An introduction to the Hasse principle through examples

Warwick's Postgraduate Seminar

I gave this talk when I was an invited speaker in the Mathematics Postgraduate Seminar of the University of Warwick on the 19th of January of 2022.

This talk is based on my master’s dissertation Local fields and the Hasse principle (Gonzalez-Hernandez, 2021).

References

2021

MSc thesis

Local fields and the Hasse principle

Alvaro Gonzalez-Hernandez

University of Oxford, 2021

Abs PDF

We discuss the main results of the theory of local fields and how it allows us to compute the ranks of elliptic curves through the method of complete 2-descent. The knowledge gained from this theory is then used to analyse a family of counterexamples to the Hasse principle depending on a prime parameter. This family of curves arises as the homogeneous spaces that appear when applying complete 2-descent to an elliptic curve with non-trivial Tate-Shafarevich group.