2003-04 VIGRE Number Theory Working Group at the University of Michigan

Organized by Brian Conrad and James Parson

This year's topic was Mazur's proof of the torsion-boundedness conjecture for elliptic curves over the rationals. We worked through the first half of Mazur's paper ``Rational isogenies of prime degree'' and then filled in the required finiteness result for a Tate-Shafarevich group (which is a bit simpler for us since we are not aiming to prove all of the results in Mazur's big IHES paper). At the end we looked at Merel's work that extended Mazur's methods to solve the problem over any number field (following a paper of P. Parent). Some talks were TeXed up. Links are given below.

Overview for Term 1: pdf.
Ruling out small primes: pdf.
Moduli of elliptic curves: pdf.
Minimal models of elliptic curves: pdf.
Variant proof for Proposition 2.1: pdf.
Formal Immersion step: pdf.
Application of Mordell-Weil finiteness: pdf.
Overview for Term 2: pdf.
Some applications of Zariski's Main Theorem: pdf.
The Eisenstein descent: pdf.
Semistable fibers of modular curves: pdf.
Finiteness of Mordell-Weil: pdf.
Analytic lemma for Parent's paper: pdf.