Local Fields and Valuations - Applications

Local fields enable explicit solution of Diophantine equations, construction of representations, and understanding of global phenomena locally.

TheoremHasse-Minkowski Theorem (Local-Global for Quadratic Forms)

A quadratic form $Q$ over $\mathbb{Q}$ represents zero nontrivially if and only if $Q$ represents zero over $\mathbb{R}$ and all $\mathbb{Q}_p$ .

Application: To determine if $ax^2 + by^2 + cz^2 = 0$ has nontrivial solutions, check:

Real condition: at most one of $a, b, c$ has the same sign as the others
$p$ -adic conditions for finitely many $p$ (those dividing $2abc$ )

This reduces an infinite problem (checking all rationals) to finitely many local checks.

ExampleSolving Conics

Does $x^2 + y^2 = 3z^2$ have nontrivial rational solutions?

Check locally:

$\mathbb{R}$ : Yes, $(1, 1, \sqrt{2/3})$ works approximately
$\mathbb{Q}_2$ : Check if $-1$ is a square times 3 in $\mathbb{Q}_2^*$
$\mathbb{Q}_3$ : Check similarly
$\mathbb{Q}_p$ for $p > 3$ : Automatic by Hensel

If all local conditions satisfy, solution exists rationally. Indeed, $(1, 1, 0)$ is trivial; need proper analysis showing solution $(2, 1, 1)$ exists.

Theorem$p$-adic Representations

Galois representations $\rho: \text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to GL_n(\mathbb{Q}_p)$ arise from:

Tate modules of abelian varieties
Étale cohomology of varieties
Modular forms (via Eichler-Shimura)

Local behavior at $p$ (restriction to $\text{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$ ) determines ramification, crystalline/semistable properties, and $L$ -function factors.

The Fontaine-Mazur conjecture characterizes geometric representations via their local properties at all primes.

ExampleComputing with $p$-adic Numbers

Modern computational algebra systems implement $p$ -adic arithmetic:

\\ Pari/GP: Solve x^2 = 2 in Q_7
localprec(20);
sqrt(2, 7)
\\ Output: 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + ...

\\ Factor polynomial over Q_5
factorpadic(x^3 - 2, 5, 10)

These computations use Hensel lifting and Newton's method $p$ -adically.

TheoremLocal Fields and Class Field Theory

Global class field theory reduces to local class field theory via product formula: $\prod_v \text{inv}_v: \prod_v H^2(K_v, \bar{K}_v^*) \to \mathbb{Q}/\mathbb{Z}$

This sum of local invariants equals zero for global elements, providing the fundamental class enabling Artin reciprocity.

Local Brauer groups $\text{Br}(K_v) \cong \mathbb{Q}/\mathbb{Z}$ are explicitly computable, making global reciprocity tractable.

ExampleCryptography and $p$-adics

Elliptic curves over $\mathbb{Q}_p$ are used in:

$p$ -adic point counting: Computing $E(\mathbb{F}_{p^n})$ via lifting to $\mathbb{Q}_p$
Formal groups: The formal group of $E$ over $\mathbb{Z}_p$ enables fast scalar multiplication
Isogenies: Rational isogenies extend to $p$ -adic isogenies, used in SIDH/CSIDH protocols

Local analysis provides efficient algorithms unavailable globally.