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We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Littlewood circle method over number fields.
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The Kloosterman circle method and weighted representation numbers of quadratic forms
Abstract We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of nonsingular integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, Salié sums, and a principle of nonstationary phase. We briefly discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local–global principle for certain Kleinian sphere packings.
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- Award ID(s):
- 2402599
- PAR ID:
- 10629762
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Research in the Mathematical Sciences
- Volume:
- 12
- Issue:
- 3
- ISSN:
- 2522-0144
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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