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Izv. Akad. Nauk SSSR Ser. Mat., 1972, Volume 36, Issue 1, Pages 19–66 (Mi izv2290)  

This article is cited in 38 scientific papers (total in 39 papers)

Parabolic points and zeta-functions of modular curves

Yu. I. Manin

Abstract: In this paper we obtain explicit formulas for the values at the center of the critical strip of Dirichlet series connected with weight 2 parabolic forms of the group $\Gamma_0(N)$. In particular, these formulas allow us to verify the Birch–Swinnerton-Dyer conjecture on the order of a zero for uniformizable elliptic curves over certain $\Gamma$-extensions. We also give applications to noncommutative reciprocity laws.

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English version:
Mathematics of the USSR-Izvestiya, 1972, 6:1, 19–64

Bibliographic databases:

UDC: 511
MSC: Primary 14G10; Secondary 14H99
Received: 05.10.1971

Citation: Yu. I. Manin, “Parabolic points and zeta-functions of modular curves”, Izv. Akad. Nauk SSSR Ser. Mat., 36:1 (1972), 19–66; Math. USSR-Izv., 6:1 (1972), 19–64

Citation in format AMSBIB
\by Yu.~I.~Manin
\paper Parabolic points and zeta-functions of modular curves
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1972
\vol 36
\issue 1
\pages 19--66
\jour Math. USSR-Izv.
\yr 1972
\vol 6
\issue 1
\pages 19--64

