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 Mosc. Math. J., 2002, Volume 2, Number 2, Pages 313–328 (Mi mmj57)

A new method of constructing $p$-adic $L$-functions associated with modular forms

A. A. Panchishkin

University of Grenoble 1 — Joseph Fourier

Abstract: We give a new method of constructing admissible $p$-adic measures associated with modular cusp eigenforms, starting from distributions with values in spaces of modular forms. A canonical projection operator is used onto the characteristic subspace of an eigenvalue $\alpha$ of the Atkin–Lehner operator $U_p$. An algebraic version of nearly holomorphic modular forms is given and used in constructing $p$-adic measures.

Key words and phrases: Modular forms, Eisenstein series, $p$-adic $L$-functions, special values.

DOI: https://doi.org/10.17323/1609-4514-2002-2-2-313-328

Full text: http://www.ams.org/.../abst2-2-2002.html
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MSC: 11F33, 11F67, 11F30
Received: December 3, 2001; in revised form February 28, 2002
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Citation: A. A. Panchishkin, “A new method of constructing $p$-adic $L$-functions associated with modular forms”, Mosc. Math. J., 2:2 (2002), 313–328

Citation in format AMSBIB
\Bibitem{Pan02} \by A.~A.~Panchishkin \paper A~new method of constructing $p$-adic $L$-functions associated with modular forms \jour Mosc. Math.~J. \yr 2002 \vol 2 \issue 2 \pages 313--328 \mathnet{http://mi.mathnet.ru/mmj57} \crossref{https://doi.org/10.17323/1609-4514-2002-2-2-313-328} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1944509} \zmath{https://zbmath.org/?q=an:1011.11026} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208593400006} \elib{http://elibrary.ru/item.asp?id=8379128} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Courtieu M., Panchishkin A., Non-Archimedean $L$-functions and arithmetical Siegel modular forms, Lecture Notes in Math., 1471, Second edition, Springer-Verlag, Berlin, 2004, viii+196 pp.
2. Colmez P., “La conjecture de Birch et Swinnerton-Dyer $p$-adique [The $p$-adic Birch–Swinnerton-Dyer conjecture]”, Astérisque, 294, 2004, 251–319
3. A. A. Panchishkin, “The Maass–Shimura differential operators and congruences between arithmetical Siegel modular forms”, Mosc. Math. J., 5:4 (2005), 883–918
4. A. A. Panchishkin, “Triple products of Coleman's families”, J. Math. Sci., 149:3 (2008), 1246–1254
5. Panchishkin A.A., “$p$-adic Banach modules of arithmetical modular forms and triple products of Coleman's families”, Pure Appl. Math. Q., 4:4 (2008), 1133–1164
6. Boecherer S., Panchishkin A.A., “p-adic Interpolation for Triple L-functions: Analytic Aspects”, Automorphic Forms and l-Functions II. Local Aspects, Contemporary Mathematics, 489, 2009, 1–39
7. Vienney M., “A New Construction of $p$-Adic Rankin Convolutions in the Case of Positive Slope”, Int J Number Theory, 6:8 (2010), 1875–1900
8. A. A. Panchishkin, “On zeta functions and families of Siegel modular forms”, J. Math. Sci., 180:5 (2012), 626–640
9. Panchishkin A., “Families of Siegel modular forms, L-functions and modularity lifting conjectures”, Israel J Math, 185:1 (2011), 343–368
10. Panchishkin A., “Analytic Constructions of P-Adic l-Functions and Eisenstein Series”, Automorphic Forms and Related Geometry: Assessing the Legacy of i.i. Piatetski-Shapiro, Contemporary Mathematics, 614, eds. Cogdell J., Shahidi F., Soudry D., Amer Mathematical Soc, 2014, 345–374
11. Wang Sh., “the System of Euler of Kato in Family (i)”, Comment. Math. Helv., 89:4 (2014), 819–865