The Hasse local-to-global principle for some genus one curves
The Hasse principle is a useful guiding philosophy in arithmetic geometry that relates "global" questions to analogous "local" questions, which are often easier to understand. A simple incarnation of the Hasse principle says that a given polynomial equation has a solution in the rational numbers (i.e., is "globally soluble") if and only if it has a solution in the real numbers and in the p-adic numbers for all primes p (i.e., is "everywhere locally soluble"). While this principle holds for many "simple" such polynomials, it is a very difficult question to classify the polynomials (or more generally, algebraic varieties) for which the principle holds or fails. In this talk, we will discuss problems related to the Hasse principle for some classes of varieties, especially certain genus one curves. We will describe how to compute the proportion of these curves that have are everywhere locally soluble (joint work with Tom Fisher and Jennifer Park), and we will explain why the Hasse principle fails for a positive proportion of these curves (joint work with Manjul Bhargava).