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Deformations of Kolyvagin systems

Content Provider	Springer Nature Link
Author	Büyükboduk, Kâzım
Copyright Year	2016
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).S’appuyant sur sa preuve antérieure montrant que les systèmes d’Euler de Beilinson-Kato pour formes modulaires s’interpolent sur des familles p-adiques quasi-ordinaires (un résultat similaire ayant été obtenu pour les points de Heegner par Howard), Ochiai a réussi à démontrer la moitié des conjectures principales à deux variables. L’objectif premier de cet article est de généraliser les travaux d’Ochiai aux systèmes de Kolyvargin afin de démontrer qu’en l’absence d’obstruction au problème de déformation en question, au sens de Mazur, les systèmes de Kolyvargin associés aux éléments de Beilinson-Kato s’interpolent sur tout l’espace de déformation (notamment au-delà du lieu quasi-ordinaire). Nous utilisons ensuite ce que nous appelons les systèmes de Kolyvargin universels pour hasarder une conjecture principale sur la courbe propre. Ce faisant, nous utilisons ces objets pour définir un faisceau quasi-cohérent sur la courbe propre qui se comporte comme une fonction L p-adique (à trois variables, dans un certain sens du terme).
Starting Page	251
Ending Page	302
Page Count	52
File Format	PDF
ISSN	21954755
Journal	Annales mathématiques du Québec
Volume Number	40
Issue Number	2
e-ISSN	21954763
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-03-02
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Euler systems Mathematics Kolyvagin systems Zeta-functions and related questions Deformations of Galois representations Iwasawa theory Algebra The eigencurve Analysis Elliptic curves over global fields Hida families $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Number Theory Abelian varieties of dimension $> 1$
Content Type	Text
Resource Type	Article
Subject	Mathematics

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