Dirichlet's Unit Theorem - Applications

Dirichlet's Unit Theorem has profound applications to Diophantine equations, transcendence theory, and computational number theory.

TheoremSolving Unit Equations

The unit equation $\epsilon_1 + \epsilon_2 = 1$ with $\epsilon_1, \epsilon_2 \in \mathcal{O}_K^*$ has only finitely many solutions.

Proof idea: The logarithmic embedding maps solutions to a finite set of hyperplanes in $\mathbb{R}^r$ . Since $\mathcal{O}_K^*$ is a lattice, intersections are finite.

This finiteness result extends to $S$ -unit equations $\sum_{i=1}^n a_i \epsilon_i = 0$ with $a_i \in K^*$ and $\epsilon_i \in \mathcal{O}_{K,S}^*$ , fundamental to Baker's theorem and effectivity in Diophantine equations.

ExampleCatalan's Equation

Catalan's conjecture (now Mihăilescu's theorem) states $3^2 - 2^3 = 1$ is the only solution to $x^p - y^q = 1$ with $x, y, p, q > 1$ .

The proof uses cyclotomic fields $\mathbb{Q}(\zeta_p, \zeta_q)$ and deep properties of their unit groups. The crucial step shows $p$ divides the class number of $\mathbb{Q}(\zeta_q)$ leads to contradiction via class field theory and properties of cyclotomic units.

TheoremABC Conjecture and Units

The ABC conjecture predicts: for coprime $a, b, c$ with $a + b = c$ : $c \leq K(\epsilon) \cdot \text{rad}(abc)^{1+\epsilon}$

for any $\epsilon > 0$ and some $K(\epsilon)$ . This would imply:

Fermat's Last Theorem (for large exponents)
Roth's theorem (effective bounds)
Finiteness of many Diophantine equations

Units play a role via the quality $q = \frac{\log c}{\log \text{rad}(abc)}$ , related to regulator growth in number fields.

ExampleThue Equations

Equations of the form $F(x, y) = m$ where $F$ is an irreducible binary form of degree $\geq 3$ have finitely many integer solutions (Thue's theorem).

The proof uses units in $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of $F(x, 1) = 0$ . Bounding solutions requires estimating regulators and using Baker's theory of linear forms in logarithms.

For $x^3 - 2y^3 = 1$ : working in $\mathbb{Q}(\sqrt[3]{2})$ and analyzing units gives unique solution $(x, y) = (1, 0)$ .

TheoremMordell-Weil Theorem

For an elliptic curve $E$ over a number field $K$ , the group $E(K)$ of $K$ -rational points is finitely generated: $E(K) \cong \mathbb{Z}^r \times T$

where $T$ is the finite torsion subgroup and $r \geq 0$ is the rank.

The proof parallels Dirichlet's unit theorem: a height function plays the role of the logarithmic embedding, and descent shows bounded height points are finite.

ExampleComputing in Magma/Pari

Modern computer algebra systems compute units via:

\\ Pari/GP example for Q(sqrt(5))
K = bnfinit(x^2 - 5);
K.fu  \\ fundamental units
K.reg \\ regulator

\\ Output: [1/2 + 1/2*sqrt(5)] (golden ratio)
\\ Regulator: 0.4812118250596...

These computations are essential for determining class numbers, solving norm equations, and verifying conjectures computationally.

TheoremGalois Module Structure

When $L/K$ is Galois with group $G$ , the unit group $\mathcal{O}_L^*$ is a $\mathbb{Z}[G]$ -module. The Galois module structure reveals:

Which units come from the base field
Relations imposed by Galois action
Connections to class groups via cohomology

For cyclotomic fields, circular units form a $\mathbb{Z}[G]$ -submodule of finite index, with the index related to $p$ -adic $L$ -functions (Iwasawa theory).

Remark

Units appear throughout arithmetic geometry:

Heights on varieties: Néron-Tate height generalizes norm
Arakelov theory: Regulators appear in intersection theory
Iwasawa theory: Growth of unit groups in towers of fields

Understanding units is thus fundamental to modern arithmetic.