We introduce a conjecture on the arithmetic sparcity of the degeneration locus of a $p$-adic local system on a smooth variety over a number field and, modulo the Bombieri–Lang conjecture, show that it follows from a conjecture on the geometry of the level varieties attached to the local system. We present a few applications of our conjecture to classical problems in arithmetic geometry. Eventually, we give some evidences and discuss a few perspectives to attack it, in particular for $p$-adic local systems arising from geometry.

中文翻译：

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Qp-局部系统的退化轨迹：猜想

我们提出一个关于a的退化轨迹的算术稀疏性的猜想$p$-数域上光滑簇上的adic局部系统，并以Bombieri-Lang猜想为模，表明它是根据局部系统上的水平簇的几何猜想得出的。我们提出了我们的猜想在算术几何经典问题上的一些应用。最终，我们给出了一些证据并讨论了一些攻击它的观点，特别是对于$p$-源自几何的adic局部系统。