MATH 894, Elliptic Surfaces (Reading Course)

Instructor: Nils Bruin
SC K 10507
(778) 782 3794
Textbook: Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, 151. Springer-Verlag, New York, 1994. xii+400 pp. ISBN: 0-387-94328-5

Background material and prerequisites:
Silverman, Joseph H. The arithmetic of elliptic curves. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986. xii+400 pp. ISBN: 0-387-96203-4

Pre/corequisite: A working knowledge of elliptic curves (as for instance presented in Silverman's "Arithmetic of elliptic curves") is essential. Knowledge and experience with algebraic geometry and algebraic number theory are also helpful. Exact prerequisites and determination if this course is suitable for any particular student can be discussed with the instructor.
Meetings: AQ 5048, Thursdays, 12:30 - 14:20
First meeting September 12
Goal and format: The group will meet weekly for a two hour session. Participants will present on different topics, which will be assigned at an organizational meeting at the start of term. In addition, there will be regular assignments (with questions taken from the course text). Some of these problems will be presented and discussed by the participants as well.

Lecturers, as a guide-line, should prepare a lecture of about an hour. Our time slot allows for overrun and discussion.

Topics: The core material to be covered in this course consists of Chapters III, IV from the course text.
  • Elliptic Surfaces
  • Heights on elliptic curves over function fields
  • Geometry of algebraic surfaces
  • Geometry of fibered surfaces
  • Geometry of elliptic curves
  • Heights and divisors on varieties
  • Specialization theorems for elliptic surfaces
  • Group varieties and schemes
  • Arithmetic surfaces
  • Neron models
  • Intersection theory, minimal models, and blowing up
  • The special fiber of a Neron model
  • Tate's algorithm
Lecture schedule:
Date Lecturer Title Sections
Sep. 12 Nils Planning and Introduction
Sep. 19 Aven Heights and split elliptic surfaces III.4-6
Sep. 26 Ahmad Geometry of of Algebraic Surfaces III.7-11
Oct. 3 Eugene Geometry of Fibered Surfaces III.7-11
Oct. 10 Pedro Elliptic surfaces and Heights and Divisors. III.7-11
Oct. 17 Ahmad Group varieties and group schemes IV.1-3
Oct. 24 Aven Arithmetic surfaces IV.4
Oct. 31 Eugene Néron models IV.5
Nov. 7 Pedro Existence of Néron models IV.6
Nov. 14 Aven Intersection Theory, Minimal Models, and Blowing-Up IV.7
Nov. 21 Eugene The special fiber of a Néron model IV.8
Nov. 28 Pedro Tate's algorithm for computing the special fiber IV.9