Spring 2015-16

**Instructor:** Dan Abramovich

**Class meeting:** Mondays, Wednesdays and Fridays at 1:00-1:50am in Kassar 105

**Office:** Kassar 118

**Telephone:** (401) 863 7968

**E-mail:** `abrmovic ( at ) math (period) brown (dot) edu`

**Web site:**
`http://www.math.brown.edu/~abrmovic/MA/s1516/254/index.html`

** Canvas:** direct link

**Preliminary Office hours:** Wednesday, 2:00-3:00, Monday and Friday 10:00-10:50

**Text:** *Diophantine Geometry* by Hindry and Silverman, Springer GTM.

**Topics:** Heights; The Mordell-Weil theorem; The theorems of Roth and Siegel; Proof of Mordell's conjecture - Bombieri's combined approach.

I'll follow the outline on page 369, except I won't cover Part A.

** Course requirements **

Homework will be assigned regularly from the book.

Students are required to organize and run a study group * Algebraic geometry for number theorists*, following Part A, supplemented by Hartshorne's * Algebraic geometry*, especially chapter 4, Mumford's * Abelian varieties*, especially Chapters 1-3, and Cornell-Silverman's *Arithmetic Geometry*, chapters on abelian varieties and Jacobian varieties by Rosen and Milne.

** Comments on the book by us**, and
** Corrections and comments on the book by the authors**

** Notes **

** Absolute values **

** Heights on projective spaces **

** Heights on varieties **

** Canonical Heights **

** Mordell-Weil**

** D.1: Dirichlet and Liouville (by C. Frechette)**

** The S-unit equation**

** Faltings's theorem (Mordell's conjecture)**

** Presentation lineup **

Hermite's Theorems: Zhou Fang

Section D.1: Claire Frechette (presented March 4)

Section D.2: David Lowrie (March 7)

Section D.3: Thomas Silverman, Laura Walton

Section D.4: Dori Bejleri, Alicia Harper

Section D.5: Giovanny Inchiostro and Kenneth Ascher

Section D.6: Shamil Asgarli and Yuwei Zhu

Section D.7: Seoyoung Kim

** Homework **

First assignment (evolving): Exercises B2, B3, B4,