Interpolation of Beilinson-Kato elements and $p$-adic $L$-functions

Our objective in this series of two articles, of which the present article is the first, is to give a Perrin-Riou-style construction of $p$-adic $L$-functions (of Bella\"iche and Stevens) over the eigencurve. As the first ingredient, we interpolate the Beilinson-Kato elements over the eigencurve (including the neighborhoods of $\theta$-critical points). Along the way, we prove \'etale variants of Bella\"iche's results describing the local properties of the eigencurve. We also develop the local framework to construct and establish the interpolative properties of these $p$-adic $L$-functions away from $\theta$-critical points.