Let $q$ be an odd prime power and $m\geq 1$. Let $Q$ be a nondegenerate quadratic form on $\mathbb F_q^{2m}$. For every sphere $S_j=\{Q=j\}$ with $j\in \mathbb F_q^\times$, we prove the sharp extension estimate $R_{S_j}^*(2\to r)\lesssim_m1$ for $r\ge 2(m+1)/m$, uniformly in $q$, $Q$, and $j$. As an application, we show that if $E\subseteq\mathbb F_q^{2m}$ satisfies $|E|/q^{m+1/3}\to\infty$, then almost every pin $y\in E$ determines $(1-o(1))q$ values of $Q(x-y)$. The proof combines two arithmetic Hecke operator estimates with an induction in the dimension, a centered sphere--cone estimate, and an orthogonal decomposition.
Let $q$ be an odd prime power, let $d\ge3$ be odd, and let $S_j=\{\xi\in \mathbb F_q^d:\xi_1^2+\cdots+\xi_d^2=j\}$, where $j\ne 0$. We prove the sharp extension estimate ${L^{4(d+1)/(3d+1)}(S_j, d\sigma)\to L^4(\mathbb F_q^d,dc)}$ whenever $\eta_q((-1)^{(d-1)/2}j)=-1$, where $\eta_q$ is the quadratic character. This co...
Let $F/\mathbf Q_p$ and $E/\mathbf Q_p$ be finite extensions, and let $G=\mathrm{GL}_2(F)$ or $D_F^\times$, where $D_F$ is the quaternion division algebra over $F$. Let $\Pi$ be an admissible Banach $E$-representation of $G$ whose locally $\mathbf Q_p$-analytic vectors $\Pi^{\mathrm{an}}$ have an infinitesimal characte...
Let ${\mathbb P}^1_{\mathfrak q}$ be the weighted projective line with coprime weights ${\mathfrak q}=(q_0,q_1)$ and $L=q_0q_1$. The weighted Veronese map $\Phi\colon {\mathbb P}^1_{\mathfrak q}\to {\mathbb P}^1$ satisfies $h\circ\Phi=L\,{\mathfrak s}$. Pulling back Roth's theorem gives the optimal exponent $2L$ for th...
Let $p$ be an odd prime, let $m\geq1$ be an integer, and let $Q$ be a nondegenerate quadratic form on $\mathbb F_p^{2m}$. For a nonempty set $E\subseteq\mathbb F_p^{2m}$, write $\Delta_Q(E)=\{Q(x-y):x,y\in E\}$. We prove that, whenever $|E|>p^m$, \[ |\Delta_Q(E)|\gg \min\left\{\frac{p}{L_E},\, \frac{|E|}{p^{m-1}L_E^2}\...
Thang Pham, Chun-Yen Shen, Dung The Tran et al.· 1 citation· ⚡1
Fix a prime $\ell$. Let $K = \mathbb{F}_q(t)$ be a global function field such that $\gcd(q,6) = 1$ and $q \equiv 1 \pmod \ell$. Let $E$ be a non-isotrivial elliptic curve over $K$. Given a fixed monic polynomial $Q$ over $\mathbb{F}_q$, and assuming some mild conditions on $E$, we show that the rank of $E$ does not cha...
Let $q=p^a$ with $p$ prime, $\lambda\in\mathbb F_p^\times$, and $\psi_q=\psi\circ\mathrm{Tr}_{\mathbb F_q/\mathbb F_p}$ for a nontrivial additive character $\psi$ of $\mathbb F_p$. For $n\ge2$ we study $\Sigma_n(h)=\sum_{u\in(\mathbb F_q^\times)^n,\sum_i h_i u_i=-\lambda}\psi_q(\sum_i u_i^{-1})$. On $\mathbb G_m^n$ we...
Hamed Ebadi· 0 citations
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