Optimal strong approximation for quadrics over $\mathbb{F}_q[t]$

DOI10.1016/J.AIM.2022.108852arXiv1907.07839MaRDI QIDQ6322298

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Publication date: 17 July 2019

Abstract: Suppose $q$ is a fixed odd prime power, $F(vecx)$ is a non-degenerate quadratic form over $mathbb{F}_q[t]$ of discriminant $Delta$ in $dgeq5$ variables $vecx$, and $f,ginmathbb{F}_q[t]$, $$. We show that whenever $extdegfgeq(4+varepsilon)extdegg+O_{varepsilon,F}(1)$, $gcd(Delt{a}^{infty},fg)=O(1)$, and the necessary local conditions are satisfied, we have a solution $vecxinmathbb{F}_q[t{]}^d$ to $F(vecx)=f$ such that $$. For $d=4$, we show that the same conclusion holds if we instead have $extdegfgeq(6+varepsilon)extdegg+O_{varepsilon,F}(1)$. This gives us a new proof (independent of the Ramanujan conjecture over function fields proved by Drinfeld) that the diameter of any $k$-regular Morgenstern Ramanujan graphs $G$ is at most $(2+varepsilon)lo{g}_{k-1}|G|+O_{varepsilon}(1)$. In contrast to the $d=4$ case, our result is optimal for $dgeq5$. Our main new contributions are a stationary phase theorem over function fields for bounding oscillatory integrals, and a notion of anisotropic cones to circumvent isotropic phenomena in the function field setting.

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