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

作者：Lei, Antonio^*; Lim, Meng Fai

来源：International Journal of Number Theory, 2022, 18(02): 303-330.

DOI：10.1142/S1793042122500208

• Iwasawa theory
• elliptic curves
• Mordell-Weil groups
• Tate-Shafarevich groups
• mixed-reduction type

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 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 lwasawa algebra, we show that the Mordell-Weil ranks of E over a subextension of the cyclotomic 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.

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