A Note on the Growth of Sha in Dihedral Extensions

TL;DR

This paper derives formulas for the size of the Tate--Shafarevich group of elliptic curves over dihedral extensions of number fields, relating it to the groups over quadratic subextensions, with specific results for odd and even degrees.

Contribution

It provides explicit formulas for the Tate--Shafarevich group over dihedral extensions, extending understanding of its behavior in these Galois extensions.

Findings

The order of the Tate--Shafarevich group over dihedral extensions relates to that over quadratic subextensions.

Formulas are valid up to fourth powers and primes dividing the extension degree.

Results apply to both odd and even degree dihedral extensions.

Abstract

We provide a formula for the order of the Tate--Shafarevich group of elliptic curves over dihedral extensions of number fields of order $2n$, up to $4^{th}$ powers and primes dividing $n$. Specifically, for odd $n$ it is equal to the order of the Tate--Shafarevich group over the quadratic subextension. A similar formula holds for even $n$.