Semi-ordinary Iwasawa theory for Rankin--Selberg products

Abstract

Chapter 1. Introduction 1.1. The setting 1.2. Conjectures and Results 1.3. A summary of our results 1.4. Layout 1.5. Notation Chapter 2. Construction of the optimized Coleman maps 2.1. Dieudonné modules and ϕ-eigenvectors 2.2. Factorization of one-variable Perrin-Riou maps 2.3. Two-variable Perrin-Riou maps and Coleman maps 2.4. The case a p (g) = 0 Chapter 3. Beilinson-Flach elements and p-adic L-functions 3.1. Beilinson-Flach elements 3.2. Euler systems of rank 2 and uniform integrality 3.3. p-adic L-functions Chapter 4. Selmer groups and main conjectures 4.1. Definitions of Selmer groups 4.2. Classical and signed Iwasawa main conjectures Chapter 5. Applications towards main conjectures 5.1. Cyclotomic main conjectures for f ⊗ g 5.2. Cyclotomic main conjectures for f ⊗ g 5.3. Main conjectures over an imaginary quadratic field where p is inert Appendix A. A divisibility criterion in regular rings Appendix B. p-adic Rankin-Selberg L-functions and universal deformations B.1. The set up B.2. The minimally ramified universal deformation representation B.3. Hida families B.4. p-adic Rankin-Selberg L-function Appendix C. Images of Galois representations attached to Rankin-Selberg convolutions C.1. Group theory C.2. Applications to families on GL 2 × Res K/Q GL 1

p-adic L-functions via local–global interpolation: the case of

“…Finally, when F = Q, variants of L p (V ) were constructed by Hida [Hid88, Theorem 5.1a] and, more recently, by Loeffler and Büyükboduk and Lei (see [BL,Section B.4]) under various local restrictions.…”