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arXiv · 1807.07604

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

Antonio Lei·Gautier Ponsinet
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Abstract

Let $F$ be a number field unramified at an odd prime $p$ and $F_\infty$ 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 $p$. B\"uy\"ukboduk 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 are torsion $\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-modules, 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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Antonio Lei, Gautier Ponsinet. 2018-07-19. On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions. https://arxiv.org/abs/1807.07604

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