Deformations of Kolyvagin systems

TL;DR

This paper extends the interpolation properties of Kolyvagin systems associated with Beilinson-Kato elements across the entire deformation space, aiming to formulate a main conjecture over the eigencurve using universal Kolyvagin systems.

Contribution

It generalizes Ochiai's results to the full deformation space and introduces universal Kolyvagin systems for a broader main conjecture over the eigencurve.

Findings

Kolyvagin systems interpolate in the full deformation space

A quasicoherent sheaf on the eigencurve behaves like a $p$-adic $L$-function

Framework for a main conjecture over the eigencurve

Abstract

Ochiai has previously proved that the Beilinson-Kato Euler systems for modular forms interpolate in nearly-ordinary $p$-adic families (Howard has obtained a similar result for Heegner points), based on which he was able to prove a half of the two-variable main conjectures. The principal goal of this article is to generalize Ochiai's work in the level of Kolyvagin systems so as to prove that Kolyvagin systems associated to Beilinson-Kato elements interpolate in the full deformation space (in particular, beyond the nearly-ordinary locus) and use what we call universal Kolyvagin systems to attempt a main conjecture over the eigencurve. Along the way, we utilize these objects in order to define a quasicoherent sheaf on the eigencurve that behaves like a $p$-adic $L$-function (in a certain sense of the word, in $3$-variables).