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

TL;DR

This paper constructs $p$-adic $L$-functions over the eigencurve by interpolating Beilinson-Kato elements, extending local properties, and developing a framework for interpolation away from critical points.

Contribution

It provides a Perrin-Riou-style construction of $p$-adic $L$-functions over the eigencurve, including neighborhoods of $ heta$-critical points, by interpolating Beilinson-Kato elements.

Findings

Interpolated Beilinson-Kato elements over the eigencurve.

Proved étale variants of Bellaïche's local properties of the eigencurve.

Developed a local framework for constructing and interpolating $p$-adic $L$-functions away from $ heta$-critical points.

Abstract

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.