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

 Topic We will study Diophantine Geometry 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.) Mathematics Department, Kassar House, Room 202 863-1124 jhs@math.brown.edu www.math.brown.edu/~jhs/MA0254/MA0254HomePage.html 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. MWF 10:00–10:50am (C hour) Kassar 105 Homework assignments are posted below. Shamil Asgarli has posted some course notes for Math 2540. See the link on his homepage. 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.

Reading assignments consist of two parts: Algebraic geometry (AG) background in Part A and Diophantine geometry (DG) reading in Parts B,C,D. I'll assign an assortment of problems, but you should look at all of the problems and try working on the ones that sound interesting.
 Classes Reading Topic HW to turn in / Due date Problems to look at 1 Jan 23–28 (AG) § A.1 (DG) § B.1–B.2 Height Functions # B.1, B.2, B.4, B.20 / Friday Feb 1 A.1.1–A.1.9 2 Jan 30–Feb 8 (AG) § A.2, A.3, A.5, A.7 (DG) § B.3, B.4, C.1 Height Functions Abelian Varieties # B.3, B.5(a,b), B.6, B.10, B.11, B.13, B.15, B23(a) / Weds Feb 20 — 3 Feb 11–Mar 1 (AG) § A.4, A.6, A.7, A.8 (DG) § B.5, B.6, C.2, D.1, D.2 Mordell-Weil Theorem Points on Curves # B.7, B.8, B.16, C.1, C.2, C.3(a,b), C.7*, 14 / Mon Mar 11 * Challenge problem — 4 Mar 11–Apr 3 (DG) § D.3–D.7 Diophantine Approximation Roth's Theorem # D.1, D.2(a,b), D.3(a), D.4, D.5, D.9, D.12 / Fri Apr 5 — 5 Apr 5–Apr 19 (DG) § D.6–D.9 Roth's Theorem Diophantine equations # D.6, D.7, D.8, D.16 (use Prop. D.9.4) / Mon Apr 22 Prove that Index(  .  , β1,  β2, r, s) is a valuation on C[X,Y]. Try to do this without looking at the book for hints. — 6 Apr 22–May 6 (DG) § E.1 – E.12 Mordell Conjecture This assignment is the take-home final exam # E.1, E.2, E.3, E.9, E.15, E.16 / Mon May 6 Extra Problem #D.∞ (click link) For #E.15 and #E.16, first read §§E.12.5–E.12.9 (If you want, it's okay to take extra time and turn it in by Mon May 13.) —

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
• Chabauty-Coleman Approach to the Mordell Conjecture
• Sketch of Faltings' Proof of the Mordell Conjecture
• Specialization Theorems
• Rational Points on K3 Surfaces