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 Uspekhi Mat. Nauk, 1976, Volume 31, Issue 1(187), Pages 5–54 (Mi umn3639)

Non-Archimedean integration and Jacquet–Langlands $p$-adic $L$-functions

Yu. I. Manin

Abstract: In 1964 Kubota and Leopoldt constructed a $p$-adic analogue of the Riemann zeta-function. Since then the class of $L$-functions with $p$-adic variants has continually been enlarged. At the beginning of the article we survey work in this direction, using the technique of the $p$-adic Mellin transform. Then we show how to apply it to the construction of non-Archimedean measures and integrals corresponding to parabolic forms relative to the Hilbert groups. The exposition is in the adele language of Jacquet and Langlands. We construct $p$-adic $L$-functions associated with representations of $GL(2)$ over completely real fields, of discrete type at infinity.

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English version:
Russian Mathematical Surveys, 1976, 31:1, 5–57

Bibliographic databases:

UDC: 517.5
MSC: 11M38, 11M26, 11M36

Citation: Yu. I. Manin, “Non-Archimedean integration and Jacquet–Langlands $p$-adic $L$-functions”, Uspekhi Mat. Nauk, 31:1(187) (1976), 5–54; Russian Math. Surveys, 31:1 (1976), 5–57

Citation in format AMSBIB
\Bibitem{Man76} \by Yu.~I.~Manin \paper Non-Archimedean integration and Jacquet--Langlands $p$-adic $L$-functions \jour Uspekhi Mat. Nauk \yr 1976 \vol 31 \issue 1(187) \pages 5--54 \mathnet{http://mi.mathnet.ru/umn3639} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=417134} \zmath{https://zbmath.org/?q=an:0336.12007|0348.12016} \transl \jour Russian Math. Surveys \yr 1976 \vol 31 \issue 1 \pages 5--57 \crossref{https://doi.org/10.1070/RM1976v031n01ABEH001444} 

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This publication is cited in the following articles:
1. Yu. I. Manin, A. A. Panchishkin, “Convolutions of Hecke series and their values at lattice points”, Math. USSR-Sb., 33:4 (1977), 539–571
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6. K Rosquist, Class Quantum Grav, 1:1 (1984), 81
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