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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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