Triangulordinary Selmer Groups

TL;DR

This paper introduces triangulordinary Galois representations over p-adic fields, describing their local conditions and proposing a variational approach to define associated Selmer groups along the Coleman--Mazur eigencurve.

Contribution

It defines triangulordinary representations, characterizes their local conditions, and proposes a new variational framework for Selmer groups on the eigencurve.

Findings

Description of Bloch--Kato local conditions for triangulordinary representations

Introduction of a variational approach to Selmer groups

Potential extension of Selmer group definitions to nonordinary loci

Abstract

Let $p$ be a prime number, and let $K$ be a $p$-adic local field. We study a class of semistable $p$-adic Galois representations of $K$, which we call {\it triangulordinary} because it includes the ordinary ones yet allows non-\'etale behavior in the associated $(\phi,\Gamma_K)$-modules over the Robba ring. Our main result provides a description of the Bloch--Kato local condition of such representations. We also propose a program, using variational techniques, that would give a definition of the Selmer group along the eigencurve of Coleman--Mazur, including notably its nonordinary locus.