Iwasawa theory for \U(r,s), Bloch-Kato conjecture and\n Functional Equation

2019/08/20 by Xin Wan, Wan, Xin
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1908.07205

openalex publication_date 2019/08/20 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

In this paper we develop a new method to study Iwasawa theory and Eisenstein\nfamilies for unitary groups \U(r,s) of general signature over a\ntotally real field F. As a consequence we prove that for a motive\ncorresponding to a regular algebraic cuspidal automorphic representation \π\non \U(r,s)_/F which is ordinary at p, twisted by a Hecke\ncharacter, if its Selmer group has rank 0, then the corresponding central\nL-value is nonzero. This generalizes a result of Skinner-Urban in their ICM\n2006 report in the special case when F=\ℚ and the motive is conjugate\nself-dual. Along the way we also obtain p-adic functional equations for the\ncorresponding p-adic L-functions and p-adic families of Klingen\nEisenstein series. Our method does not involve computing Fourier-Jacobi\ncoefficients (as opposed to previous work which only work in low rank cases,≠.g. \U(1,1), \U(2,0) and \U(1,0)) whose\nautomorphic interpretation is unclear in general.\n

Citations

• Triple product p-adic L-functions associated to finite slope p-adic families of modular forms, with an appendix by Eric Urban
• The doubling Archimedean zeta integrals for p-adic interpolation

Related

• The Iwasawa Main Conjecture for universal families of modular motives
• Iwasawa's main conjecture for Rankin-Selberg motives in the anticyclotomic case
• Iwasawa Theory for GU(2,1) at inert primes
• Non-abelian p-adic L-functions and Eisenstein series of unitary groups; the CM method
• Aspects of Iwasawa theory over function fields