A field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are always possible. The rational numbers $\mathbb{Q}$, the real...
| Section | Title |
|---|---|
| 1 | Chapter 4. Algebraic Number Theory |
| 2 | Splitting Fields |
| 3 | Galois Groups |
| 4 | Finite Fields |
| 5 | Cyclotomic Fields |
| 6 | Ramification |
| 7 | Absolute Values |
| 8 | -Adic Numbers |
| 9 | Completion of Fields |
| 10 | Hensel’s Lemma |
| 11 | Local-Global Principles |
| 12 | Adeles and Ideles |
| 13 | Abelian Extensions |
| 14 | Reciprocity Maps |
| 15 | Hilbert Class Fields |
| 16 | Local Class Field Theory |
| 17 | Global Class Field Theory |
| 18 | Modular Groups |
| 19 | Modular Functions |
| 20 | Modular Forms |
| 21 | Eisenstein Series |
| 22 | Cusp Forms |
| 23 | Hecke Operators |
| 24 | Modular Curves |
| 25 | Elliptic Curves and Modularity |
| 26 | The Modularity Theorem |
| 27 | Automorphic Forms |
| 28 | Automorphic Representations |
| 29 | The Langlands Program |
| 30 | Galois Representations |
| 31 | Functoriality |
| 32 | Automorphic -Functions |
| 33 | Trace Formulas |
| 34 | Shimura Varieties |
| 35 | Geometric Langlands Theory |
Chapter 4. Algebraic Number TheoryA field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are always possible. The rational numbers $\mathbb{Q}$, the real...
Splitting FieldsA central problem in algebra is to determine where a polynomial factors completely into linear terms. Consider the polynomial
Galois GroupsA polynomial equation may possess several roots related by hidden algebraic symmetries. Consider
Finite FieldsThe familiar fields
Cyclotomic FieldsOne of the most important classes of number fields arises from the solutions of the equation
RamificationOne of the central ideas of algebraic number theory is that prime numbers may behave differently after passing to a larger field.
Absolute ValuesThe ordinary absolute value on the real numbers measures magnitude:
$p$-Adic NumbersThe real numbers arise by completing the rational numbers with respect to the ordinary absolute value. This completion produces a field suited to Euclidean geometry and...
Completion of FieldsThe rational numbers form a field rich enough for arithmetic, yet insufficient for many limiting processes.
Hensel’s LemmaOne of the central ideas of number theory is that congruences modulo powers of a prime often approximate genuine arithmetic solutions.
Local-Global PrinciplesA central problem in number theory is determining whether an equation possesses rational or integral solutions.
Adeles and IdelesThe rational numbers may be studied through their completions:
Abelian ExtensionsA central goal of algebraic number theory is to understand field extensions of a given base field, especially extensions of the rational numbers
Reciprocity MapsOne of the oldest themes in number theory is reciprocity: the phenomenon that solvability conditions for one prime are controlled by arithmetic involving another prime.
Hilbert Class FieldsOne of the central discoveries of algebraic number theory is that unique factorization may fail in rings of algebraic integers.
Local Class Field TheoryGlobal class field theory studies finite abelian extensions of number fields such as
Global Class Field TheoryOne of the central goals of algebraic number theory is to classify field extensions of a number field
Modular GroupsModular forms begin with the action of certain matrix groups on the complex upper half-plane.
Modular FunctionsThe modular group acts on the upper half-plane by fractional linear transformations:
Modular FormsModular forms are among the central objects of modern number theory.
Eisenstein SeriesAmong all modular forms, Eisenstein series are the most explicit and computationally accessible.
Cusp FormsModular forms satisfy strong symmetry conditions under the modular group. Among them, cusp forms form the deepest and most arithmetic subclass.
Hecke OperatorsModular forms already possess symmetry under the modular group. Yet a deeper arithmetic structure emerges through another family of operators: the Hecke operators.
Modular CurvesThe modular group acts on the upper half-plane by fractional linear transformations:
Elliptic Curves and ModularityAn elliptic curve is simultaneously:
The Modularity TheoremFor centuries, elliptic curves and modular forms were studied as separate objects.
Automorphic FormsModular forms are functions on the upper half-plane satisfying symmetry conditions under the modular group
Automorphic RepresentationsClassical modular form theory begins with analytic functions satisfying symmetry conditions.
The Langlands ProgramThe Langlands program is one of the most ambitious and influential theories in modern mathematics.
Galois RepresentationsGalois groups encode the symmetries of algebraic equations and field extensions.
FunctorialityThe Langlands program predicts that many different arithmetic objects are connected by systematic transfers.
Automorphic $L$-FunctionsThe Riemann zeta function
Trace FormulasFourier analysis decomposes functions into harmonic frequencies.
Shimura VarietiesModular curves parameterize elliptic curves and connect modular forms with arithmetic geometry.
Geometric Langlands TheoryThe classical Langlands program relates: