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Euler Systems and Arithmetic Geometry
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.
Author(s): Barry Mazur and Tom Weston
168 Pages 
Investigations in Two Dimensional Arithmetic Geometry
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.
Author(s): Matthew Morrow
182 Pages
Orientation Theory in Arithmetic Geometry
This note explains the following topics : Notations and conventions, Absolute cohomology and purity, Functoriality instable homotopy, Absolute cohomology, Absolute purity, Analytical invariance, Orientation and characteristic classes, Orientation theory and Chern classes, Thom classes and MGL modules, Fundamental classes, Intersection theory, Gysin morphisms and localization long exact sequence, Residues and the case of closed immersions, Projective lci morphisms, Uniqueness, Riemann Roch formulas, Todd classes, The case of closed immersions, The general case, Principle of computation, Change of orientation, Universal formulas and the Chern character, Residues and symbols, Residual Riemann Roch formula The axiomatic of Panin revisited Axioms for arithmetic cohomologies and Etale cohomology.
Author(s): Frederic Deglise
82 Pages
Introduction to Arithmetic Geometry by Andrew V. Sutherland
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, Fermat’s Last Theorem, Jacobians of curves.
Author(s): Andrew V. Sutherland
36 Pages
Arithmetic Geometry Lecture Notes
This lecture note explains everything about Arithmetic Geometry.
Author(s): Arizona University
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