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

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.

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.

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {45},
     number = {2},
     year = {1995},
     pages = {317-374},
     doi = {10.5802/aif.1457},
     zbl = {0821.11036},
     mrnumber = {96j:11089},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1995__45_2_317_0}
}
Tan, Ki-Seng. Refined theorems of the Birch and Swinnerton-Dyer type. Annales de l'Institut Fourier, Volume 45 (1995) no. 2, pp. 317-374. doi : 10.5802/aif.1457. http://www.numdam.org/item/AIF_1995__45_2_317_0/

[AT] E. Artin and J. Tate, Class Field Theory, Benjamin, New York, 1967. | Zbl 0176.33504

[BS] Z. Borevich and I.R. Shafarevich, Number Theorey, English translation, Academic Press, New York, 1966. | Zbl 0145.04902

[D] P. Deligne, Les constants, etc., Séminaire Delange-Poisot-Poitou, 11e année 19, 1970. | Numdam | Zbl 0215.08301

[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 89h:11071 | Zbl 0681.12005

[GSt] R. Greenberg and G. Stevens, p-adic L-functions and p-adic modular forms, Invent. Math., 111 (1993), 407-447. | MR 93m:11054 | Zbl 0778.11034

[K] H. Kisilevsky, Multiplicative independence in function fields, J. Number Theory, 44 (1993), 352-355. | MR 94k:11125 | Zbl 0780.11058

[L] S. Lang, Algebraic Number Theory, Graduate Texts in Mathematics, Vol. 110, Springer-Verlag, New York, 1986. | MR 95f:11085 | Zbl 0601.12001

[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 54 #2659 | Zbl 0343.14005

[Ml2] J. Milne, Arithmetic Duality Theorems, Academic Press, New York, 1986. | MR 88e:14028 | Zbl 0613.14019

[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 85j:14081 | Zbl 0574.14036

[MT2] B. Mazur and J. Tate, Refined conjectures of the Birch and Swinnerton-Dyer type, Duke Math. J., 54/2 (1987), 711-750. | MR 88k:11039 | Zbl 0636.14004

[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 87e:11076 | Zbl 0699.14028

[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 55 #10550 | Zbl 0358.16006

[S] J. Silverman, Arithmetic of Elliptic Curves, Graduate Texts in Math., Vol. 106, Springer-Verlag, New York, 1986. | MR 87g:11070 | Zbl 0585.14026

[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 0237.00013

[T1] J. Tate, Duality theorems in Galois cohomology over number fields, in Proc. Intern. Congress Math., Stockholm (1962), 231-241. | Zbl 0126.07002

[T2] J. Tate, On the conjecture of Birch and Swinnerton-Dyer and a geometric analogue, Séminaire Bourbaki n° 306 (1966). | Numdam | Zbl 0199.55604

[T3] J. Tate, The arithmetic of elliptic curves, Invent. Math., 23 (1974), 179-206. | MR 54 #7380 | Zbl 0296.14018

[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 95d:11158 | Zbl 0802.11026

[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 0820.11069

[W1] A. Weil, Basic Number Theory, Grundl. Math. Wiss. Bd. 144, Springer-Verlag, New York, 1967. | Zbl 0176.33601

[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 0254.14012