Dissertation

Selmer growth and a “triangulordinary” local condition

Pottharst, Jonathan Livaudais

2008

Overview

We present two results about Selmer groups. Given a torsion p-adic Galois representation A of a number field K, the Selmer group of A over K is the subspace of Galois cohomology H1( GK, A) consisting of cycles c satisfying certain local conditions, i.e. such that the restrictions res v(c) ∈ H1( Gv, A) to decomposition groups Gv (for places v of K) lie in distinguished subspaces Lv ⊆ H 1(Gv, A). These groups are conjecturally related to algebraic cycles (à la Shafarevich–Tate) on the one hand, and on the other to special values of L-functions (à la Bloch–Kato). Our first result shows how, using a global symmetry (the sign of functional equation under Tate global duality), one can produce increasingly large Selmer groups over the finite subextensions of a [special characters omitted]-extension of K. Our second result gives a new characterization of the Selmer group, namely of the local condition Lv for v | p. It uses (ϕ, [special characters omitted])-modules over Berger's Robba ring [special characters omitted] to give a vast generalization of the well-known \"ordinary\" condition of Greenberg to the nonordinary setting. We deduce a definition of Selmer groups for overconvergent modular forms (of finite slope). 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.

• Publisher: ProQuest Dissertations & Theses
• Subject: Mathematics
• ISBN: 9780549615279, 054961527X