MATH 83530: Algebra II

Course Outline:  

This course is for students who have the level of Algebra I as it is taught here at the Graduate Center. Vector spaces over a field,  modules and ideals over a ring,  noetherian rings and the basic theorems of Hilbert (finite basis theorem and Nullstellensatz) will be considered are known or will be very quicly reviewed. WE will use the basic notions of Algebraic Number Theory like: Number Fields , Ring of Integers, Class group. We will recall without proofs the facts we need at the beginning of the class to make it accessible to a  student of Algebra I (Sources: P.Samuel “Algebraic Numbers”, My notes “Basic Aritmetic Geometry” on my web site at gc.cuny.edu – the French version is more complete).

We will teach the following:

1. Tensor products and localisation

2. The Picard Group 

3. Affine and projective schemes. Blow ups

4. Differentials , differents and discriminants

5. Height of rational points of a scheme over a number field

We will then introduced the canonical height associated to a dynamical system on the Riemann Sphere. We will study such dynamical systems from an algebraic point of view. In particular we will look at the dynamics associated to the multiplication by 2 in an elliptic curve . We will relate these notions and the questions they raised to the abc conjecture and the Lehmer conjecture.

 

Books:

Hindry – Silvermann “Introduction to Diophantine Geometry”(Springer)

Everest - Ward“Heights of Polynomials and Entropy in Algebraic Dynamics”

 

Article:

With J.Pineiro and T.Tucker “Mahler measure for Dynamical systems on P1 and Intersection Theory on a Singular Arithmetic Surface” Progress in Math Birkhauser  (235).  Accessible on my web page.

Last Modified on: 09/09/2006

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