\(\mathfrak p\)-rigidity and Iwasawa \(\mu\)-invariants
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 \(c\) is the complex conjugation on \(\mathbb{C}\) which induces the unique non-trivial element of Gal\((K/F)\)). Such a CM type can be identified with the set of infinite places of \(F\). Let \(K^-_\infty\) (resp. \(K^+_\infty\)) be the anticyclotomic \(\mathbb{Z}^d_p\)-extension (resp. cyclotomic \(\mathbb{Z}_p\)-extension) of \(K\) and \(K^-_{{\mathfrak p},\infty}\subset K^-_\infty\) the \({\mathfrak p}\)-anticyclotomic subextension, i.e., the maximal subextension unramified outside the primes above \({\mathfrak p}\) in \(K\). Let \(K_{{\mathfrak p},\infty}= K^-_{{\mathfrak p},\infty}\), \(K^+_\infty\), \(\Gamma^{\pm}= \text{Gal}(K^{\pm}_\infty/K)\), \(\Gamma^-_{\mathfrak p}= \text{Gal}(K^-_{{\mathfrak p},\infty}/K)\) and \(\Gamma_{\mathfrak p}= \text{Gal}(K_{{\mathfrak p},\infty}/K)\). Let \(\lambda\) be an arihmetic Hecke character over \(K\), with prime-to-\(p\) conductor. Associated to this data, a natural \((d+1)\)-variable Katz \(p\)-adic \(L\)-function \(L_{\Sigma,\lambda}(T_1,\dots,T_d,S)\in\overline{\mathbb{Z}}_p\Gamma\) can be constructed, where the \(T_i\) are the anticyclotomic variables and \(S\) the cyclotomic one. Let \(L^-_{\Sigma,\lambda}\), \(L^-_{\Sigma,\lambda,{\mathfrak p}}\) and \(L_{\Sigma,\lambda,{\mathfrak p}}\) be obtained by projecting \(\overline{\mathbb{Z}}_p\Gamma\) onto resp. \(\overline{\mathbb{Z}}_p\Gamma^-\), \(\overline{\mathbb{Z}}_p[[\Gamma^-_{\mathfrak p}]]\) and \(\overline{\mathbb{Z}}_p[[\Gamma_{\mathfrak p}]]\). If \(K/F\) is \(p\)-ordinary, the authors show that the anti-cyclotomic \(\mu\)-invariants verify \(\mu(L^-_{\Sigma,\lambda,{\mathfrak p}})= \mu(L^-_{\Sigma,\lambda})\). Note that in most cases, the latter \(\mu\)-invariant has been determined explicitly ([the second author, loc. cit.; \textit{M.-L. Hsieh}, J. Reine Angew. Math. 688, 67--100 (2014; Zbl 1294.11195)]). When \(\lambda\) is self-dual with root number \(-1\), all the Hecke \(L\)-values appearing in the interpolation property of \(L^-_{\Sigma,\lambda}\) vanish, so that \(L^-_{\Sigma,\lambda}\) and \(L^-_{\Sigma,\lambda,{\mathfrak p}}\) identically vanish. However, one can look at the cyclotomic derivatives \(L_{\Sigma,\lambda}'= ({\partial\over\partial S}L_{\Sigma,\lambda}(T_1,\dots,T_d, S))|_{S=0}\) and \(L_{\Sigma,\lambda,{\mathfrak p}}'\) (defined analogously). The \(\mu\)-invariants of \(L_{\Sigma,\lambda,{\mathfrak p}}'\) and \(L_{\Sigma,\lambda}'\) are then related as follows: if \(K/F\) is \(p\)-ordinary and \(p\) does not divide the relative class number \(h^-_K\), then \(\mu(L_{\Sigma,\lambda,{\mathfrak p}}')= \mu(L_{\Sigma,\lambda}')\). Note again that in most cases, the latter \(\mu\)-invariant has been determined explicitly ([the first author, J. Inst. Math. Jussieu 14, No. 1, 131--148 (2015; Zbl 1323.11087)]). Finally the authors prove the following \({\mathfrak p}\)-version of a conjecture by \textit{R. Gillard} [Sémin. Théor. Nombres Bordx., Sér. II 3, No. 1, 13--26 (1991; Zbl 0732.11060)] regarding the vanishing of the \(\mu\)-invariant of the Katz \(p\)-adic \(L\)-function: if \(K/F\) is a \(p\)-ordinary CM quadratic extension and \(\lambda\) is an arithmetic Hecke character over \(K\), then \(\mu(L_{\Sigma,\lambda,{\mathfrak p}})=0\).
Katz \({\mathfrak p}\)-adic \(L\)-function
scientific article; zbMATH DE number 6806365
scientific article