p-adic Hodge theory

p-adic Hodge theory is a branch of mathematics that studies relationships between algebraic objects defined over p-adic fields (extensions of the rationals with p-adic valuations) and geometric structures like ‘vibes’ or ‘cohomologies’ associated with algebraic varieties. It seeks to understand how p-adic properties encode geometric and arithmetic information, connecting different perspectives—such as algebraic, geometric, and Galois representations—through a framework that reveals deep insights into number theory and algebraic geometry. Essentially, it links discrete p-adic data to continuous geometric structures, enriching our understanding of solutions to polynomial equations in number theory.