Root number in integer parameter families of elliptic curves

Abstract

In a previous article [7], the author proves that the value of the root number varies in a non-isotrivial family of elliptic curves indexed by one parameter t running through \({\mathbb {Q}}\). However, a well-known example of Washington has root number \(-1\) for every fiber when t runs through \({\mathbb {Z}}\). Such examples are rare since, as proven in this paper, the root number of the integer fibers varies for a large class of families of elliptic curves. This result depends on the squarefree conjecture and Chowla’s conjecture, and is unconditional in many cases.