🔢Arithmetic Geometry Unit 1 – Algebraic Number Theory Basics

Key Concepts and Definitions

• Algebraic number theory studies algebraic structures related to algebraic numbers, which are roots of polynomials with integer coefficients
• Number fields are finite extensions of the rational numbers $\mathbb{Q}$ obtained by adjoining algebraic numbers
• Algebraic integers are elements of a number field that satisfy a monic polynomial equation with integer coefficients
• Rings of integers are the integral closure of $\mathbb{Z}$ in a number field, consisting of all algebraic integers in that field
• Ideals are subsets of rings that absorb multiplication by ring elements and play a crucial role in studying the structure of rings
• Prime ideals are ideals that cannot be written as the product of two smaller ideals, analogous to prime numbers in $\mathbb{Z}$
• Norm and trace are functions that map elements of a number field to rational numbers, providing important arithmetic information
• The Dirichlet Unit Theorem describes the structure of the unit group of a number field, which consists of elements with multiplicative inverses

Number Fields and Algebraic Integers

• A number field $K$ is a finite extension of $\mathbb{Q}$ obtained by adjoining an algebraic number $\alpha$ to $\mathbb{Q}$, denoted as $K = \mathbb{Q}(\alpha)$
  • Example: $\mathbb{Q}(\sqrt{2})$ is a number field obtained by adjoining $\sqrt{2}$ to $\mathbb{Q}$
• The degree of a number field $[K:\mathbb{Q}]$ is the dimension of $K$ as a vector space over $\mathbb{Q}$
• An algebraic integer is an element $\alpha \in K$ that satisfies a monic polynomial equation with integer coefficients: $\alpha^n + a_{n-1}\alpha^{n-1} + \cdots + a_1\alpha + a_0 = 0$, where $a_i \in \mathbb{Z}$
• The set of all algebraic integers in a number field $K$ forms a ring, denoted as $\mathcal{O}_K$
  • Example: In $\mathbb{Q}(\sqrt{2})$, the algebraic integers are of the form $a + b\sqrt{2}$, where $a, b \in \mathbb{Z}$
• Algebraic integers have important properties, such as forming a lattice in the complex plane and having a well-defined norm and trace
• The ring of integers $\mathcal{O}_K$ is a Dedekind domain, which means it has unique factorization of ideals into prime ideals

Rings of Integers and Ideals

• The ring of integers $\mathcal{O}_K$ of a number field $K$ is the integral closure of $\mathbb{Z}$ in $K$, consisting of all algebraic integers in $K$
• An ideal $I$ of $\mathcal{O}_K$ is a subset of $\mathcal{O}_K$ that is closed under addition and multiplication by elements of $\mathcal{O}_K$
  • Example: In $\mathbb{Z}[\sqrt{-5}]$, the ideal $(2, 1 + \sqrt{-5})$ consists of elements of the form $2a + (1 + \sqrt{-5})b$, where $a, b \in \mathbb{Z}[\sqrt{-5}]$
• Principal ideals are ideals generated by a single element, i.e., of the form $(\alpha) = \{\alpha \beta : \beta \in \mathcal{O}_K\}$ for some $\alpha \in \mathcal{O}_K$
• Prime ideals are ideals that cannot be written as the product of two smaller ideals, playing a role similar to prime numbers in $\mathbb{Z}$
• The ideal class group of $\mathcal{O}_K$ is the quotient group of fractional ideals modulo principal ideals, measuring the failure of unique factorization of elements in $\mathcal{O}_K$
• Dedekind domains, such as $\mathcal{O}_K$, have the property that every ideal can be uniquely factored into a product of prime ideals

Prime Factorization in Number Fields

• In a number field $K$, prime ideals take the place of prime numbers in the factorization of elements and ideals
• Every non-zero ideal $I$ in the ring of integers $\mathcal{O}_K$ can be uniquely factored as a product of prime ideals: $I = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_r^{e_r}$
  • Example: In $\mathbb{Q}(\sqrt{-5})$, the ideal $(6)$ factors as $(6) = (2, 1 + \sqrt{-5})^2 (3, 1 + \sqrt{-5}) (3, 1 - \sqrt{-5})$
• The factorization of a prime number $p \in \mathbb{Z}$ in $\mathcal{O}_K$ is determined by the splitting behavior of the minimal polynomial of $\alpha$ modulo $p$, where $K = \mathbb{Q}(\alpha)$
• Primes can split, remain inert, or ramify in $\mathcal{O}_K$, depending on the number of prime ideal factors and their exponents
  • Split: $p\mathcal{O}_K = \mathfrak{p}_1 \cdots \mathfrak{p}_r$, with distinct prime ideals $\mathfrak{p}_i$
  • Inert: $p\mathcal{O}_K$ remains prime in $\mathcal{O}_K$
  • Ramified: $p\mathcal{O}_K = \mathfrak{p}^e$, with $e > 1$
• The Dedekind-Kummer theorem relates the splitting of primes to the factorization of the minimal polynomial modulo $p$

