Deformations of Kolyvagin systems

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), assuming that the deformation problem at hand is unobstructed in the sense of Mazur. We then 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).