Horizontal variation of Tate--Shafarevich groups
Skip to main content
eScholarship
Open Access Publications from the University of California

Horizontal variation of Tate--Shafarevich groups

  • Author(s): Burungale, Ashay A
  • Hida, Haruzo
  • Tian, Ye
  • et al.
No data is associated with this publication.
Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $\iota: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field. For a class group character $\chi$ over $K$, let $\mathbb{Q}(\chi)$ be the field generated by the image of $\chi$ and $\mathfrak{p}_{\chi}$ the prime of $\mathbb{Q}(\chi)$ above $p$ determined via $\iota_p$. Under mild hypotheses, we show that the number of class group characters $\chi$ such that the $\chi$-isotypic Tate--Shafarevich group of $E$ over $H_{K}$ is finite with trivial $\mathfrak{p}_{\chi}$-part increases with the absolute value of the discriminant of $K$.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content

This item is under embargo until December 20, 2020.