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arXiv · 2106.15213

Values of Inhomogeneous Forms at S-integral points

Anish Ghosh·Jiyoung Han
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Abstract

We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic form and the shift are allowed to vary. In order to do so, we prove analogues of Rogers' moment formulae for $S$-arithmetic congruence quotients as well as for the space of affine lattices. We believe the latter results to be of independent interest.

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Anish Ghosh, Jiyoung Han. 2021-06-29. Values of Inhomogeneous Forms at S-integral points. https://arxiv.org/abs/2106.15213

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