Arithmetic Geometry & p-adic Hodge Theory
Arithmetic geometry bridges number theory and algebraic geometry, studying solutions to polynomial equations over number fields. p-adic Hodge theory, revolutionized by Peter Scholze's perfectoid spaces, provides powerful tools for understanding Galois representations and arithmetic phenomena.
Fundamental Framework
p-adic Numbers
The p-adic numbers $\mathbb{Q}_p$ are the completion of $\mathbb{Q}$ with respect to the p-adic absolute value:
$$|x|_p = p^{-v_p(x)}$$where $v_p(x)$ is the p-adic valuation (highest power of p dividing x).
Scholze's Tilting Equivalence
For a perfectoid space $X$ over a perfectoid field $K$, there exists a perfectoid space $X^♭$ over $K^♭$ such that:
$$X_{\text{ét}} \simeq X^♭_{\text{ét}}$$This equivalence of étale sites is fundamental to modern p-adic geometry.
p-adic Number Visualization
Perfectoid Spaces
Definition (Scholze)
A perfectoid space is an adic space $X$ over a perfectoid field such that:
- $X$ is uniform
- The Frobenius map $\Phi: \mathcal{O}_X^+ / p \to \mathcal{O}_X^+ / p$ is surjective
Galois Representations & p-adic Hodge Theory
de Rham Cohomology
Crystalline Cohomology
Fontaine's Theory
The category of p-adic Galois representations admits a hierarchy:
$$\text{Rep}_{\text{cris}}(G_K) \subset \text{Rep}_{\text{st}}(G_K) \subset \text{Rep}_{\text{dR}}(G_K) \subset \text{Rep}_{\mathbb{Q}_p}(G_K)$$Prismatic Cohomology
Recent Breakthrough (Bhatt-Scholze, 2019)
Prismatic cohomology unifies various p-adic cohomology theories through the prismatic site, providing a single framework for:
- Crystalline cohomology
- de Rham cohomology
- Étale cohomology
- Hodge-Tate cohomology
p-adic Valuation Calculator
Compute p-adic Properties
Groundbreaking Research
Fields Medal winning work introducing perfectoid spaces and tilting equivalence.
Revolutionary unification of p-adic cohomology theories.
Geometric approach to p-adic Hodge theory via the fundamental curve.
Comprehensive introduction to modern p-adic geometry.
Leading Research Centers
- University of Bonn - Peter Scholze's group, epicenter of perfectoid theory
- IAS Princeton - Bhargav Bhatt, prismatic cohomology
- Harvard University - Mark Kisin, p-adic Hodge theory
- UC Berkeley - Arithmetic geometry group
- IHES Paris - Laurent Fargues, diamonds and adic spaces
Applications & Connections
Langlands Program
p-adic methods provide local components for the global Langlands correspondence.
Diophantine Equations
Understanding rational points via p-adic approximation and local-global principles.
Iwasawa Theory
Study of arithmetic objects in infinite p-adic towers of number fields.