On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions

Lei, Antonio; Ponsinet, Gautier

doi:10.1090/bproc/43

On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions
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by Antonio Lei and Gautier Ponsinet HTML | PDF

Proc. Amer. Math. Soc. Ser. B 7 (2020), 1-16

Abstract:

Let $F$ be a number field unramified at an odd prime $p$ and let $F_\infty$ be the $\mathbf {Z}_p$-cyclotomic extension of $F$. Let $A$ be an abelian variety defined over $F$ with good supersingular reduction at all primes of $F$ above $p$. Büyükboduk and the first named author have defined modified Selmer groups associated to $A$ over $F_\infty$. Assuming that the Pontryagin dual of these Selmer groups is a torsion $\mathbf {Z}_p[[\textrm {Gal}(F_\infty /F)]]$-module, we give an explicit sufficient condition for the rank of the Mordell-Weil group $A(F_n)$ to be bounded as $n$ varies.

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