Sparsity of rational points on torsion level covers of Hilbert modular varieties

Abstract

Let $F$ be a totally real field of degree $n$ and discriminant $\Delta_F$, and let $X_1(\eta)$ 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 $\eta$. Let $L=K_{\overline X_1(\eta)}+D$ 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 $X_1(\eta)$ become sparser as $|\mathrm{Nm}(\eta)|\to\infty$, with $(\eta,\Delta_F)=1$. More precisely, for every number field $K$, set $$ N_{\eta,K}(B)=\#\{x\in X_1(\eta)(K):H_L(x)\leq B\}. $$ If $|\mathrm{Nm}(\eta)|\ge 5^n$ and $(\eta,\Delta_F)=1$, we prove $$ \limsup_{B\to\infty}\frac{\log\max\{1,N_{\eta,K}(B)\}}{\log B} \leq\delta_{\eta,K,n},\qquad \delta_{\eta,K,n}\ll_{[K:\mathbb Q],n}|\mathrm{Nm}(\eta)|^{-1/(2n)}. $$ In particular, $\delta_{\eta,K,n}\to0$ uniformly when $n$ and $[K:\mathbb Q]$ are bounded and $|\mathrm{Nm}(\eta)|\to\infty$. The main geometric result is a uniform lower bound, growing with the level, for the log-canonical degree of subvarieties of $X_1(\eta)$. 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 $|\mathrm{Nm}(\eta)|$, every subvariety of $X_1(\eta)$ 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.