🔢Arithmetic Geometry Unit 8 – p–adic analysis and geometry

Foundations of p-adic Numbers

• p-adic numbers extend the rational numbers $\mathbb{Q}$ by completing with respect to the p-adic norm $|\cdot|_p$ for a prime $p$
• The p-adic norm measures the divisibility of an integer by the prime $p$, with $|x|_p = p^{-k}$ if $p^k$ is the highest power of $p$ dividing $x$
• p-adic numbers have a unique representation as a power series $\sum_{i=k}^\infty a_i p^i$ with coefficients $a_i \in \{0, 1, \ldots, p-1\}$ and $k \in \mathbb{Z}$
• The set of p-adic integers $\mathbb{Z}_p = \{x \in \mathbb{Q}_p : |x|_p \leq 1\}$ forms a subring of the p-adic numbers $\mathbb{Q}_p$
  • $\mathbb{Z}_p$ consists of power series with non-negative exponents
• p-adic numbers are complete with respect to the p-adic metric, meaning Cauchy sequences converge
• Hensel's lemma lifts solutions of polynomial equations from modulo $p$ to $\mathbb{Z}_p$ under certain conditions
  • Enables solving polynomial equations in $\mathbb{Q}_p$ that may not have solutions in $\mathbb{Q}$

p-adic Topology and Metrics

• The p-adic norm induces a topology on $\mathbb{Q}_p$ distinct from the usual Euclidean topology
• Open sets in the p-adic topology are unions of balls $B_r(a) = \{x \in \mathbb{Q}_p : |x - a|_p < r\}$ for $a \in \mathbb{Q}_p$ and $r > 0$
• The p-adic topology is ultrametric, satisfying the strong triangle inequality $|x + y|_p \leq \max(|x|_p, |y|_p)$
  • Leads to unique properties like every point of a ball being its center
• $\mathbb{Z}_p$ is the unit ball in $\mathbb{Q}_p$ and is both open and closed
• Continuous functions in the p-adic topology are uniformly continuous due to the ultrametric property
• Sequences converge in the p-adic topology if and only if their terms become increasingly divisible by $p$
• Completions of algebraic extensions of $\mathbb{Q}_p$ yield local fields with analogous topological properties

Analytic Functions in p-adic Fields

• Analytic functions over p-adic fields have convergent power series expansions
• Mahler expansions express continuous functions on $\mathbb{Z}_p$ as series in binomial coefficients
• The Tate algebra $\mathbb{Q}_p\langle T_1, \ldots, T_n \rangle$ consists of power series with coefficients in $\mathbb{Q}_p$ converging on the unit polydisk
  • Nonarchimedean analog of holomorphic functions on the complex unit polydisk
• Rigid analytic spaces are built from p-adic analytic functions analogously to complex analytic spaces
  • Enables a form of algebraic geometry over p-adic fields
• Fourier transforms and Poisson summation formulas have p-adic analogs for studying p-adic analytic functions
• p-adic L-functions interpolate special values of complex L-functions and are constructed using p-adic analysis
  • Crucial tool in modern number theory connecting arithmetic and analytic properties

p-adic Differential Calculus

• Derivatives of p-adic analytic functions are defined by formal differentiation of power series
• Higher derivatives $\frac{d^n}{dx^n}f(x)$ exist for all $n \geq 0$ and p-adic analytic functions are smooth
• The p-adic exponential function $\exp_p(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$ converges for $|x|_p < p^{-1/(p-1)}$
  • Locally invertible with inverse the p-adic logarithm $\log_p(1+x) = \sum_{n=1}^\infty (-1)^{n-1}\frac{x^n}{n}$ for $|x|_p < 1$
• p-adic analytic functions satisfy a form of the inverse function theorem
• Newton polygons analyze zeros and poles of p-adic meromorphic functions
• p-adic differential equations have a rich theory analogous to complex differential equations
  • Used to study p-adic dynamical systems and p-adic cohomology theories

Geometry of p-adic Spaces

• Algebraic varieties over $\mathbb{Q}_p$ can be studied geometrically using rigid analytic spaces or Berkovich spaces
• Rigid analytic spaces are ringed spaces locally modeled on affinoid algebras like the Tate algebra
  • Captures p-adic analytic geometry but lacks some nice topological properties
• Berkovich spaces associate to $\mathbb{Q}_p$-varieties a locally ringed space that is Hausdorff and contractible
  • Provides a more flexible framework for p-adic analytic geometry
• Formal schemes are a p-adic analog of complex analytic spaces built from formal power series rings
  • Models "infinitesimal" neighborhoods in p-adic geometry
• Étale cohomology of p-adic varieties connects to p-adic Hodge theory and Galois representations
• p-adic uniformization describes certain p-adic analytic spaces in terms of p-adic symmetric spaces
  • Analogous to complex uniformization (Schottky, Mumford curves)

Applications in Arithmetic Geometry

• p-adic Hodge theory studies p-adic Galois representations and their relation to p-adic cohomology theories
  • De Rham, crystalline, and semi-stable representations capture arithmetic data
• p-adic variation of Hodge structures describes families of Galois representations over p-adic analytic spaces
• p-adic Siegel modular forms are p-adic analytic functions on Siegel upper half spaces with arithmetic significance
• The Langlands program seeks to relate Galois representations to automorphic forms and has important p-adic aspects
  • p-adic automorphic forms and p-adic Langlands correspondence
• p-adic methods are used to study rational points on algebraic varieties
  • Chabauty's method, Coleman integration, non-abelian Chabauty
• Iwasawa theory studies the growth of arithmetic objects (class groups, Selmer groups) in towers of number fields
  • Formulated in terms of p-adic analytic objects (p-adic L-functions, Iwasawa algebras)

Advanced Topics and Open Problems

• p-adic Hodge theory beyond $\mathbb{Q}_p$, such as for p-adic fields with imperfect residue fields
• Relative p-adic Hodge theory for families of Galois representations over p-adic analytic bases
• p-adic Langlands correspondence for groups beyond $\mathrm{GL}_2(\mathbb{Q}_p)$, such as for $\mathrm{GL}_n(\mathbb{Q}_p)$ or quaternion algebras
• Geometrization of the local Langlands correspondence using moduli spaces of Galois representations
• Iwasawa main conjectures relating p-adic L-functions to Selmer groups in towers of number fields
  • Formulated for elliptic curves, modular forms, Hilbert modular forms, etc.
• Analogues of the Birch and Swinnerton-Dyer conjecture in p-adic settings
• Cohomological methods to study rational points and Diophantine equations using p-adic techniques

Key Theorems and Proofs

• Ostrowski's theorem classifies non-trivial absolute values on $\mathbb{Q}$ as equivalent to $|\cdot|_p$ or $|\cdot|_\infty$
• Hensel's lemma for lifting solutions of polynomial equations from modulo $p$ to $\mathbb{Z}_p$
• Weierstrass preparation theorem for factoring p-adic analytic functions into polynomials and units
• Tate's theorem on p-adic uniformization of elliptic curves with split multiplicative reduction
• Fontaine's periods rings $B_\mathrm{dR}, B_\mathrm{cris}, B_\mathrm{st}$ for classifying p-adic Galois representations
• Faltings' theorem (Tate conjecture) relating Hodge-Tate weights to Galois actions on étale cohomology
• Theorems on the existence and interpolation properties of p-adic L-functions
• Comparison theorems between p-adic cohomology theories (de Rham, crystalline, étale)