Higher dimensional geometry of $p$-jets
Lance Edward Miller, Jackson S. Morrow
TL;DR
The paper proves a quantitative unramified Manin–Mumford bound for smooth subvarieties $X$ of abelian varieties $A$ with ample cotangent bundle, showing finiteness of $X(F^{\mathrm{alg}}) \cap A(F^{\mathrm{alg}})[\mathrm{non-}p\text{-tors}]$ for large $p$ and giving an explicit bound expressed via Segre classes on the reduction $X_0$. The method follows Buium’s arithmetic jet–space program, establishing the affineness of the special fiber of the first jet space and performing effective intersection-theoretic computations in the Chow ring to bound torsion cosets. A key novelty is the higher-dimensional generalization of the jet-space affineness and the explicit bound, which extend the curve case and yield polynomial-in-$p$ control in special complete-intersection cases via results of Debarre. The combination yields both a precise quantitative result for unramified torsion and an effective global Manin–Mumford statement in higher dimensions, with applications to complete intersections in abelian varieties and explicit bounds tied to intersection data and theta divisors.
Abstract
In this work, we prove a quantitative version of the prime-to-$p$ Manin--Mumford conjecture for varieties with ample cotangent bundle. More precisely, let $A$ be an abelian variety defined over a number field $F$, and let $X$ be a smooth projective subvariety of $A$ with ample cotangent bundle. We prove that for every prime $p\gg 0$, the intersection of $X(F^{\text{alg}})$ and the geometric prime-to-$p$ torsion of $A$ is finite and explicitly bounded by a summation involving cycle classes in the Chow ring of the reduction of $X$ modulo $p$. This result is a higher dimensional analogue of Buium's quantitative Manin--Mumford for curves. Our proof follows a similar outline to Buium's in that it heavily relies on his theory of arithmetic jet spaces. In this context, we prove that the special fiber of the arithmetic jet space associated to a model of $X$ is affine as a scheme over $\mathbb{F}_p^{\text{alg}}$. As an application of our results, we use a result of Debarre to prove that when $X$ is $\mathbb{Q}^{\text{alg}}$-isomorphic to a complete intersection of $c > \text{dim}(A)/2$ many general hypersurfaces of $A_{\mathbb{Q}^{\text{alg}}}$ of sufficiently large degree, the intersection of $X(F^{\text{alg}})$ and the geometric prime-to-$p$ torsion of $A$ is bounded by a polynomial that depends only on $p$, the dimension of the ambient abelian variety, and intersection numbers of certain products of the hypersurfaces.