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Course Goal
State the Birch and Swinnerton-Dyer Conjecture for modular abelian varieties and explain some of the main theorems and computational evidence for it. Along the way build up foundations needed to understand modular abelian varieties starting from basic algebraic geometry and number theory. Give lots of examples.
Evaluation
There will be homework (60%) and a final project (40%), but no exams.
- Homework (60% of grade). Weekly problem sets that will be graded.
- Final Project (40% of grade). A paper about something related to the course. This could be a computation, discussion of examples, or a proof of something not proved in the lectures.
Reading
The textbooks for the course are Seminar on Fermat’s Last Theorem (ed. Murty) and Arithmetic Geometry (ed. Cornell and Silverman). The following articles from those two books will be very relevant to the course material, and I recommend you at least skim them all:
- Rosen: Abelian Varieties over C
- Milne: Abelian Varieties (Sections 1, 2, 7, 8, 9, 10, 11, 12, 16, 19)
- Silverman: The Theory of Height Functions (Sections 1-5)
- Milne: Jacobian Varieties (Sections 1-6 and 10)
- Artin: Neron Models (Sections 1 and 2)
- Murty: Modular Elliptic Curves (Sections 1-4)
- Diamond-Im: Modular Forms and Modular Curves (Part I, Part II sections 7, 9.1, 9.2, 10.1)
- Ribet-Stein: Lectures on Serre’s Conjecture
- Kolyvagin: Bounding Selmer Groups via the Theory of Euler Systems (Section 0 only)
The direction of the course will follow the first 11 chapters of a book I’m writing with Ken Ribet that grew out of a graduate course he taught at Berkeley in 1996. Ribet’s course gave some of the foundations needed to understand Wiles’s proof of Fermat’s Last Theorem. In contrast, Math 252 will be aimed at laying the groundwork so that you can understand the Birch and Swinnerton-Dyer conjecture for modular abelian varieties.