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    This publication is cited in the following articles:
    1. Yu. I. Manin, “Cyclotomic fields and modular curves”, Russian Math. Surveys, 26:6 (1971), 7–78  mathnet  crossref  mathscinet  zmath
    2. P. F. Kurchanov, “Elliptic curves of infinite rank over $\Gamma$-extensions”, Math. USSR-Sb., 19:2 (1973), 320–324  mathnet  crossref  mathscinet  zmath
    3. Yu. I. Manin, “Periods of parabolic forms and $p$-adic Hecke series”, Math. USSR-Sb., 21:3 (1973), 371–393  mathnet  crossref  mathscinet  zmath
    4. V. G. Berkovich, “The rational points on the Jacobians of modular curves”, Math. USSR-Sb., 30:4 (1976), 478–500  mathnet  crossref  mathscinet  zmath  isi
    5. Yu. I. Manin, “Non-Archimedean integration and Jacquet–Langlands $p$-adic $L$-functions”, Russian Math. Surveys, 31:1 (1976), 5–57  mathnet  crossref  mathscinet  zmath
    6. V. V. Shokurov, “Modular symbols of arbitrary weight”, Funct. Anal. Appl., 10:1 (1976), 85–86  mathnet  crossref  mathscinet  zmath
    7. P. F. Kurchanov, “Ryady Dirikhle, svyazannye s parabolicheskimi formami Zhake–Lenglendsa”, UMN, 32:2(194) (1977), 215–216  mathnet  mathscinet  zmath
    8. P. F. Kurchanov, “The zeta-function of elliptic curves over certain Abelian extensions of imaginary quadratic fields”, Math. USSR-Sb., 31:1 (1977), 49–62  mathnet  crossref  mathscinet  zmath  isi
    9. P. F. Kurchanov, “Cohomology of discrete groups and Dirichlet series connected with Jacquet–Langlands cusp Forms”, Math. USSR-Izv., 12:3 (1978), 543–555  mathnet  crossref  mathscinet  zmath
    10. M. A. Kenku, “The modular curves X0(65) and X0(91) and rational isogeny”, Math Proc Camb Phil Soc, 87:1 (1980), 15  crossref  mathscinet  zmath
    11. V. V. Shokurov, “Shimura integrals of cusp forms”, Math. USSR-Izv., 16:3 (1981), 603–646  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. O. M. Fomenko, “Applications of the theory of modular forms to number theory”, J Math Sci, 14:4 (1980), 1307  crossref
    13. A. J. Scholl, “Fourier coefficients of Eisenstein series on non-congruence subgroups”, Math Proc Camb Phil Soc, 99:1 (1986), 11  crossref  mathscinet  zmath
    14. V. Kumar Murty, Dinakar Ramakrishnan, “The Manin—Drinfeld theorem and Ramanujan sums”, Proc Math Sci, 97:1-3 (1987), 251  crossref  mathscinet  zmath
    15. Dimitrios Poulakis, “La courbe modulaire X0(125) et sa Jacobienne”, Journal of Number Theory, 25:1 (1987), 112  crossref
    16. V. A. Kolyvagin, “On the Mordell–Weil and Shafarevich–Tate groups for Weil elliptic curves”, Math. USSR-Izv., 33:3 (1989), 473–499  mathnet  crossref  mathscinet  zmath
    17. Glenn Stevens, “Stickelberger elements and modular parametrizations of elliptic curves”, Invent Math, 98:1 (1989), 75  crossref
    18. Min Ho Lee, “Periods of mixed cusp forms”, manuscripta math, 73:1 (1991), 163  crossref  mathscinet  zmath  isi
    19. J. E. Cremona, “Modular symbols for 1( N) and elliptic curves with everywhere good reduction”, Math Proc Camb Phil Soc, 111:2 (1992), 199  crossref  mathscinet  zmath
    20. 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
    21. Toshikazu Takagi, “The Cuspidal Class Number Formula for the Modular CurvesX0(M) withMSquare-Free”, Journal of Algebra, 193:1 (1997), 180  crossref
    22. S. P. Strunkov, K.-P. Shum, “On a Burnside-type problem for finite semigroups”, Russian Math. Surveys, 53:1 (1998), 223–224  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    23. Paul E Gunnells, “Modular Symbols for -Rank One Groups and VoronıReduction”, Journal of Number Theory, 75:2 (1999), 198  crossref
    24. Ernst-Ulrich Gekeler, “A note on the finiteness of certain cuspidal divisor class groups”, Isr J Math, 118:1 (2000), 357  crossref  mathscinet  zmath  isi
    25. Paul E. Gunnells, “Computing Heeke Eigenvalues Below the Cohomologieal Dimension”, Experimental Mathematics, 9:3 (2000), 351  crossref
    26. János A. Csirik, “The Kernel of the Eisenstein Ideal”, Journal of Number Theory, 92:2 (2002), 348  crossref
    27. V. A. Bykovskii, “Generating Elements of the Annihilating Ideal for Modular Symbols”, Funct. Anal. Appl., 37:4 (2003), 263–272  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    28. Matilde Marcolli, “Limiting modular symbols and the Lyapunov spectrum”, Journal of Number Theory, 98:2 (2003), 348  crossref
    29. Yu. I. Manin, “Iterated Shimura integrals”, Mosc. Math. J., 5:4 (2005), 869–881  mathnet  crossref  mathscinet  zmath
    30. Henri Darmon, Robert Pollack, “Efficient calculation of Stark-Heegner points via overconvergent modular symbols”, Isr J Math, 153:1 (2006), 319  crossref  isi
    31. V. A. Bykovskii, “Teoriya Eikhlera – Shimury – Manina”, Dalnevost. matem. zhurn., 9:1-2 (2009), 29–37  mathnet  elib
    32. J.-M. Couveignes, “Linearizing torsion classes in the Picard group of algebraic curves over finite fields”, Journal of Algebra, 321:8 (2009), 2085  crossref
    33. Cécile Armana, “Sur les symboles modulaires de Manin–Teitelbaum pour”, Comptes Rendus Mathematique, 347:15-16 (2009), 845  crossref
    34. Petra Forster, Hendrik Kasten, “A Stickelberger index for the Tate-Shafarevich group revisited”, manuscripta math, 2010  crossref
    35. Kentaro Ihara, “Special values of multiple Hecke L-functions”, J. Appl. Math. Comput, 2012  crossref
    36. Byungheup Jun, Jungyun Lee, “Equidistribution of generalized Dedekind sums and exponential sums”, Journal of Number Theory, 137 (2014), 67  crossref
    37. V. A. Bykovskii, “Sootnosheniya Eikhlera – Shimury dlya teta-funktsii”, Dalnevost. matem. zhurn., 17:2 (2017), 152–157  mathnet  elib
    38. N. M. Glazunov, “$p$-adic $L$-functions and $p$-adic multiple zeta values”, Chebyshevskii sb., 20:1 (2019), 112–130  mathnet  crossref
    39. E. Kresh, Yu. Chinkel, “Biratsionalnye tipy algebraicheskikh orbifoldov”, Matem. sb., 212:3 (2021), 54–67  mathnet  crossref  mathscinet
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