The Hasse principle and twists of the modular curve X(p).

The Hasse principle is the idea that a Diophantine equation over the rational numbers should have a rational solution if and only if it has solutions in all of its completions, namely, the real numbers and all p-adic fields. In recent work of Lorenzo and Vullers, they give twists of the modular curve X(7) that are counterexamples to the Hasse principle. In this talk, we will discuss generalizations of their result, for example, that there are infinitely many counterexamples to the Hasse principle that are twists of the modular curve X(p) for primes p congruent to 1 mod 4.

This is joint work with Diana Mocanu.