Norm and Trace

• The norm $N_{K/\mathbb{Q}}(\alpha)$ of an element $\alpha$ in a number field $K$ is the product of all conjugates of $\alpha$, i.e., the roots of its minimal polynomial
  • Example: For $\alpha = a + b\sqrt{d} \in \mathbb{Q}(\sqrt{d})$, $N_{K/\mathbb{Q}}(\alpha) = (a + b\sqrt{d})(a - b\sqrt{d}) = a^2 - db^2$
• The trace $Tr_{K/\mathbb{Q}}(\alpha)$ of an element $\alpha$ in a number field $K$ is the sum of all conjugates of $\alpha$
  • Example: For $\alpha = a + b\sqrt{d} \in \mathbb{Q}(\sqrt{d})$, $Tr_{K/\mathbb{Q}}(\alpha) = (a + b\sqrt{d}) + (a - b\sqrt{d}) = 2a$
• Norm and trace are multiplicative and additive, respectively: $N_{K/\mathbb{Q}}(\alpha\beta) = N_{K/\mathbb{Q}}(\alpha)N_{K/\mathbb{Q}}(\beta)$ and $Tr_{K/\mathbb{Q}}(\alpha + \beta) = Tr_{K/\mathbb{Q}}(\alpha) + Tr_{K/\mathbb{Q}}(\beta)$
• The norm of an ideal $I$ in $\mathcal{O}_K$ is defined as the index $[\mathcal{O}_K : I]$, which is the size of the quotient ring $\mathcal{O}_K/I$
• Norms of ideals are multiplicative: $N(IJ) = N(I)N(J)$ for ideals $I$ and $J$ in $\mathcal{O}_K$
• The norm and trace of elements and ideals provide important arithmetic information and are used in various applications, such as solving Diophantine equations and studying the distribution of prime ideals

Dirichlet Unit Theorem

• The unit group $\mathcal{O}_K^{\times}$ of the ring of integers $\mathcal{O}_K$ consists of elements with multiplicative inverses in $\mathcal{O}_K$
• The Dirichlet Unit Theorem describes the structure of $\mathcal{O}_K^{\times}$ as a finitely generated abelian group
• For a number field $K$ with $r_1$ real embeddings and $r_2$ pairs of complex embeddings, $\mathcal{O}_K^{\times} \cong \mu_K \times \mathbb{Z}^{r_1 + r_2 - 1}$, where $\mu_K$ is the group of roots of unity in $K$
  • Example: For $K = \mathbb{Q}(\sqrt{2})$, $r_1 = 2$, $r_2 = 0$, and $\mathcal{O}_K^{\times} \cong \{\pm 1\} \times \mathbb{Z}$, generated by $-1$ and $1 + \sqrt{2}$
• The generators of the free part of $\mathcal{O}_K^{\times}$ are called fundamental units and can be computed using algorithms such as the Voronoi algorithm or the Buchmann-Lenstra algorithm
• The regulator $R_K$ of a number field $K$ is a positive real number that measures the density of the unit group $\mathcal{O}_K^{\times}$ and appears in the class number formula
• The Dirichlet Unit Theorem has applications in solving Diophantine equations, studying the distribution of prime ideals, and computing class numbers

Class Groups and Class Numbers

• The ideal class group $Cl_K$ of a number field $K$ is the quotient group of fractional ideals modulo principal ideals
• The class number $h_K$ is the order of the ideal class group $Cl_K$ and measures the failure of unique factorization of elements in $\mathcal{O}_K$
  • Example: For $K = \mathbb{Q}(\sqrt{-5})$, $h_K = 2$, indicating that there are two ideal classes in $Cl_K$
• The class group is a finite abelian group, and its structure can be computed using algorithms such as the Buchmann-Lenstra algorithm or the Hafner-McCurley algorithm
• The class number formula relates the class number $h_K$ to other invariants of the number field, such as the regulator $R_K$, the discriminant $\Delta_K$, and the Dedekind zeta function $\zeta_K(s)$
• The Brauer-Siegel theorem provides an asymptotic estimate for the class number and regulator of a number field as the discriminant grows
• Class groups and class numbers have applications in solving Diophantine equations, studying the distribution of prime ideals, and constructing abelian extensions of number fields

Applications in Arithmetic Geometry

• Algebraic number theory provides a foundation for arithmetic geometry, which studies geometric objects defined over number fields and their arithmetic properties
• Elliptic curves over number fields are a central object of study in arithmetic geometry, and their properties are closely related to the arithmetic of the underlying number field
  • Example: The Mordell-Weil theorem states that the group of rational points on an elliptic curve over a number field is finitely generated
• The Birch and Swinnerton-Dyer conjecture relates the rank of the group of rational points on an elliptic curve to the behavior of its L-function, connecting arithmetic and analytic properties
• The Shafarevich-Tate group of an elliptic curve measures the failure of the local-to-global principle for rational points and is conjectured to be finite (Shafarevich-Tate conjecture)
• Modular forms and Galois representations are powerful tools in arithmetic geometry that have led to significant advances, such as the proof of Fermat's Last Theorem by Wiles and Taylor-Wiles
• The Langlands program is a vast network of conjectures that relate arithmetic properties of geometric objects to automorphic forms and Galois representations, providing a unifying framework for arithmetic geometry
• Other important topics in arithmetic geometry include abelian varieties, Shimura varieties, p-adic Hodge theory, and arithmetic intersection theory, all of which rely on the foundations of algebraic number theory.