Geometry of \(p\)-jets
In the author's previous work on diophantine problems over function fields, methods of differential algebra were used. In this present work, the author concentrates on number fields where, of course, no non-trivial derivations exist. The author replaces these with certain ``natural'' non-additive maps called ``\(p\)-derivations''. The main result shown is the following quantitative version of the Manin-Mumford conjecture: Theorem: Let \(X\) be a curve defined over a number field \(K\) and let \(X\to J\) be the associated Abel map into the Jacobian of \(X\). Let \({\mathfrak p}\) be a prime of \(K\) with \(p= \text{char} {\mathfrak p} >2g\) with \(g\) the genus of \(X\). Assume that \(K/ \mathbb{Q}\) is unramified at \({\mathfrak p}\) and that \(X/K\) has good reduction at \({\mathfrak p}\). Let \(K^{ac}\) be the algebraic closure of \(K\). Then \[ \# (X(K^{ac}) \cap J(K^{ac})_{\text{tors}}) \leq p^{4g} 3^g[p(2g-2) +6g]\cdot g!. \]
scientific article; zbMATH DE number 908694
scientific article