论文标题
P-ADIC积分和线性依赖于曲线家族I
p-adic integrals and linearly dependent points on families of curves I
论文作者
论文摘要
我们证明，在足够大的曲线家族的足够大的纤维力量上，“低等级”点并不是Zariski密集。 Dimitrov-Gao-Habegger和Kühne（和Yuan）的最新工作暗示了在等级中指数的界限，而Zilber-Pink的猜想意味着等级是线性的结合。我们的主要结果是（稍弱）的线性结合了“低级”。我们还证明了各向同性家族（等级的条件放松）和$ s $单位方程的解决方案的结果相似，在该方程中，现在的边界现在在等级中为次指数。我们的证明涉及家庭中chabauty-coleman（-kim）方法的概念（或从某种意义上说，对于简单的连接品种）。对于Zariski非密度，我们使用Blàzquez-Sanz，Casale，Freitag和Nagloo在Ax-Schanuel定理上的最新作品，用于主要捆绑的叶子。
We prove that the set of `low rank' points on sufficiently large fibre powers of families of curves are not Zariski dense. The recent work of Dimitrov-Gao-Habegger and Kühne (and Yuan) imply the existence of a bound which is exponential in the rank, and the Zilber-Pink conjecture implies a bound which is linear in the rank. Our main result is a (slightly weaker) linear bound for `low ranks'. We also prove analogous results for isotrivial families (with relaxed conditions on the rank) and for solutions to the $S$-unit equation, where the bounds are now sub-exponential in the rank. Our proof involves a notion of the Chabauty-Coleman(-Kim) method in families (or, in some sense, for simply connected varieties). For Zariski non-density, we use the recent work of Blàzquez-Sanz, Casale, Freitag and Nagloo on Ax-Schanuel theorems for foliations on principal bundles.
