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

## TL;DR

This paper investigates the growth of Mordell-Weil ranks of abelian varieties in $p$-adic Lie extensions, providing asymptotic upper bounds and conjectures relating to their behavior over cyclotomic and more general extensions.

## Contribution

It establishes asymptotic upper bounds for Mordell-Weil rank growth in $p$-adic Lie extensions and proposes conjectures linking these bounds to ranks over cyclotomic extensions.

## Key findings

- Derived upper bounds expressed via cyclotomic invariants
- Formulated conjecture relating ranks in $p$-adic Lie extensions to cyclotomic ranks
- Extended discussion to non-ordinary reduction cases and supersingular elliptic curves

## Abstract

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 analogue conjectural upper bound for the growth of Mordell-Weil ranks of an elliptic curve with good supersingular reduction at the prime $p$ over a $\mathbb{Z}_p^2$-extension of an imaginary quadratic field.