标题： 上同调与同余 显示英文标题

标题： Cohomology and congruences

摘要： 这些是我于2024年夏季在ICTP的数论与物理学校以及在圣保罗的IME-USP举行的第27届巴西代数会议上的讲座课程笔记。 我们对代数超曲面的德拉姆上同调中的$p$有理方法进行了基础性的阐述，包括具体的例子和在数论与组合学中的应用。 这些讲座基于我们与弗里茨·博克尔斯共同撰写的系列论文，题为\emph{德沃克晶体}(\cite{DCI,DCII,DCIII})。 这些方法在数学物理和算术几何（\cite{IN,Cartier0}）中也有应用，我们在最后对此进行了概述。 我感谢两个学校的组织者以及参加我课程的学员，他们的问题激发了这些笔记的撰写。 显示英文摘要

摘要： These are notes of my lecture courses given in the summer of 2024 in the School on Number Theory and Physics at ICTP in Trieste and in the 27th Brazilian Algebra Meeting at IME-USP in S\~ao Paulo. We give an elementary account of $p$-adic methods in de Rham cohomology of algebraic hypersurfaces with explicit examples and applications in number theory and combinatorics. These lectures are based on the series of our joint papers with Frits Beukers entitled \emph{Dwork crystals} (\cite{DCI,DCII,DCIII}). These methods also have applications in mathematical physics and arithmetic geometry (\cite{IN,Cartier0}), which we overview here towards the end. I am grateful to the organisers of both schools and to the participants of my courses whose questions stimulated writing these notes.