Hida, Haruzo Modules of congruence of Hecke algebras and \(L\)-functions associated with cusp forms. (English) Zbl 0645.10029 Am. J. Math. 110, No. 2, 323-382 (1988). Let \(f=\sum^{\infty}_{n=1}a_nq^n\) be a primitive cusp form of weight \(k\) for the congruence subgroup \(\Gamma_0(M)\) of \(\mathrm{SL}(2, \mathbb Z)\) and the Dirichlet character \(\psi\) modulo \(M\). For each prime number \(\ell\) there are complex numbers \(\alpha_{\ell}\), \(\beta_{\ell}\) such that \[ \sum^{\infty}_{n=1}a_nn^{-s}=\prod_{\ell}[(1- \alpha_{\ell}\ell^{-s})(1-\beta_{\ell}\ell^{-s})]^{-1}. \] The author provides a \(p\)-adic interpolation \((p\geq 5)\) of one variable of the “canonical algebraic part” of special values at certain integer arguments \(s=1\) of \[ \mathcal D(s,f)=\prod_{\ell}[(1-{\bar \psi}_0(\ell)\alpha^2_{\ell}\ell^{-s})(1-{\bar \psi}_0(\ell)\alpha_{\ell}\beta_{\ell}\ell^{-s})(1-{\bar \psi}_ 0(\ell)\beta^2_{\ell}\ell^{-s})]^{-1} \] \((\psi_0 = \) primitive character which induces \(\psi)\), which converges absolutely for sufficiently large \(\operatorname{Re}(s)\) and has a meromorphic continuation to the whole \(s\)-plane. This interpolation is achieved by varying \(f\) along the spectrum of each irreducible component of the \(p\)-adic Hecke algebra. Reviewer: Hans Opolka (Braunschweig) Cited in 1 ReviewCited in 32 Documents MSC: 11F85 \(p\)-adic theory, local fields 11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols 11F12 Automorphic forms, one variable Keywords:Iwasawa theory; L-function; Euler product; primitive cusp form; Dirichlet character; p-adic interpolation; special values; p-adic Hecke-algebra PDFBibTeX XMLCite \textit{H. Hida}, Am. J. Math. 110, No. 2, 323--382 (1988; Zbl 0645.10029) Full Text: DOI