Let $f$ be a newform of weight $2$, square-free level and trivial character, let $A_f$ be the abelian variety attached to $f$ and for every good ordinary prime $p$ for $f$ let $\boldsymbol f^{(p)}$ be the $p$-adic Hida family through $f$. We prove that, for all but finitely many primes $p$ as above, if $A_f$ is an elliptic curve such that $A_f(\mathbb Q)$ has rank $1$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\mathbb Q$ is finite then all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite ($p$-primary) Shafarevich-Tate group and $1$-dimensional image of the relevant $p$-adic \'etale Abel-Jacobi map... (read more)

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