Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms

Abstract

In this article, we discuss Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms. After the overview on the situation of the ordinary case of Hida family, we will introduce a Coleman map for Coleman family for the non-ordinary case (Coleman family) which was obtained as a joint work with Filippo Nuccio [NO16] and we give some results on Iwasawa Theory for Coleman families as applications of . First, we give a construction of a two-variable $p$-adic $L$-function for a Coleman family thanks to the ingredients given in [NO16] (Theorem 5.1). Combining this result and Coleman map obtained in [NO16], we also construct Beilinson-Kato Euler systems over a Coleman family (Thorem 5.3). Finally we formulate Iwasawa Main conjecture for a Coleman family and prove the half of Iwasawa Main Conjecture (Theorem 5.6).

References in corpus (5)

• Iwasawa theory of overconvergent modular forms, I: Critical $p$-adic $L$-functions
• The cyclotomic Iwasawa main conjecture for Hilbert cuspforms with complex multiplication
• Le système d'Euler de Kato en famille (I)
• Coleman Map in Coleman Families
• Specialization Method in Krull Dimension two and Euler System Theory over Normal Deformation Rings

Cited by in corpus (2)

• $p$-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou
• Zeta morphisms for rank two universal deformations