This note covers
the following topics: Rational points on varieties, Heights, Arakelov Geometry,
Abelian Varieties, The Brauer-Manin Obstruction, Birational Geomery, Statistics
of Rational Points, Zeta functions.
note explains the following topics: Galois Modules, Discrete Valuation
Rings, The Galois Theory of Local Fields, Ramification Groups, Witt Vectors,
Projective Limits of Groups of Units of Finite Fields, The Absolute Galois Group
of a Local Field, Group Cohomology, Galois Cohomology, Abelian Varieties, Selmer
Groups of Abelian Varieties, Kummer Theory, Torsors for Algebraic Groups, The
Main Theorem, Operators on Modular Curves, Heegner Points, Hecke Operators on
Heegner Points and Local Behavior of Cohomology Classes.
This note explains the following topics: Diophantine equations ,
Algebraic curves, The projective plane , Genus, Birational equivalence, The
elliptic curve group law , Rational points on elliptic curves, The Sato-Tate
conjecture, The Birch and Swinnerton-Dyer conjecture, Fermatís Last Theorem,
Jacobians of curves.
Major topics topics coverd are:
Absolute values on fields, Ostrowski's classification of absolute values on U,
Cauchy sequences and completion, Inverse limits,Properties of Zp, The field of P
-Adic numbers, P-adic expansions, Hensel's lemma, Finite fields, Profinite
groups, Affine varieties, Morphisms and rational maps, Quadratic forms, Rational
points on conics and Valuations on the function field of a curve.