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MA939 - Topics in Number Theory

Lecturer: Damiano Testa

Term(s): Term 2

Commitment: 30 lectures

2 live events each week, online at least until the end of January:

  • Tuesdays: 9-10am -- live, online event
  • Wednesdays: 9-10am -- live, online event

I will also post videos with further material for the module, throughout the term.

Assessment: Oral exam

Prerequisites: Galois Theory, Algebraic Number Theory, Algebraic Geometry.

Of course, only select topics from each of these modules will be useful. However, a general understanding of the basic results of these modules is beneficial. If you are willing to put in the extra effort to read up on missing prerequisites, you might pick up enough background material to fill the gaps.

Content : The main focus of the module is to give techniques for determining the set of rational points of an algebraic variety.

There are several specialized tools and approaches that can be used in this context:

  • local and global solutions and the Hasse principle,
  • rationality and unirationality over general fields,
  • del Pezzo surfaces,
  • Brauer groups,
  • Galois cohomology and Brauer-Manin obstructions.

The end goal is to discuss “local-to-global” problems and obstruction theory in the context of algebraic surfaces. The guiding objective is the computation the Brauer group in concrete cases and its use to obstruct the existence of rational points.

References: The main reference is draft book that Martin Bright, Ronald van Luijk and I are writing on the topic. If you find any errors, typos or have any form of feedback, please, email me and let me know!