Number Theory – Mathematics 2540
Brown University – Spring, 2018
Professor Joseph Silverman

Topic We will study Diophantine Geometry
Text I'll be covering material from:
Diophantine Geometry: An Introduction (Graduate Texts in Mathematics) , Marc Hindry and Joseph H. Silverman,
Springer-Verlag, ISBN: 978-0387989815 – 1st edition, © 2000. ($74.99 softcover, $59.99 eBook)
(These are the prices listed on the Springer website. But the book should be available for much less through Brown.)
Office Mathematics Department, Kassar House, Room 202
Phone 863-1124
Email jhs@math.brown.edu
Web Site www.math.brown.edu/~jhs/MA0254/MA0254HomePage.html
Office Hours By appointment, send me an email and I'll find a time we can meet. I'm generally on campus on MWF, seldom on TTh.
Course Time MWF 10:00–10:50am (C hour)
Course Location Kassar 105
Homework Homework assignments are posted below.
Note on Using Computers in Math 254 I tend to use a computer program called PARI-GP to do number theory calculations. The good news about PARI is that it is free and very fast and powerful at doing number theoretic computations. The bad news is that it's not tremendouly user friendly, and also somewhat limited at doing calculations in algebraic number fields. You can download PARI by clicking here. An alternative to PARI is SAGE, which is also free. You can get SAGE at the SAGE web site. (You can also run PARI from within SAGE.) I've also found Magma to be good for computations in algebraic number theory and algebraic geometry.

Schedule, Reading Assignments, HW Assignments
Week Chapter Sections Topic HW to turn in / Due date Problems to look at

Course Goals: To learn fundamental methods and results in Diophantine geometry, which is the study of points on on algebraic varieties that are defined over number fields and their rings of integers.

Learning Activities and Time Allocation: Learning activities include class attendance, frequent problem sets, and a take-home final exam. The time to complete these activities are (1) attending lectures, approximately 3 hours/week; (2) working on the problem sets and the final exam, approximately 9 hours/week.

Assessment: Course grades will be determined by the quantity and quality of problem sets submitted (80% of grade) and by the grade on the takehome final exam (20% of grade).

Expectations of Students: It is expected that students will attend all lectures and participate in class discussion in an appropriate manner. Assignments are due on the listed dates. All students are expected to abide by Brown's academic code, which may found here

Tentative Syllabus:

  1. The Geometry of Curves and Abelian Varieties
  2. Height Functions
  3. Abelian Varieties and the Mordell-Weil Theorem
  4. Diophantine Approximation
  5. Integral Points on Curves of Genus ≥ 1
  6. Rational Points on Curves of Genus ≥ 2
Further topics as time permits:
  • Geometry, Arithmetic, and Vojta's Conjectures
  • Local/Global Obstructions
  • Rational Points on K3 Surfaces

Go to Professor Silverman's Home Page.