Mordell-Weil ranks and Tate-Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

Abstract

Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic ${\mathbb{Z}}_p$-extension of a finite extension $F$ of $K$ where $p$ is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell-Weil ranks of $E$ over a subextension of the cyclotomic ${\mathbb{Z}}_p$-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions.

20 pages

References in corpus (1)

• Tamagawa numbers of some crystalline representations

Cited by in corpus (5)

• Euler Characteristics and their Congruences for Multi-signed Selmer Groups
• On fine Selmer groups and signed Selmer groups of elliptic modular forms
• Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p
• On the control theorem for fine Selmer groups and the growth of fine Tate-Shafarevich groups in ${\mathbb{Z}}_p$-extensions
• On the Mordell-Weil Ranks of supersingular abelian varieties over ${\mathbb{Z}}_p^2$-extensions