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Iwasawa theory of elliptic curves and Galois module structure. (English) Zbl 0802.11051
Let $$E$$ be an elliptic curve with complex multiplication by the ring of integers of $$K$$, an imaginary quadratic field. Assume the class number of $$K$$ is 1, and $$E$$ is defined over $$K$$ and has everywhere good reduction over $$F$$, an abelian extension of $$K$$. Let $$p$$ be an odd prime which splits as $$p= \pi\pi^*$$ over $$K$$. Let $${\mathcal B}_ i$$ be the $${\mathcal O}_ F$$-Hopf algebra representing $$\ker [\pi^{*i}]: E(\mathbb{Q}^ c)\to E(\mathbb{Q}^ c)$$, and $${\mathcal A}_ i$$ the Cartier dual of $${\mathcal B}_ i$$. For $$Q$$ in $$E(F)$$ let $$G_ Q(i)$$ be the preimage of $$Q$$ under $$[\pi^{*i}]$$, and let $$C_ Q(i)$$ be the maximal $${\mathcal A}_ i$$-stable submodule of the integral closure of $$\mathbb{Z}$$ in the Kummer algebra $$F_ Q(i)= \text{Map}_{\Omega_ F} (G_ Q(i), \mathbb{Q}^ c)$$. Then, as M. Taylor [Ill. J. Math. 32, 428-452 (1988; Zbl 0648.14019)] shows, $$C_ Q(i)$$ is a $${\mathcal B}_ i$$-principal homogeneous space, hence a locally free $${\mathcal A}_ i$$-module. Thus, for $${\mathcal M}_ i$$ the maximal order of $${\mathcal A}_ i$$, there is a map $$\psi_ i: E(F)/ [\pi^{*i} ]E(F)\to C\ell({\mathcal M}_ i)$$ into the locally free class group of $${\mathcal M}_ i$$, by $$\psi_ i(Q)= (C_ Q (i)\cdot {\mathcal M}_ i)$$.
The author shows that if the algebraic $$p$$-adic height pairing for $$E/F$$ of B. Perrin-Riou [Invent. Math. 70, 369-398 (1983; Zbl 0547.14025)] is non-degenerate modulo torsion, that the $$p$$-primary component of the Tate-Shafarevich group of $$E/F$$ is finite and that the rank of a certain eigenspace obtained from $$E(F)$$ is 1, then there is some point $$Q$$ of infinite order with $$\psi_ i(Q)=0$$ for all $$i$$.

##### MSC:
 11R23 Iwasawa theory 11G05 Elliptic curves over global fields 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
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##### References:
 [1] A. Agboola, Abelian varieties and Galois module structure in global function fields , to appear in Math. Z. · Zbl 0863.11078 · doi:10.1007/BF02571951 · eudml:174697 [2] A. Agboola and M. J. Taylor, Class invariants of Mordell-Weil groups , to appear in J. Reine Angew. Math. · Zbl 0799.11049 [3] A. Brumer, On the units of algebraic number fields , Mathematika 14 (1967), 121-124. · Zbl 0171.01105 · doi:10.1112/S0025579300003703 [4] N. P. Byott and M. J. Taylor, Hopf orders and Galois module structure , Group Rings and Class Groups, DMV Sem., vol. 18, Birkhäuser, Basel, 1992, pp. 153-210. · Zbl 0811.11068 [5] Ph. Cassou-Noguès and A. Srivastav, On Taylor’s conjecture for Kummer orders , Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 2, 349-363. · Zbl 0726.11038 · doi:10.5802/jtnb.32 · numdam:JTNB_1990__2_2_349_0 · eudml:93521 [6] Ph. Cassou-Noguès and M. J. Taylor, Elliptic Functions and Rings of Integers , Progr. Math., vol. 66, Birkhäuser, Boston, 1987. · Zbl 0608.12013 [7] J. Coates, Infinite descent on elliptic curves with complex multiplication , Arithmetic and Geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser, Boston, 1983, pp. 107-137. · Zbl 0541.14026 [8] J. Coates and M. J. Taylor, $$L$$-functions and Arithmetic , London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, Proceedings of the Durham Symposium. · Zbl 0718.00005 [9] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer , Invent. Math. 39 (1977), no. 3, 223-251. · Zbl 0359.14009 · doi:10.1007/BF01402975 · eudml:142468 [10] A. Fröhlich, Galois Module Structure of Algebraic Integers , Ergeb. Math. Grenzgeb. (3), vol. 1, Springer-Verlag, Berlin, 1983. · Zbl 0501.12012 [11] R. Greenberg, Iwasawa theory for $$p$$-adic representations , Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, in Honor of K. Iwasawa, pp. 97-137. · Zbl 0739.11045 [12] B. H. Gross, Heegner points on $$X_ 0(N)$$ , Modular forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 87-105. · Zbl 0559.14011 [13] A. Plater, Height pairings on elliptic curves , Ph.D. thesis, Cambridge Univ., 1991. · Zbl 0752.11026 [14] B. Perrin-Riou, Descente infinie et hauteur $$p$$-adique sur les courbes elliptiques à multiplication complexe , Invent. Math. 70 (1982/83), no. 3, 369-398. · Zbl 0547.14025 · doi:10.1007/BF01391797 · eudml:142980 [15] B. Perrin-Riou, Arithmétique des courbes elliptiques et théorie d’Iwasawa , Mém. Soc. Math. France (N.S.) (1984), no. 17, 130. · Zbl 0599.14020 · numdam:MSMF_1984_2_17__1_0 · eudml:94856 [16] K. Rubin, $$p$$-adic $$L$$-functions and rational points on elliptic curves with complex multiplication , Invent. Math. 107 (1992), no. 2, 323-350. · Zbl 0770.11033 · doi:10.1007/BF01231893 · eudml:143969 [17] J. Silverman, The Arithmetic of Elliptic Curves , Graduate Texts in Math., vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026 [18] A. Srivastav and M. J. Taylor, Elliptic curves with complex multiplication and Galois module structure , Invent. Math. 99 (1990), no. 1, 165-184. · Zbl 0705.14031 · doi:10.1007/BF01234415 · eudml:143754 [19] M. J. Taylor, Mordell-Weil groups and the Galois module structure of rings of integers , Illinois J. Math. 32 (1988), no. 3, 428-452. · Zbl 0631.14033 [20] M. J. Taylor, Classgroups of Group Rings , London Math. Soc. Lecture Note Ser., vol. 91, Cambridge Univ. Press, Cambridge, 1984. · Zbl 0597.13002 [21] M. J. Taylor, Rings of integers of fields obtained by the division of Heegner points , handwritten manuscript. · Zbl 0974.81018 [22] M. J. Taylor, The Galois module structure of certain arithmetic principal homogeneous spaces , J. Algebra 153 (1992), no. 1, 203-214. · Zbl 0776.11065 · doi:10.1016/0021-8693(92)90153-D
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