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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 
Modular forms and arithmetic geometry
The aim of these notes is to describe modular forms whose Fourier coefficients involve quantities from arithmetic algebraic geometry. The examples suggest an arithmetic analogue of theta series, and a formula that relates averages of them to special values of derivatives of Eisenstein series.
Author(s): Stephen S. Kudla
57 Pages
Lectures on Local Systems in Algebraic Arithmetic Geometry
This note contains below topics, General Introduction, Kronecker s Rationality Criteria and Grothendieck s pCurvature Conjecture, Density of Special Loci, Rigid Local Systems and FIsocrystals, Rigid Local Systems FontaineLaffaille modules and Crystalline Local Systems.
Author(s): Hélène Esnault
69 Pages
p adic Analysis in Arithmetic Geometry
This note covers introduction, p adic numbers, Newton polygons, Multiplicative seminorms and berkovich space, The berkovich affine and projective line, Analytic spaces and function, Berkovich spaces of curves and integration.
Author(s): Michael Stoll
52 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 and nbsp,Etale cohomology.
Author(s): Frederic Deglise
82 Pages
Arithmetic and Algebraic Geometry
This PDF Lectures covers the following topics related to Arithmetic and Algebraic Geometry, Rings, Spectra, Affine Varieties, Projective Varieties, Regularity, Curves.
Author(s): Shou-Wu Zhang
74 Pages
Arithmetic Geometry by Prof Szpiro
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.
Author(s): Prof Szpiro
87 Pages