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Modules of congruence of Hecke algebras and \(L\)-functions associated with cusp forms. (English) Zbl 0645.10029
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.

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
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