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.

中文翻译：

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关于Mordell-Weil的增长，在$ p $ -adic Lie扩展名中

假设$ p $是一个奇数质数，$ F_ \ infty $是一个数字字段$ F $的$ p $ -adic Lie扩展。假设$ A $是超过$ F $的阿贝尔变种，在$ p $以上的每个素数都具有通常的减少量。在各种假设下，我们确定了在所说的$ p $ -adic Lie扩展中，澳元Abel品种的Mordell-Weil等级的增长的渐近上限。我们的上限可以用来自环原子水平的不变性表示。受此公式的激励，我们对$ p $ -adic Lie扩展上的Mordell–Weil秩的增长的渐近上限进行了猜想，这是根据阿贝利亚变种的Mordell–Weil秩相对于圈数$ \ mathbb {Z} _p $-扩展名。最后，当阿贝尔变种具有非平凡归约时，自然会问是否存在这样的猜想上限。为了这，