Greenberg, Ralph On the Birch and Swinnerton-Dyer conjecture. (English) Zbl 0546.14015 Invent. Math. 72, 241-265 (1983). The main result is: Let \(E\) be an elliptic curve with complex multiplication. If \(L(E/\mathbb Q,s)\) has a zero of odd order at \(s=1\), then either the Mordell-Weil group \(E(\mathbb Q)\) has rank at least one, or the \(p\)-primary components of the Tate-Shafarevich group are infinite for all primes \(p\) (\(\neq 2,3)\) where at which \(E\) has ordinary good reduction. This complements a theorem of J. Coates and A. Wiles [Invent. Math. 39, 223–251 (1977; Zbl 0359.14009)] and provides further evidence for the Birch–Swinnerton-Dyer conjectures. The author records a variant of these conjectures suggested by Coates in case the Tate-Shafarevich groups have finite order. The proof is via a Hecke \(L\)-series associated with a Grössencharacter \(\psi\). The \(L\)-series obeys a functional equation \(L(s)=wL(2-s)\). Both cases \(w=\pm 1\) are needed in the proof. In particular the author shows that \(L(\Psi^{2k+1},k+1)=0\) for only finitely many \(k\) when \(\Psi^{2k+1}\) is a character with \(w=1\). This non-vanishing theorem leads via some intricate work with the anti-cyclotomic \(p\)-extension of the complex multiplication field to a proof that the Selmer groups are infinite in the case \(w=-1\) and thence to the main theorem. Reviewer: G.Horrocks Cited in 10 ReviewsCited in 39 Documents MSC: 11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G15 Complex multiplication and moduli of abelian varieties 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14H45 Special algebraic curves and curves of low genus 14K22 Complex multiplication and abelian varieties 14H52 Elliptic curves 14G05 Rational points Keywords:elliptic curve with complex multiplication; Mordell-Weil group; Birch Swinnerton-Dyer conjectures; Hecke L-series; Selmer groups Citations:Zbl 0359.14009 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Arthaud, N.: On Birch and Swinnerton-Dyer’s conjecture for elliptic curves with complex multiplication. I. Comp. Math.37, 209-232 (1978) · Zbl 0396.12011 [2] Brumer, A.: On the units of algebraic number fields. Mathematika14, 121-124 (1967) · Zbl 0171.01105 · doi:10.1112/S0025579300003703 [3] Coates, J.: Infinite descent on elliptic curves with complex multiplication. To appear in Progress in Mathematics. Boston: Birkhauser · Zbl 0541.14026 [4] Coates, J.: Elliptic curves with complex multiplication. Herman Weyl lectures, 1979, Annals of Math. Studies, to appear [5] Coates, J., Wiles, A.: On the conjecture of Birch and Swinnerton-Dyer. Invent. Math.39, 223-251 (1977) · Zbl 0359.14009 · doi:10.1007/BF01402975 [6] Damarell, R.:L-functions of elliptic curves with complex multiplication I. Acta Arith.17, 287-301 (1970) [7] Deuring, M.: Die Zetafunktionen einer algebraischen Kurve vom Geschlecht Eins I, II, III, IV. Nachr. Akad. Wiss. Göttingen, 85-94 (1953); 13-42 (1955); 37-76 (1956); 53-80 (1957) · Zbl 0064.27401 [8] Frohlich, A., Queyrut, J.: On the functional equation of the ArtinL-function for characters of real representations. Invent. Math.14, 173-183 (1971) · Zbl 0229.13006 · doi:10.1007/BF01418887 [9] Greenberg, R.: Onp-adicL-functions and Iwasawa’s theory for imaginary quadratic fields Proceedings of conference on modern trends in Alg. No. theory related to Fermat’s last theorem, pp. 215-286. Boston: Birkhäuser [10] Greenberg, R.: Functional equations, root numbers, and Iwasawa theory. In preparation [11] Gross, B., Zagier, D.: On the critical values of HeckeL-series. Soc. Math. de France, Memoire No.2, 49-54 (1980) · Zbl 0462.14015 [12] Hardy, G.H.: Divergent Series. Clarendon Press (1949) · Zbl 0032.05801 [13] Iwasawa, K.: On ? t -extension of algebraic number fields. Ann. of Math.98, 246-326 (1973) · Zbl 0285.12008 · doi:10.2307/1970784 [14] Katz, N.:p-adic interpolation of real analytic Eisenstein series. Ann. of Math.104, 459-571 (1976) · Zbl 0354.14007 · doi:10.2307/1970966 [15] Kohnen, W., Zagier, D.: Values ofL-series of modular forms at the center of the critical strip. Invent. Math.64, 175-198 (1981) · Zbl 0468.10015 · doi:10.1007/BF01389166 [16] Manin, J., Vishik, S.:p-adic Hecke series for imaginary quadratic fields. Math. Sbornik95 (137) (1974), No. 3 · Zbl 0329.12016 [17] Martinet, J.: Character theory and ArtinL-functions. Alg. No. fields. (Frohlich, A. (ed.) New York: Academic Press (1977) · Zbl 0359.12015 [18] Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Invent. Math.18, 183-266 (1972) · Zbl 0245.14015 · doi:10.1007/BF01389815 [19] Ogg, A.: Modular forms and dirichlet series. New York: Benjamin 1969 · Zbl 0191.38101 [20] Perrin-Riou, B.: Groupe de Selmer d’une courbe elliptique a multiplication complexe. Comp. Math.43, 387-417 (1981) · Zbl 0479.14019 [21] Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. Math.64, 455-470 (1981) · Zbl 0506.14039 · doi:10.1007/BF01389277 [22] Rubin, K., Wiles, A.: Mordell-Weil groups of elliptic curves over cyclotomic fields. Proceedings of conference on modern trends in Alg. No. theory related to Fermat’s last theorem, pp. 237-254. Boston: Birkhäuser [23] Serre, JP.: Classes de corps cyclotomique. Sem. Bourbaki174 (1958) [24] Serre, JP, Tate, J.: Good reduction of abelian varieties. Ann. Math.88, 492-517 (1968) · Zbl 0172.46101 · doi:10.2307/1970722 [25] Shimura, G.: On elliptic curves with complex multiplication as factors of the Jacobian of modular function fields. Nagoya Math. J.43, 199-208 (1971) · Zbl 0225.14015 [26] Tate, J.: Fourier analysis in number fields and Hecke’s zeta functions. Alg. No. theory. (Cassels, Frohlich, eds.). London: Academic Press 305-347 (1967) [27] Tate, J.: Arithmetic of elliptic curves. Invent. Math.23, 179-206 (1974) · Zbl 0296.14018 · doi:10.1007/BF01389745 [28] Tate, J. Local constants. Alg. No. fields, (Frohlich, A., ed.). New York: Academic Press 1977 · Zbl 0425.12019 [29] Weil, A.: On a certain type of character of the idele-class groups of an algebraic number field. Proc. Int’l. Symp. Tokyo-Nikko, 1-7 1955 [30] Weil, A.: Dirichlet series and automorphic forms. Lecture notes in math. Vol. 189. Berlin-Heidelberg-New York: Springer 1971 · Zbl 0218.10046 [31] Yager, R.: On two-variablep-adicL-functions. Ann. of Math.115, 411-449 (1982) · Zbl 0496.12010 · doi:10.2307/1971398 [32] Yager, R.:p-adic measures on Galois groups. Preprint · Zbl 0555.12006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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