The Iwasawa \(\mu \)-invariant of \(p\)-adic Hecke \(L\)-functions
For an odd prime \(p,\) the author computes the \(\mu\)-invariant of the anticyclotomic Katz \(p\)-adic \(L\)-function of a \(p\)-ordinary \(CM\) field \(M\) if the conductor of the branch character is a product of primes split over the maximal real subfield \(F.\) Except for rare cases where the root number of the \(p\)-adic functional equation is congruent to \(-1 \bmod p\), the \(\mu\)-invariant vanishes. More precisely, assume the ``ordinary hypothesis'', i.e., every \(p\)-place of \(F\) splits in \(M.\) Fix a prime to \(p\) \({\mathcal O}_M\)-ideal \({\mathfrak G} = {\mathcal F} {\mathcal F}' {\mathcal J},\) where \({\mathcal F} {\mathcal F'}\) (resp. \({\mathcal J})\) consists of split (resp. inert or ramified) primes of \(F,\) \({\mathcal F} + {\mathcal F}' = {\mathcal O}_F\) and \({\mathcal F} \subset ({\mathcal F}')^c\) (where \(c\) denotes complex conjugation). Following Katz (who did the case \({\mathfrak G} = (1)),\) the author and \textit{J. Tilouine} [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 2, 189--259 (1993; Zbl 0778.11061)] constructed a unique measure \(\varphi\) on the ray class group \(Z ({\mathfrak G})\) mod \({\mathfrak G} p^\infty\) of \( M\) which interpolates (in a precise way which we do not recall here) Hecke \(L\)-values. Fix a \(p\)-adic valuation ring \(W\) which is finite flat over the ring of Witt vectors \(W (\overline{\mathbb F}_p).\) Let \(\Delta\) be the maximal torsion subgroup of \(Z({\mathfrak G}).\) A character \(\psi : \Delta \to W^\times\) is called a branch character. The \(\psi\)-branch \(\varphi_\psi\) of the measure \(\varphi\) is defined on \(\Gamma = Z ({\mathfrak G})/\Delta\) by \(\int_\Gamma \;\Phi \;d \;\varphi_\psi = \int_{Z({\mathfrak G})} \;\psi \;\Phi\;d \;\varphi.\) The main result of this paper (theorem 5.1) computes \(\mu(\varphi^-_\psi)\) explicitly in terms of the branch character \(\psi\) when \({\mathcal J} = (1).\) This implies, when \(p\) is unramified in \(F,\) that \(\mu (\varphi^-_\psi) = 0\) unless the root number of the \(p\)-adic functional equation is congruent to \(-1 \bmod p\) (which rarely happens). The author's methods are deep and technically demanding. To give just an idea, let us say that he starts from Sinnott's algebro-geometric proof of the vanishing of the cyclotomic \(\mu\)-invariant (Ferrero-Washington's theorem) relying on the analysis of rational functions on \({\mathfrak G}_{m/{\mathbb F}_p}\) (under transcendental automorphisms of the formal group \(\widehat G_m).\) His idea is to use Hilbert modular Shimura varieties and Eisenstein series in place of \({\mathfrak G}_m\) and rational functions. It is then necessary to go through an extensive study of the \(q\)-expansion of Eisenstein series and the geometry of the moduli space of abelian varieties with real multiplication by \({\mathcal O}_F.\) On the one hand, the ``\(q\)-expansion principle'' is equivalent to the geometric irreducibility of the mod \(p\) fiber of the variety, which was shown by Ribet. On the other hand, the datum of an ordinary CM-type gives rise to an abelian scheme \(A\) of the given CM-type over \(W.\) The author constructs Eisenstein series \(E_a\) indexed by \(a \in \Omega\) for an appropriate finite subset \(\Omega\) of automorphisms of the deformation space of \(A\) with the following properties : {\parindent=5mm \begin{itemize}\item[1.]\(E_a\) is congruent to an arithmetic Eisenstein series mod \(p\). \item[2.]Elements of \(\Omega\) are disjoint modulo the stabilizer of \(A\) inside the automorphism group of the moduli space (i.e. the Hilbert modular Shimura variety). \item[3.]The functions \(a(E_a) = E_a \circ a\) for \(a \in \Omega\) with \(E_a \not\equiv 0 \bmod p\) are linearly independent modulo \(p\). \item[4.]The expansion of a non zero linear combination of \(\{ a (E_a)\}_{a \in \Omega}\) with respect to the canonical variable \(t\) of the Serre-Tate deformation space of \(A\) coincides with the power series expansion of a given branch of the anticyclotomic Katz measure under study. \end{itemize}} The author stresses that the \(q\)-expansion of his Eisenstein series fully reflects divisibility by \(p\) of the Katz measure. After proving his main theorem, he discusses what happens when \({\mathcal J} \neq (1)\). But the computation of \(\mu\) when the branch character is ramified and primitive at a non split prime seems far more demanding than in the case of split-prime level.
anticyclotomic Katz \(p\)-adic \(L\)-function
scientific article; zbMATH DE number 5779355
scientific article