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A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture. (English) Zbl 0444.14015

MSC:
14G05 Rational points
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25 Arithmetic ground fields for curves
14K15 Arithmetic ground fields for abelian varieties
14K05 Algebraic theory of abelian varieties
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References:
[1] Lang, S.: Les formes bilinéaires de Néron et Tate. Sem. Bourbaki, no 274, 1964 · Zbl 0138.42101
[2] Manin, Ju., Zarkin, Y.G.: Heights on families of abelian varieties. Mat. Sbornik89, 171-181 (1972)
[3] Néron, A.: Quasi-fonctions et hauteurs sur les variétés abéliennes. Annals of Math.82, 249-331 (1965) · Zbl 0163.15205 · doi:10.2307/1970644
[4] Ono, T.: Arithmetic of algebraic tori. Ann. of Math.,74, 101-139 (1961) · Zbl 0119.27801 · doi:10.2307/1970307
[5] Ono, T.: On the Tamagawa number of algebraic tori. Ann. of Math.,78, 47-73 (1963) · Zbl 0122.39101 · doi:10.2307/1970502
[6] Sansuc, Thèse, Paris (1978)
[7] Tate, J.: The arithmetic of elliptic curves. Invent. Math.23, 179-206 (1974) · Zbl 0296.14018 · doi:10.1007/BF01389745
[8] Tate, J.: On the conjecture of Birch and Swimmerton-Dyer and a geometric analog. Sem. Bourbaki No.306, Feb. 1966 · Zbl 0199.55604
[9] Tate, J.: Letter to Serre, June 21, 1968
[10] Weil, A.: Adèles and algebraic groups. Institute for Advanced Study, Princeton, 1961 · Zbl 0109.02101
[11] Mazur, B., Messing, W.: Universal extensions and one-dimensional crystalline cohomology. Lecture Notes in Math. No.370, Berlin-Heidelberg-New York: Springer 1974 · Zbl 0301.14016
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