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On Quadric Surfaces Weighted by Arbitrary Coefficients
Abstract
For a non-singular quadratic form F in four variables, we study integer solutions to F=0 with an arbitrary weight attached to one of the variables. This allows us to investigate genuinely sparse subsets of the corresponding variety. Our main input is to obtain cancellation in sums of character sums over the moduli arising in the application of the DFI's delta method to the quadratic hypersurface. This cancellation relies crucially on the equidistribution of quadratic roots and the Weil bound for Kloosterman sums.
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BibTeX
@misc{guria2026quadric,
title = {{On Quadric Surfaces Weighted by Arbitrary Coefficients}},
author = {Rachita Guria and Prahlad Sharma and Saurabh Singh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12944},
eprint = {2610.12944},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12944},
}
RIS
TY - UNPB
TI - On Quadric Surfaces Weighted by Arbitrary Coefficients
AU - Rachita Guria
AU - Prahlad Sharma
AU - Saurabh Singh
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12944
UR - https://arxiv.org/abs/2610.12944
AB - For a non-singular quadratic form F in four variables, we study integer solutions to F=0 with an arbitrary weight attached to one of the variables. This allows us to investigate genuinely sparse subsets of the corresponding variety. Our main input is to obtain cancellation in sums of character sums over the moduli arising in the application of the DFI's delta method to the quadratic hypersurface. This cancellation relies crucially on the equidistribution of quadratic roots and the Weil bound for Kloosterman sums.
ER -