Kobin A. Number Theory 2024

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Kobin A. Number Theory 2024

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Textbook in PDF format Elementary Number Theory Introduction Divisibility The Division Algorithm Greatest Common Divisors The Prime Numbers The Fundamental Theorem of Arithmetic The Infinitude of Primes Special Primes Linear Congruence Modular Arithmetic Linear Congruence Fermat's and Euler's Theorems Fermat's Little Theorem Euler's and Wilson's Theorems Public Key Cryptography Higher Order Congruence Finding Roots Primitive Roots Power Residues Reciprocity Quadratic Residues Quadratic Reciprocity Applications of Quadratic Reciprocity Analytic Number Theory Introduction Preliminaries Basic Analysis Euler-Maclaurin Summation The Bernoulli Numbers Euler's Work On the Sums of Series of Reciprocals Newton's Identities Euler's Product Form The Prime Number Theorem Complex Analysis Arithmetic Functions and Limits Line Integrals Differentiability Integration in the Complex Plane Singularities and the Residue Theorem Zeta Functions and L-Series The Functional Equation Finding the Zeros Sketch of the Prime Number Theorem Dirichlet Series Algebraic Number Theory Introduction Attempting Fermat's Last Theorem Algebraic Number Fields Integral Extensions of Rings Norm and Trace The Discriminant Factorization of Ideals Ramification Cyclotomic Fields and Quadratic Reciprocity Lattices Norms of Ideals The Class Group The Unit Theorem Local Fields Discrete Valuation Rings The p-adic Numbers Absolute Values Local Fields Henselian Fields Ramification Theory Extensions of Valuations Galois Theory of Valuations Higher Ramification Groups Discriminant and Different Adèlic Number Theory Restricted Direct Products Adèles and Idèles Idèle Class Group Class Field Theory Global Class Field Theory The Hilbert Class Field Orders Frobenius Automorphisms Ray Class Groups L-series and Dirichlet Density The Frobenius Density Theorem The Second Fundamental Inequality The Artin Reciprocity Theorem The Conductor Theorem The Existence and Classification Theorems The Cebotarev Density Theorem Ring Class Fields Quadratic Forms and n-Fermat Primes Binary Quadratic Forms The Form Class Group n-Fermat Primes Adèlic Class Field Theory Frobenius Elements Artin Reciprocity Kronecker-Weber Theorem Elliptic Curves Introduction Geometry and Number Theory Rational Curves Algebraic Geometry Affine and Projective Space Morphisms of Affine Varieties Morphisms of Projective Varieties Products of Varieties Blowing Up Dimension of Varieties Complete Varieties Tangent Space Local Parameters Curves Divisors Morphisms Between Curves Linear Equivalence Differentials The Riemann-Hurwitz Formula The Riemann-Roch Theorem The Canonical Map Bézout's Theorem Rational Points of Conics Elliptic Curves Weierstrass Equations Moduli Spaces The Group Law The Jacobian Rational Points on Elliptic Curves Isogenies The Dual Isogeny The Weil Conjectures Elliptic Curves over Local Fields Jacobians of Hyperelliptic Curves The Mordell-Weil Theorem Some Galois Cohomology Selmer and Tate-Shafarevich Groups Twists, Covers and Homogeneous Spaces Descent Heights Elliptic Curves and Complex Analysis Elliptic Functions Elliptic Curves The Classical Jacobian Jacobians of Higher Genus Curves Complex Multiplication Classical Complex Multiplication Torsion and Rational Points Class Field Theory with Elliptic Curves L-Functions Introduction Locally Compact Groups Topological Vector Spaces Banach Algebras The Gelfand Transform Spectral Theorems Unitary Representations Duality Functions of Positive Type Fourier Inversion Pontrjagin Duality Functional Equations Local -Functions Adèlic and Idèlic Characters Schwartz-Bruhat Functions and Riemann-Roch Global Zeta Functions and Functional Equations Hecke L-Functions Modular Forms Modular Forms The Upper Half-Plane Modular Functions and Modular Forms Modular Functions as Sections q-Expansions Hecke Operators Hecke Operators on Lattices Hecke Operators on Modular Functions Eigenfunctions Petersson Inner Product Theta Series Level Structure Congruence Subgroups Modular Curves Automorphic Forms Galois Cohomology (in progress)