This
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
covers the following topics: Integration on valuation fields over local fields,
Integration on product spaces and GLn of a valuation field over a local field,
Fubinis theorem and non linear changes of variables over a two dimensional local
field, Two dimensional integration la Hrushovski Kazhdan, Ramification, Fubinis
theorem and Riemann Hurwitz formulae and an explicit approach to residues on and
canonical sheaves of arithmetic surfaces.
This PDF Lectures covers the
following topics related to Arithmetic Geometry : Operations with modules,
Schemes and projective schemes, Rings of dimension one, The compactified Picard
group of an order of a number field, Different, discriminant and conductor, The
classic theorems of the algebraic number theory, Heights of rational points on a
scheme over a number field.
The aim
of these notes is to describe some examples of modular forms whose Fourier
coefficients involve quantities from arithmetical algebraic geometry.
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.