Introductory description

The “Topics in Number Theory” module will rotate from year to year, according to the research interests of the module lecturer. Examples of areas which may be covered include

• Diophantine equations
• Rational points on algebraic varieties
• Modular and automorphic forms
• Galois representations and Iwasawa theory
• Analytic number theory and zeta-functions
• Combinatorial number theory and arithmetic statistics
  In the initial year 2020-21 the module will focus on rational points on algebraic varieties.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The topics for this module will rotate from year to year there will be no fixed syllabus. Examples of areas which may be covered include

• Rational points on algebraic varieties
• Modular and automorphic forms
• Galois representations and Iwasawa theory
• Analytic number theory and zeta-functions
• Combinatorial number theory and arithmetic statistics

Learning outcomes

By the end of the module, students should be able to:

• By the end of the module, students should have acquired advanced knowledge of a specific area of number theory of contemporary research interest. For instance, in 2020-21 students will be familiar with “local-to-global” problems and obstruction theory in the context of algebraic surfaces, and will be able to compute examples of Brauer groups in concrete cases arising from del Pezzo surfaces.

Indicative reading list

The module will not follow any one book particularly closely. For the 2020-21 instance of the module, indicative references include:

• H. Cohen, “Number Theory Volume I: Tools and Diophantine Equations”
• A. Varilly-Alvarado, “Lectures on the arithmetic of Del Pezzo surfaces”, available from https://math.rice.edu/~av15/Files/LeidenLectures.pdf
• M. Bright, D. Testa, R. van Luijk, “Geometry and Arithmetic of Surfaces”, lecture notes for a mini-course available from http://www.boojum.org.uk/maths/book.html

Subject specific skills

This will vary from year to year according to the topics chosen. In 2020-1 students will acquire the following skills:

• Ability to investigate and solve Diophantine problems involving existence of rational or integral points on algebraic varieties, focusing on curves and surfaces
• Interplay between geometric and arithmetic properties of algebraic varieties
• Fluency in working with local and global fields
• Understanding of the distribution of these solutions and their relation to local solvability over p-adic fields

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organizing information
• presentation skills (both oral and written)