I have the following setting:


1.) A Galois extension of number fields $K\hookrightarrow L$, with $\operatorname{Gal}(L/K)=\mathbb{Z}\_{p}$. In my terminology, number field does not imply finiteness over $\mathbb{Q}$. Let $K\hookrightarrow {K}\_{{{p}^{n}}}\hookrightarrow L$ be such that $\operatorname{Gal}({K}\_{{{p}^{n}}}/K)=\mathbb{Z}/{{p}^{n}}\cdot \mathbb{Z}$.


2.) An Abelian variety (in my case: a modular Jacobian or an elliptic curve) A, defined over $K$, such that $A(K\_{{p}^{n}})$ is finitely generated, but $A(L)$ is not. Furthermore, we assume $A(K)$ has nonzero rank and $A(L)$ has finite torsion.


Does that alone imply the existence of a strictly increasing sequence $\left({m}\_{n}\right)\_{n\in\mathbb{N}}$ with $m\_{1}\geq 1$ and a family of nontorsion points ${{\alpha}\_{n}}\in {A({{K}\_{{{p}^{{m}\_{n}}}}})}$ and a natural number $t\geq 1$ such that:


A.) $\alpha\_{n+1}\notin A({{K}\_{{{p}^{{m}\_{n}}}}})$ for all but finitely many $n$.


B.) $\operatorname{tr}\_{K\_{{{p}^{{m}\_{n+t}}}}/K\_{{p}^{{m}\_{n+t-1}}}}{\alpha}\_{n+t}={f}\_{n}{\alpha}\_{n}+T\_{n},$


where $T\_{n}$ is torsion of order prime to $p(p^2-1)$, and $\nu\_{p}(f\_{n})=0$, where ${\nu}\_{p}$ denotes the valuation at $p$.



Points $\alpha_n$ of $A$ over the $m_n$-th layer in a $\mathbb{Z}_p$-ext. of $K$, where $A$ is an Ab. var. and $m_n$ is strictly increasing
I suspect that the answer is "no". If you take an elliptic curve with supersingular reduction at $p$, $K$ a sufficiently carefully chosen imaginary quadratic field, and $L / K$ the anticyclotomic $\mathbf{Z}\_p$-extension, then you can rig things so that $E(L)$ is not finitely generated, because there are non-torsion Heegner points for all but finitely many of the characters of $L / K$. But these can't be massaged into a trace-compatible sequence, and in fact no such trace-compatible sequence exists, because of $E$ being supersingular.


