\(\mathfrak p\)-rigidity and Iwasawa \(\mu\)-invariants
type: Item
instance of: scholarly article
title: \(\mathfrak p\)-rigidity and Iwasawa \(\mu\)-invariants
zbMATH Open document ID: 1405.11083
author: Haruzo Hida
author: Ashay A. Burungale
published in: Algebra \& Number Theory
publication date: 2017-11-13T00:00:00Z
review text: According to the authors' introduction, ``zeta values'' seem to mysteriously encode deep arithmetical information, and also to suggest surprising modular and Iwasawa-theoretic phenomena. Sometimes, they are a sum of evaluations of modular forms at CM points. Such an expression for critical Hecke \(L\)-values and conjectural nontriviality of the corresponding anticyclotomic \(p\)-adic \(L\)-function, suggested to the second author a linear independence of mod\(\,p\) Hilbert modular forms, which he proved based on Chai's theory of Hecke-stable subvarieties of a Shimura variety [Ann. Math. (2) 172, No. 1, 41--137 (2010; Zbl 1223.11131)]. Here, given a totally real field \(F\) of degree \(d\) with ring of integers \(O\), an odd\(\,p\) unramified in \(F\) and a prime \({\mathfrak p}\) of \(F\) above \(p\), the expression and conjectural nontriviality of the corresponding anticyclotomic \({\mathfrak p}\)-adic \(L\)-function lead the authors to show a rather surprising rigidity property of mod\(\,p\) Hilbert modular forms. Partly based on this rigidity, they also obtain intriguing equalities between Iwasawa \(\mu\)-invariants of seemingly independent \(p\)-adic \(L\)-functions. Let us give a few details. The authors' \({\mathfrak p}\)-rigidity theorem states that a nonzero mod\(\,p\) Hilbert modular form over \(F\) does not vanish identically on the partial Serre-Tate deformation space \(\widehat{\mathbb{G}}_m\otimes O_{\mathfrak p}\). In particular, such a form is determined by its restriction to \(\widehat{\mathbb{G}}_m\otimes O_{\mathfrak p}\). Let then \(K\) be a totally imaginary quadratic extension of \(F\) such that every prime of \(F\) above \(p\) splits in \(K\). The splitting condition guarantees the existence of a \(p\)-adic CM type, i.e., a CM type \(\Sigma\) of \(K\) such that the \(p\)-adic places induced by elements in \(\Sigma\) via \(\iota_p:\overline{{\mathbb Q}}\to\mathbb{C}\) are disjoint from the ones induced by \(\Sigma c\) via \(\iota_\infty\) (where…
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 6806365
zbMATH Keywords: Hilbert \(p\)-modular forms
zbMATH Keywords: \({\mathfrak p}\)-rigidity
zbMATH Keywords: Shimura variety
zbMATH Keywords: Hecke stable subvariety
zbMATH Keywords: Iwasawa \(\mu\)-invariants
zbMATH Keywords: Katz \({\mathfrak p}\)-adic \(L\)-function
MaRDI profile type: Publication
full work available at URL: ant.2017.11.1921
OpenAlex ID: W2765794593
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DOI: 10.2140/ANT.2017.11.1921
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description: scientific article; zbMATH DE number 6806365