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Uspekhi Mat. Nauk, 1971, Volume 26, Issue 6(162), Pages 7–71 (Mi umn5277)  

This article is cited in 27 scientific papers (total in 28 papers)

Cyclotomic fields and modular curves

Yu. I. Manin

Abstract: The first chapter of this article contains an exposition of the work of Iwasawa and Mazur on the arithmetic of Abelian varieties over cyclotomic fields. The study of questions arising here leads us in the second chapter to the use of the zeta-function apparatus, and the conjectures of Weil and Birch–Swinnerton-Dyer; this permits us to obtain conditional formulae for the order of the Tate–Shafarevich group.

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English version:
Russian Mathematical Surveys, 1971, 26:6, 7–78

Bibliographic databases:

UDC: 517.5+519.4
MSC: 11T22, 11R60, 11R34, 12G05, 11G40, 14G10
Received: 31.05.1971

Citation: Yu. I. Manin, “Cyclotomic fields and modular curves”, Uspekhi Mat. Nauk, 26:6(162) (1971), 7–71; Russian Math. Surveys, 26:6 (1971), 7–78

Citation in format AMSBIB
\by Yu.~I.~Manin
\paper Cyclotomic fields and modular curves
\jour Uspekhi Mat. Nauk
\yr 1971
\vol 26
\issue 6(162)
\pages 7--71
\jour Russian Math. Surveys
\yr 1971
\vol 26
\issue 6
\pages 7--78

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    This publication is cited in the following articles:
    1. Yu. G. Zarhin, Yu. I. Manin, “Height on families of Abelian varieties”, Math. USSR-Sb., 18:2 (1972), 169–179  mathnet  crossref  mathscinet  zmath
    2. M. I. Bashmakov, “The cohomology of Abelian varieties over a nuvber field”, Russian Math. Surveys, 27:6 (1972), 25–70  mathnet  crossref  mathscinet  zmath
    3. Yu. I. Manin, “Parabolic points and zeta-functions of modular curves”, Math. USSR-Izv., 6:1 (1972), 19–64  mathnet  crossref  mathscinet  zmath
    4. M. I. Bashmakov, N. Zh. Al'-Nader, “Behavior of the curve $x^3+y^3=1$ in a cyclotomic $\Gamma$-extension”, Math. USSR-Sb., 19:1 (1973), 117–130  mathnet  crossref  mathscinet  zmath
    5. P. F. Kurchanov, “Elliptic curves of infinite rank over $\Gamma$-extensions”, Math. USSR-Sb., 19:2 (1973), 320–324  mathnet  crossref  mathscinet  zmath
    6. Yu. I. Manin, “Periods of parabolic forms and $p$-adic Hecke series”, Math. USSR-Sb., 21:3 (1973), 371–393  mathnet  crossref  mathscinet  zmath
    7. O. N. Vvedenskii, “On “universal norms” of formal groups defined over the ring of integers of a local field”, Math. USSR-Izv., 7:4 (1973), 733–747  mathnet  crossref  mathscinet  zmath
    8. P. F. Kurchanov, “On the rank of elliptic curves over $\Gamma$-extensions”, Math. USSR-Sb., 22:3 (1974), 465–472  mathnet  crossref  mathscinet  zmath
    9. V. G. Berkovich, “On the division by an isogeny of the points of an elliptic curve”, Math. USSR-Sb., 22:3 (1974), 473–492  mathnet  crossref  mathscinet  zmath
    10. A. G. Nasybullin, “Ellipticheskie krivye s supersingulyarnoi reduktsiei nad $\Gamma$-rasshireniyami”, UMN, 32:2(194) (1977), 221–222  mathnet  mathscinet  zmath
    11. A. A. Panchishkin, “On Dirichlet series connected with modular forms of integral and half-integral weight”, Math. USSR-Izv., 15:2 (1980), 373–385  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. Peter Schneider, “p-adic height pairings I”, Invent math, 69:3 (1982), 401  crossref  mathscinet  zmath  isi
    13. J. S. Cassels, “Explicit results on the arithmetic of curves of higher genus”, Russian Math. Surveys, 40:4 (1985), 43–48  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    14. V. A. Kolyvagin, “Finiteness of $E(\mathbf Q)$ and $Ш(E,\mathbf Q)$ for a subclass of Weil curves”, Math. USSR-Izv., 32:3 (1989), 523–541  mathnet  crossref  mathscinet  zmath
    15. Yuri Zarhin, “Torsion of Abelian varieties over GL(2)-extensions of number fields”, Math Ann, 284:4 (1989), 631  crossref  mathscinet  zmath  isi  elib
    16. A.K. Bousfield, “A classification of K-local spectra”, Journal of Pure and Applied Algebra, 66:2 (1990), 121  crossref
    17. Min Ho Lee, “Periods of mixed cusp forms”, manuscripta math, 73:1 (1991), 163  crossref  mathscinet  zmath  isi
    18. Yu. G. Zarhin, “Torsion and endomorphisms of Abelian varieties over infinite extensions of number fields”, Math. USSR-Izv., 38:3 (1992), 647–657  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    19. Heiko Knospe, “Iwasawa-theory of abelian varieties at primes of non-ordinary reduction”, manuscripta math, 87:1 (1995), 225  crossref  mathscinet  zmath  isi
    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. I. S. Rakhimov, “Arithmetic Invariants for a Class of Elliptic Curves”, Siberian Adv. Math., 14:2 (2004), 79–91  mathnet  mathscinet  zmath  elib
    22. Kazuo Matsuno, “Finite Λ-submodules of Selmer groups of abelian varieties over cyclotomic -extensions”, Journal of Number Theory, 99:2 (2003), 415  crossref
    23. M. V. Bondarko, S. V. Vostokov, F. Lorenz, “The Hilbert pairing for formal groups over $\sigma$-rings”, J. Math. Sci. (N. Y.), 134:6 (2006), 2445–2476  mathnet  crossref  mathscinet  zmath  elib  elib
    24. [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
    25. Yu. V. Nesterenko, “On arithmetic properties of values of theta-constants”, J. Math. Sci., 146:2 (2007), 5697–5716  mathnet  crossref  mathscinet  zmath  elib
    26. I. S. Rakhimov, “O povedenii arifmeticheskikh invariantov nekotorogo klassa ellipticheskikh krivykh v krugovykh $\Gamma$-rasshireniyakh”, Matem. tr., 8:1 (2005), 122–134  mathnet  mathscinet  zmath  elib
    27. E. V. Ferens-Sorotskiy, “Weierstrass preparational theorem for noncommutative rings”, J. Math. Sci. (N. Y.), 161:4 (2009), 597–601  mathnet  crossref  zmath
    28. AMOD AGASHE, “SQUARENESS IN THE SPECIAL L-VALUE AND SPECIAL L-VALUES OF TWISTS”, Int. J. Number Theory, 06:05 (2010), 1091  crossref
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