On the Iwasawa main conjectures for modular forms at non-ordinary primes

2018/04/29 by Francesc Castella, Castella, Francesc, Mirela Çiperiani +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1804.10993

openalex publication_date 2018/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove under mild hypotheses the Iwasawa main conjectures of Lei--Loeffler--Zerbes for modular forms of weight 2 at non-ordinary primes. Our proof is based on the study of the two-variable analogues of these conjectures formulated by Büyükboduk--Lei for imaginary quadratic fields in which p splits, and on anticyclotomic Iwasawa theory. As application of our results, we deduce the p-part of the Birch and Swinnerton-Dyer formula in analytic ranks 0 or 1 for abelian varieties over ℚ of \rm GL₂-type for non-ordinary primes p>2.

Citations

• On ordinary ?-adic representations associated to modular forms
• On the Asymptotic Growth ofBloch-Kato-Shafarevich-Tate Groups ofModular Forms Over CyclotomicExtensions

Cited by

• Ranks of elliptic curves over Z_p²-extensions

Related

• Iwasawa Main Conjecture for Non-Ordinary Modular Forms
• Iwasawa theory for quadratic Hilbert modular forms
• Iwasawa theory for symmetric powers of CM modular forms at nonordinary primes, II
• Anticyclotomic p-ordinary Iwasawa Theory of Elliptic Modular Forms
• Iwasawa theory of twists of elliptic modular forms over imaginary quadratic fields at inert primes