Deformations of Kolyvagin systems

Abstract

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).

On the Iwasawa theory of CM fields for supersingular primes

“…• prove that the divisibility in the previous item may be upgraded to an equality using the structure of the module of Λ-adic Kolyvagin systems, as described in [Büy16] (we provide a detailed account of this in Section 4 below).…”

“…Corollary 4.5.2 of [MR04] asserts that the module of Kolyvagin systems KS(F , T ) is a k-vector space of dimension one , thanks to the second part of Lemma 2.31. On the other hand, it follows from the main theorem of [Büy16] that these residual Kolyvagin systems deform to X (where X = T, T(E), T cyc or T cyc (E)) and that the module KS(F , X) is free of rank one over the corresponding coefficient ring. The elements of these modules (namely, Kolyvagin systems) are used to bound the characteristic ideal of H 1 F * (F, X) ∨ .…”

Main conjectures for higher rank nearly ordinary families -- I

“…For a Selmer structure F on T and s as above, we may define the module of Kolyvagin systems on the artinian module T s . We will not include its precise definition in this note and refer the reader to [MR04,Büy16]. Given a Selmer structure F on T cyc , we let KS(T s , F, P s ) denote the module of Kolyvagin systems for the Selmer triple (T s , F, P s ) and set…”