no. 2
Refined theorems of the Birch and Swinnerton-Dyer type
Annales de l'Institut Fourier, Tome 45 (1995) no. 2, pp. 317-374.

Dans cet article nous généralisons le contexte de la conjecture de Mazur-Tate et dans une certaine mesure en donnons un énoncé plus fin. Nous prouvons ces nouvelles conjectures en supposant vraies les conjectures classiques de Birch et Swinnerton-Dyer. Ceci est remarquable dans le cas du corps des fonctions où ces résultats constituent une amélioration de travaux antérieurs de Tate et Milne.

In this paper, we generalize the context of the Mazur-Tate conjecture and sharpen, in a certain way, the statement of the conjecture. Our main result will be to establish the truth of a part of these new sharpened conjectures, provided that one assume the truth of the classical Birch and Swinnerton-Dyer conjectures. This is particularly striking in the function field case, where these results can be viewed as being a refinement of the earlier work of Tate and Milne.

@article{AIF_1995__45_2_317_0,
     author = {Tan, Ki-Seng},
     title = {Refined theorems of the {Birch} and {Swinnerton-Dyer} type},
     journal = {Annales de l'Institut Fourier},
     pages = {317--374},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {2},
     year = {1995},
     doi = {10.5802/aif.1457},
     mrnumber = {96j:11089},
     zbl = {0821.11036},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1457/}
}
TY  - JOUR
AU  - Tan, Ki-Seng
TI  - Refined theorems of the Birch and Swinnerton-Dyer type
JO  - Annales de l'Institut Fourier
PY  - 1995
SP  - 317
EP  - 374
VL  - 45
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1457/
DO  - 10.5802/aif.1457
LA  - en
ID  - AIF_1995__45_2_317_0
ER  - 
%0 Journal Article
%A Tan, Ki-Seng
%T Refined theorems of the Birch and Swinnerton-Dyer type
%J Annales de l'Institut Fourier
%D 1995
%P 317-374
%V 45
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1457/
%R 10.5802/aif.1457
%G en
%F AIF_1995__45_2_317_0
Tan, Ki-Seng. Refined theorems of the Birch and Swinnerton-Dyer type. Annales de l'Institut Fourier, Tome 45 (1995) no. 2, pp. 317-374. doi : 10.5802/aif.1457. http://www.numdam.org/articles/10.5802/aif.1457/

[AT] E. Artin and J. Tate, Class Field Theory, Benjamin, New York, 1967. | Zbl

[BS] Z. Borevich and I.R. Shafarevich, Number Theorey, English translation, Academic Press, New York, 1966. | Zbl

[D] P. Deligne, Les constants, etc., Séminaire Delange-Poisot-Poitou, 11e année 19, 1970. | Numdam | Zbl

[G] B. Gross, On the value of abelian L-functions at s = 0, J. Fac. Sci. Univ. Tokyo Sect. IA, Math., 35 (1988), 177-197. | MR | Zbl

[GSt] R. Greenberg and G. Stevens, p-adic L-functions and p-adic modular forms, Invent. Math., 111 (1993), 407-447. | MR | Zbl

[K] H. Kisilevsky, Multiplicative independence in function fields, J. Number Theory, 44 (1993), 352-355. | MR | Zbl

[L] S. Lang, Algebraic Number Theory, Graduate Texts in Mathematics, Vol. 110, Springer-Verlag, New York, 1986. | MR | Zbl

[M] B. Mazur, Letter to J. Tate, 1987.

[Ml1] J. Milne, On a conjecture of Artin and Tate, Annals of Math., 102 (1975), 517-533. | MR | Zbl

[Ml2] J. Milne, Arithmetic Duality Theorems, Academic Press, New York, 1986. | MR | Zbl

[Mu] D. Mumford, Biextensions of formal groups, in the Proceedings of the Bombay Colloquium on Algebraic Geometry, Tata Institute of Fundamental Research Studies in Mathematics 4, London, Oxford University Press, 1968.

[MT1] B. Mazur and J. Tate, Canonical pairing via biextensions, in Arithmetic and Geometry, Progr. Math., Vol. 35 (1983), 195-237, Birkhäuser, Boston-Basel-Stuttgart. | MR | Zbl

[MT2] B. Mazur and J. Tate, Refined conjectures of the Birch and Swinnerton-Dyer type, Duke Math. J., 54/2 (1987), 711-750. | MR | Zbl

[MTT] B. Mazur, J. Tate and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math., 84 (1986), 1-84. | MR | Zbl

[PV] I.B.S. Passi and L.R. Vermani, The associated graded ring of an integral group ring, Math. Proc. Camb. Phil. Soc., 82 (1977), 25-33. | MR | Zbl

[S] J. Silverman, Arithmetic of Elliptic Curves, Graduate Texts in Math., Vol. 106, Springer-Verlag, New York, 1986. | MR | Zbl

[GA7 I] A. Grothendieck et al., Séminaire de géométrie algébrique du Bois Marie, 1967/1969, Groupes de monodromie en géométrie algébrique, Lecture Notes in mathematics 288, Springer, Berlin-Heidelberg-New York, 1972. | Zbl

[T1] J. Tate, Duality theorems in Galois cohomology over number fields, in Proc. Intern. Congress Math., Stockholm (1962), 231-241. | Zbl

[T2] J. Tate, On the conjecture of Birch and Swinnerton-Dyer and a geometric analogue, Séminaire Bourbaki n° 306 (1966). | Numdam | Zbl

[T3] J. Tate, The arithmetic of elliptic curves, Invent. Math., 23 (1974), 179-206. | MR | Zbl

[T4] J. Tate, Letter to B. Mazur, 1988.

[T5] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular Functions of One Variable IV, Lecture Notes in Math. 476 (1975), p. 33-53, Springer-Verlag, Berlin-Heidelberg-New York.

[Tn1] K.-S. Tan, Refined conjectures of the Birch and Swinnerton-Dyer Type, Harvard University, Dept. of Mathematics, Ph. D. Thesis, 1990.

[Tn2] K.-S. Tan, Modular elements over function fields, Journal of Number Theory, 45 (1993, n° 3), 295-311. | MR | Zbl

[Tn3] K.-S. Tan, On the p-adic height pairings, AMS Proceedings on the p-adic Monodromy, to appear.

[Tn4] K.-S. Tan, On the special values of abelian L-function, submitted to J. Fac. Sci. Univ. Tokyo. | Zbl

[W1] A. Weil, Basic Number Theory, Grundl. Math. Wiss. Bd. 144, Springer-Verlag, New York, 1967. | Zbl

[W2] A. Weil, Adèles and Algebraic Groups, Birkhauser, Boston, 1982.

[Z] J.G. Zarhin, Néron pairing and quasicharacters, Izv. Akad. Nauk. SSSR Ser. Mat. 36 (3), 497-509, 1972. (Math. USSR Izvestija, Vol. 6, No. 3, 491-503). | Zbl

Cité par Sources :