Beauville surfaces without real structures -- Couniformization of curves over number fields -- On the V-filtration of -modules -- Hecke orbits on Siegel modular varieties -- Ax-Kochen-Eršov Theorems for p-adic integrals and motivic integration -- Nested sets and Jeffrey-Kirwan residues -- Counting extensions of function fields with bounded discriminant and specified Galois group -- Classical a…
Modular Forms, Elliptic Curves, and Modular Curves -- Modular Curves as Riemann Surfaces -- Dimension Formulas -- Eisenstein Series -- Hecke Operators -- Jacobians and Abelian Varieties -- Modular Curves as Algebraic Curves -- The Eichler-Shimura Relation and L-functions -- Galois Representations. .This book introduces the theory of modular forms with an eye toward the Modularity Theorem: All r…
Field extensions -- Roots -- Galois theory -- A bit of group theory -- Applications -- Algebraic theory of differential equations.This unique textbook focuses on the structure of fields and is intended for a second course in abstract algebra. Besides providing proofs of the transcendance of pi and e, the book includes material on differential Galois groups and a proof of Hilbert's irreducibilit…
Infinite Galois Theory and Profinite Groups -- Valuations and Linear Disjointness -- Algebraic Function Fields of One Variable -- The Riemann Hypothesis for Function Fields -- Plane Curves -- The Chebotarev Density Theorem -- Ultraproducts -- Decision Procedures -- Algebraically Closed Fields -- Elements of Algebraic Geometry -- Pseudo Algebraically Closed Fields -- Hilbertian Fields -- The Cla…
A Fundamental Theorem -- Topics in Algebra -- Some Quadratic Problems -- The Genus of an Algebraic Curve -- Miscellany."... The exposition is not only clear, it is friendly, philosophical, and considerate even to the most naive or inexperienced reader. And it proves that the philosophical orientation of an author really can make a big difference. The mathematical content is intensely classical.…
Complex Numbers in Algebraic Form -- Complex Numbers in Trigonometric Form -- Complex Numbers and Geometry -- More on Complex Numbers and Geometry -- Olympiad-Caliber Problems -- Answers, Hints and Solutions to Proposed Problems.It is impossible to imagine modern mathematics without complex numbers. Complex Numbers from A to . . . Z introduces the reader to this fascinating subject that, from t…
Parte 1 -- Un cifrario del XVIII secolo -- La prima macchina cifrante al mondo -- Damm, Hagelin e Gyldén -- Intercettazione di segnali radio e analisi crittografica prima del 1939 -- Guerra -- Entra in scena Arne Beurling -- La flotta russa del Baltico -- Segnali misteriosi -- Telescriventi -- L’analisi di Beurling -- La G-Schreiber e gli “app” -- Crittoanalisi sistematica -- Esce di sce…
A Brief History of Prime -- Diophantine Equations -- Quadratic Diophantine Equations -- Recovering the Fundamental Theorem of Arithmetic -- Elliptic Curves -- Elliptic Functions -- Heights -- The Riemann Zeta Function -- The Functional Equation of the Riemann Zeta Function -- Primes in an Arithmetic Progression -- Converging Streams -- Computational Number Theory.An Introduction to Number Theor…