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