Introduction to Arithmetic Geometry by Andrew V. Sutherland
Introduction to Arithmetic Geometry by Andrew V. Sutherland
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.
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.
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.