Practical information

This course took place in Bristol, and online on Teams (as part of TCC) in May and June 2025.

Questions, comments, and remarks are encouraged. My contact details are on my main page.

Abstract

The Birch and Swinnerton-Dyer conjecture is a millennium problem originally stated for elliptic curves over the rationals and later generalised to abelian varieties over number fields. In this course, we will learn about the different geometric and arithmetic invariants that occur in the conjecture and their significance in the broader context of arithmetic geometry.

Notes

You can find some notes here (TeX). Any errors in it are mine. Please feel encouraged to report such errors to me. Other suggestions are also welcome.

Prerequisites

Certainly you don't need to know all of these topics to join the course, but if you do, it might help you connect it to other things you know about. There might not be enough time in the course to discuss these topics in the full detail they deserve, so you might find it beneficial to read a bit about these topics yourself between the lectures.

• Elliptic curves / abelian varieties (over the complex numbers, over number, local and finite fields)
• Algebraic geometry (curves, schemes)
• Number fields, local / p-adic fields (extensions, Galois theory, class groups)
• Cohomology (sheaf cohomology, group cohomology, Galois cohomology)

Possible topics

Below you can find a list of topics covered.

• Introduction to the conjecture and abelian varieties (lecture 1)
• Models of curves and abelian varieties (lecture 2)
• Periods and Néron differential (lecture 3/4)
• Regulators, heights, and Arakelov theory (lecuture 4/5)
• Tate-Shafarevich group, Selmer groups, descent, local-global principle, torsors and H1 (lecture 5/6)
• Tamagawa numbers (lecture 7)
• L-functions, Tate module, and conductor (lecture 7)
• Known results: modularity, Heegner points, Gross-Zagier, Euler systems, invariance under isogeny and Weil restriction, parity (lecture 8)