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p-adic L-functions for modular forms. (English) Zbl 0618.10027
Let K be a number field, and let G be the group of points of GL(2)/K with values in the adeles of K. Beginning with a weight two Hecke eigenform F on G of level $$\Gamma_ 0(a)$$ the author constructs an $${\mathcal S}$$-adic L-function $$L_{F,{\mathcal S}}(\omega)$$ where $${\mathcal S}$$ is a finite set of primes away from the level of the form. The $${\mathcal S}$$-adic L-function is obtained as a Mellin transform of a measure on the Galois group $${\mathcal G}_{{\mathcal S}}$$ of the maximal unramified away from $${\mathcal S}$$ abelian extension of K. This $${\mathcal S}$$-adic L-function is seen to interpolate the critical values of the zeta function of the twists of F by finite characters whose conductors are supported on $${\mathcal S}$$. Moreover, $$L_{F,{\mathcal S}}(\omega)$$ is shown to satisfy a functional equation relating $$L_{F,{\mathcal S}}(\omega)$$ to $$L_{F,{\mathcal S}}(\omega^{-1})$$ for $${\mathcal S}$$-adic characters $$\omega$$ of $${\mathcal G}_{{\mathcal S}}$$.
Reviewer: S.Kamienny

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11S40 Zeta functions and $$L$$-functions 11R56 Adèle rings and groups
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