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

This article is cited in 19 scientific papers (total in 20 papers)

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
Received: 30.07.1975

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
\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
\jour Russian Math. Surveys
\yr 1976
\vol 31
\issue 1
\pages 5--57

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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  mathnet  crossref  mathscinet  zmath  isi
    2. Nicholas M. Katz, “p-AdicL-functions for CM fields”, Invent math, 49:3-4 (1978), 199  crossref  mathscinet  zmath
    3. P. F. Kurchanov, “Dirichlet series of Jacquet–Langlands cusp forms over fields of $CM$-type”, Math. USSR-Izv., 14:1 (1980), 61–78  mathnet  crossref  mathscinet  zmath  isi
    4. A. A. Panchishkin, “Symmetric squares of Hecke series and their values at integral points”, Math. USSR-Sb., 36:3 (1980), 365–387  mathnet  crossref  mathscinet  zmath  isi
    5. P. F. Kurchanov, “Local measures connected with Jacquet–Langlands cusp forms over fields of $CM$-type”, Math. USSR-Sb., 36:4 (1980), 449–467  mathnet  crossref  mathscinet  zmath  isi
    6. K Rosquist, Class Quantum Grav, 1:1 (1984), 81  crossref  mathscinet  adsnasa  isi
    7. A. A. Panchishkin, “Convolutions of Hilbert modular forms and their non-Archimedean analogues”, Math. USSR-Sb., 64:2 (1989), 571–584  mathnet  crossref  mathscinet  zmath
    8. A. A. Panchishkin, “Non-Archimedean Rankin $L$-functions and their functional equations”, Math. USSR-Izv., 32:2 (1989), 339–358  mathnet  crossref  mathscinet  zmath
    9. A. Yu. Khrennikov, “Mathematical methods of non-Archimedean physics”, Russian Math. Surveys, 45:4 (1990), 87–125  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    10. V. G. Drinfeld, V. A. Iskovskikh, A. I. Kostrikin, A. N. Tyurin, I. R. Shafarevich, “Yurii Ivanovich Manin (on his 60th birthday)”, Russian Math. Surveys, 52:4 (1997), 863–873  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    11. Min Ho Lee, “Mixed Hilbert modular forms and families of abelian varieties”, Glasgow Math J, 39:2 (1997), 131  crossref  mathscinet  zmath  isi
    12. Min Ho Lee, “Hilbert cusp forms and special values of Dirichlet series of Rankin type”, Glasgow Math J, 40:1 (1998), 71  crossref  mathscinet  zmath  isi
    13. Yann-Henri Le Bras, A. A. Panchishkin, “SUR LES PRODUITS TRIPLES Λ-ADIQUES”, Communications in Algebra, 29:9 (2001), 3727  crossref
    14. A. A. Panchishkin, “A new method of constructing $p$-adic $L$-functions associated with modular forms”, Mosc. Math. J., 2:2 (2002), 313–328  mathnet  crossref  mathscinet  zmath  elib
    15. Panchishkin A., “Two Variable P-Adic l Functions Attached to Eigenfamilies of Positive Slope”, Invent. Math., 154:3 (2003), 551–615  crossref  isi
    16. [Anonymous], “Non-Archimedean l-Functions and Arithmetical Siegel Modular Forms”, Non-Archimedean l-Functions and Arithmetical Siegel Modular Forms, 2nd Augmented Ed, Lecture Notes in Mathematics, 1471, Springer-Verlag Berlin, 2004, 13+  isi
    17. A. A. Panchishkin, “The Maass–Shimura differential operators and congruences between arithmetical Siegel modular forms”, Mosc. Math. J., 5:4 (2005), 883–918  mathnet  crossref  mathscinet  zmath
    18. A. A. Panchishkin, “Triple products of Coleman's families”, J. Math. Sci., 149:3 (2008), 1246–1254  mathnet  crossref  mathscinet  zmath  elib
    19. Fabian Januszewski, “Modular symbols for reductive groups andp-adic Rankin–Selberg convolutions over number fields”, Journal für die reine und angewandte Mathematik (Crelles Journal), 2010, -  crossref
    20. S. O. Gorchinskiy, Vik. S. Kulikov, A. N. Parshin, V. L. Popov, “Igor Rostislavovich Shafarevich and His Mathematical Heritage”, Proc. Steklov Inst. Math., 307 (2019), 1–21  mathnet  crossref  crossref  mathscinet  isi  elib
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