%

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On representations of spinor genera

Chan¹,

Xu²

2004

Compositio Math.

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“…Some effort has been made in [EH82] and [HSX98]. In this paper, we make some progress on this subject.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Put i(M, K) = p<∞ i p ∈ I F . The ideal associated to i(M, K) is p<∞ p rp which is a generalization of the intersection ideal defined in [HSX98].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On representations of spinor genera

Chan¹,

Xu²

2004

Compositio Math.

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“…Hsia studied the problem of whether a quadratic form with coefficients in the ring of integers C7~ of a number field 1~ is represented by another quadratic form of the same kind ( [9], [10]). …”

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confidence: 99%

Applications of spinor class fields: embeddings of orders and quaternionic lattices

Arenas-Carmona¹

2003

Annales De L’institut Fourier

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Quadratic Forms over $${\mathbb{Z}}$$ from Diophantus to the 290 Theorem

Hahn

2008

AACA

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What follows 1 is a brief historical survey of the representation theory of quadratic forms over the integers. It starts with questions considered by Diophantus about 1800 years ago, dashes past classical contributions of Euler, Lagrange, and Sylvester, considers work of Ramanujan and Dickson, continues with theorems of Conway and Schneeberger, and ends with a short sketch of the proof of the 290-Theorem by Bhargava and Hanke. This survey is self-contained in the sense that all the basic definitions and concepts are provided.Let R be a commutative ring with 1. A quadratic form in n variables over R is a homogeneous polynomial q = q(X 1 , . . . , X n ) = 1≤i≤j≤n a i,j X i X j of degree two with coefficients a i,j in R. An element s ∈ R is a value of q if there is a vector (r 1 , . . . , r n ) ∈ R n such that q(r 1 , . . . , r n ) = s. In this case q is said to represent s. If q represents 0 non-trivially then q is isotropic, and q is anisotropic if 0 is represented only by the zero vector.The Babylonians in the 18th century B.C. already knew about triples (a, b, c) of positive integers that satisfy the equation a 2 + b 2 = c 2 . So they had insight into the ways that the quadratic form X 2 1 + X 2 2 − X 2 3 over the integers Z represents 0. In the 6th century B.C., the Pythagoreans appear to have been unsettled by the fact that the quadratic form 2X 2 1 − X 2 2 over Z is anisotropic. However, the story of quadratic forms, in any reasonably systematic sense, begins with Diophantus and his Arithmetica [11] in the 3rd century A.D. For example, wanting to find two squares of rational numbers having a given difference, Diophantus saw that 60 = (x + 3) 2 − x 2 with x = 8 1 2 . In particular, the quadratic form X 2 1 − X 2 2 over 1 This article is dedicated to a wonderful and exceptional mathematician, human being, and friend: to Timothy O'Meara on the occasion of his 80-th birthday.

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Representations of indefinite quadratic forms

Cited by 28 publications

References 0 publications

On representations of spinor genera

On representations of spinor genera

Applications of spinor class fields: embeddings of orders and quaternionic lattices

Quadratic Forms over $${\mathbb{Z}}$$ from Diophantus to the 290 Theorem

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