This paper concerns the arithmetic of certain $p$-adic families of elliptic modular forms. We relate, using a formula of Rubin, some Iwasawa-theoretic aspects of the three items in the title of this paper. In particular, we examine several conjectures, three of which assert the non-triviality of an Euler system, a $p$-adic regulator, and the derivative of a $p$-adic $L$-function. We investigate sufficient conditions for the first conjecture to hold and show that, under additional assumptions, the first...

Let $K$ be a $p$-adic local field with residue field $k$ such that $[k:k^p]=p^e\<+\infty$ and $V$ be a $p$-adic representation of $\text{Gal}(\overline{K}/K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K}/K)$ if and only if $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K^{\text{pf}}}/K^{\text{pf}})$ where $K^{\text{pf}}/K$ is a certain $p$-adic local field with residue field the smallest perfect field $k^{\text{pf}}$ containing $k$.