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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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