Root number of the twists of an elliptic curve

TL;DR

This paper provides an explicit description of how the root number varies in families of elliptic curve twists by polynomial values, including criteria for constant root number and cases with bad reduction.

Contribution

It offers a detailed analysis of root number behavior in elliptic curve twists, extending previous work to cases with bad reduction and special j-invariants.

Findings

Criteria for constant root number families over 

Complete description of root number behavior with bad reduction

Analysis of cases with j-invariant 0 and 1728

Abstract

We give an explicit description of the behaviour of the root number in the family given by twists of an elliptic curve $E$ by the rational values of a polynomial $f(T)$. In particular, we give a criterion (on $f$ depending on $E$) for the family to have a constant root number over $\mathbb{Q}$. This completes a work of Rohrlich: we detail the behaviour of the root number when $E$ has bad reduction over $\mathbb{Q}^{ab}$ and we treat the cases $j(E)=0,1728$ which were not considered previously.