Hilbert 模簇的挠点层覆盖上有理点的稀疏性
Sparsity of rational points on torsion level covers of Hilbert modular varieties
- Purdue University（普渡大学）
- University of Toronto（多伦多大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明 Hilbert 模簇挠点层覆盖上有理点随水平范数增大而稀疏，给出对数典范度下界，并推广了几何挠点定理。
AI中文摘要：
设 $F$ 为次数 $n$、判别式 $\Delta_F$ 的全实域，$X_1(\eta)$ 为 Hilbert 模簇的覆盖，参数化具有 $\mathcal O_F$ 实乘法的阿贝尔簇，以及一个零化子为 $\eta$ 的挠点。设 $L=K_{\overline X_1(\eta)}+D$ 为光滑环面紧化上的对数典范丛，$H_L$ 为相应的乘法高度。我们证明，当 $|\mathrm{Nm}(\eta)|\to\infty$ 且 $(\eta,\Delta_F)=1$ 时，$X_1(\eta)$ 上的有理点变得稀疏。更精确地，对每个数域 $K$，设 $$ N_{\eta,K}(B)=\\#\{x\in X_1(\eta)(K):H_L(x)\leq B\}. $$ 若 $|\mathrm{Nm}(\eta)|\ge 5^n$ 且 $(\eta,\Delta_F)=1$，我们证明 $$ \limsup_{B\to\infty}\frac{\log\max\{1,N_{\eta,K}(B)\}}{\log B} \leq\delta_{\eta,K,n},\qquad \delta_{\eta,K,n}\ll_{[K:\mathbb Q],n}|\mathrm{Nm}(\eta)|^{-1/(2n)}. $$ 特别地，当 $n$ 和 $[K:\mathbb Q]$ 有界且 $|\mathrm{Nm}(\eta)|\to\infty$ 时，$\delta_{\eta,K,n}\to0$ 一致成立。主要几何结果是对 $X_1(\eta)$ 的子簇的对数典范次数给出随水平增长的一致下界。将该估计与由 Ellenberg--Lawrence--Venkatesh 和 Brunebarbe--Maculan 基于行列式方法的最新进展相结合，我们得到上述稀疏性结果。我们还证明，对于充分大的 $|\mathrm{Nm}(\eta)|$，$X_1(\eta)$ 的每个子簇都是一般型的，并建立了 Bakker--Tsimerman 几何挠点定理的高维推广。即，对于任意维拟射影基上的具有实乘法的阿贝尔簇族，我们用基的典范体积来界定其 Mordell--Weil 群的挠子群，且在固定次数的全实乘法域上一致成立。
英文摘要：
Let $F$ be a totally real field of degree $n$ and discriminant $Δ_F$, and let $X_1(η)$ be the cover of the Hilbert modular variety parametrizing abelian varieties with real multiplication by $\mathcal O_F$, together with a torsion point having annihilator $η$. Let $L=K_{\overline X_1(η)}+D$ be the log-canonical bundle on a smooth toroidal compactification, and let $H_L$ be an associated multiplicative height. We prove that rational points on $X_1(η)$ become sparser as $|\mathrm{Nm}(η)|\to\infty$, with $(η,Δ_F)=1$. More precisely, for every number field $K$, set $$ N_{η,K}(B)=\#\{x\in X_1(η)(K):H_L(x)\leq B\}. $$ If $|\mathrm{Nm}(η)|\ge 5^n$ and $(η,Δ_F)=1$, we prove $$ \limsup_{B\to\infty}\frac{\log\max\{1,N_{η,K}(B)\}}{\log B} \leqδ_{η,K,n},\qquad δ_{η,K,n}\ll_{[K:\mathbb Q],n}|\mathrm{Nm}(η)|^{-1/(2n)}. $$ In particular, $δ_{η,K,n}\to0$ uniformly when $n$ and $[K:\mathbb Q]$ are bounded and $|\mathrm{Nm}(η)|\to\infty$. The main geometric result is a uniform lower bound, growing with the level, for the log-canonical degree of subvarieties of $X_1(η)$. Combining this estimate with recent progress derived from determinant-method, due to Ellenberg--Lawrence--Venkatesh and Brunebarbe--Maculan, we obtain the sparsity result above. We also prove that, for sufficiently large $|\mathrm{Nm}(η)|$, every subvariety of $X_1(η)$ is of general type, and establish a higher-dimensional generalization of the geometric torsion theorem of Bakker--Tsimerman. Namely, for a family of abelian varieties with real multiplication over a quasi-projective base of arbitrary dimension, we bound the torsion subgroup of its Mordell--Weil group in terms of the canonical volume of the base, uniformly in the totally real multiplication field of fixed degree.
