Sparsity of \(p\)-divisible unramified liftings for subvarieties of abelian varieties with trivial stabilizer (Q2636544)

The proof of the Manin-Mumford conjecture by Raynaud relies on the following local lemma: let \(X\) be a curve of genus \(\geq 2\) in an abelian variety \(A\) over a number field \(K\), extending to \({\mathcal X}\) and \({\mathcal A}\) over a subscheme \(U\subset\text{Spec}{\mathcal O}_K\) not containing any ramified primes. For \(\wp\in U\), if \(R_n\) is the ring of Witt vectors of length \(n+1\) with coefficients in \(R_0=\overline{k(\wp)}\), we denote by \(X_{\wp^n}\) the lift \({\mathcal X}\times_U\text{Spec}R_n\), and similarly for \({\mathcal A}\). In this situation, the map \(pA_{\wp^1}(R_1)\cap X_{\wp^1}(R_1)\to X_{\wp^0}(R_0)\) has finite (i.e.\ not Zariski dense) image (where \(p\) is the maximal ideal of the ring of all Witt vectors). In other words, most \(\overline{k(\wp)}\) points of \(X\) will not lift to \(p\)-divisible points in \(A\) over the first Witt ring. This paper proves the same thing for \(X\) of arbitrary dimension, provided that \(X\) has trivial stabiliser (essentially this means that it does not contain an abelian variety) and \(p\) is not too small. The proof of Raynaud's lemma is that if the image is dense then the Frobenius \(F_{X_{\wp^0}}\) iduces an automorphism of \(X_{\wp^0}\) That would give a nontrivial map \(F^*_{X_{\wp^0}}\Omega_{X_{\wp^0}}\to \Omega_{X_{\wp^0}}\), which is impossible for degree reasons. Here, the degree is replaced by data taken from the Harder-Narasimhan filtration, following [\textit{D. Rössler}, ``Strongly semistable sheaves and the Mordell-Lang conjecture over function fields'', Preprint, \url{arXiv:1412.7303}]. The techniques used to check that this works also allow the author to give a bound on the number of components of the first critical scheme (whose \(R_0\)-points are precisely \(pA_{\wp^1}(R_1)\cap X_{\wp^1}(R_1)\)) following the corresponding step in the effective version of Manin-Mumford due to \textit{A. Buium} [Duke Math. J. 82, No. 2, 349--367 (1996; Zbl 0882.14007)].