On the growth of Mordell–Weil ranks in $p$-adic Lie extensions

Let $p$ be an odd prime and $F_\infty$ a $p$-adic Lie extension of a number field $F$. Let $A$ be an abelian variety over $F$ which has ordinary reduction at every primes above $p$. Under various assumptions, we establish asymptotic upper bounds for the growth of Mordell–Weil rank of the abelian variety of $A$ in the said $p$-adic Lie extension. Our upper bound can be expressed in terms of invariants coming from the cyclotomic level. Motivated by this formula, we make a conjecture on an asymptotic upper bound of the growth of Mordell–Weil ranks over a $p$-adic Lie extension which is in terms of the Mordell–Weil rank of the abelian variety over the cyclotomic $\mathbb{Z}_p$-extension. Finally, it is then natural to ask whether there is such a conjectural upper bound when the abelian variety has non-ordinary reduction. For this, we can at least modestly formulate an analogous conjectural upper bound for the growth of Mordell–Weil ranks of an elliptic curve with good supersingular reduction at the prime $p$ over a $\mathbb{Z}^2_p$-extension of an imaginary quadratic field.

Keywords

Mordell–Weil ranks, $p$-adic Lie extensions, $\mathfrak{M}_H (G)$-conjecture

2010 Mathematics Subject Classification

11G10, 11R23

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Received 22 April 2019

Accepted 25 October 2019

Published 18 February 2021