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Derived heights and generalized Mazur-Tate regulators. (English) Zbl 0853.14013
Let \(E\) be an elliptic curve defined over a number field \(K\), and let \(L/K\) be an abelian extension with Galois group \(G\). B. Mazur and J. Tate [in: Arithmetic and geometry, Vol. I, Prog. Math. 35, 195-237 (1983; Zbl 0574.14036) and Duke Math. J. 54, 711-750 (1987; Zbl 0636.14004)] have defined a height pairing \(\langle \;,\;\rangle_{MT} : E_L (K) \times E(K) \to G\), where \(E_L (K)\) is a subgroup of finite index of \(E(K)\), consisting of the points of \(E(K)\) that are local norms from \(E(L)\). Let \(I\) denote the augmentation ideal in the integral group ring \(\mathbb{Z} [G]\). There is a canonical identification \(G = I/I^2\), allowing us to view the Mazur-Tate pairing as taking values in \(I/I^2\). Let \(P_1, \dots, P_r\) (resp. \(Q_1, \dots, Q_r)\) denote integral bases for \(E_L (K)\) (resp. \(E (K))\) modulo torsion. The matrix \((\langle P_i, Q_j\rangle_{MT})\) is an \(r \times r\) matrix with entries in \(I/I^2\), and its determinant gives an element of \(I^r/I^{r + 1}\). Let \(\Lambda_{MT}\) denote this element; it is the Mazur-Tate regulator associated to \((E,L/K)\). The goal of this paper is to define (under certain conditions) a lift \(\widetilde \Lambda\) of \(\Lambda_{MT}\) to \(I^r\). This lift depends on some choices, but the following are independent of the choices:
1. the order of vanishing of \(\widetilde \Lambda\), defined to be the least \(\rho\) (possibly \(\infty)\) such that \(\widetilde \Lambda\) belongs to \(I^\rho\) but not to \(I^{\rho + 1}\),
2. the image \(\Lambda\) of \(\widetilde\Lambda\) in \(I^\rho/I^{\rho + 1}\).
We call \(\Lambda\) the generalized Mazur-Tate regulator associated to \((E,L/K)\). It is equal to the Mazur-Tate regulator when \(\rho=r\), but provides extra information when \(\Lambda_{MT} = 0\). In particular, it can be used to formulate a refined conjecture in the spirit of Mazur and Tate [cf. the Duke paper cited above]. The conjecture in that paper relates the Mazur-Tate regulator to the leading coefficient of a \(\theta\)-element interpolating special values of the Hasse-Weil \(L\)-function of \(E/K\). In particular, it predicts that the order of vanishing of this element is at least \(r\), but that some extra vanishing may arise from degeneracies in the Mazur-Tate height (i.e., when \(\Lambda_{MT} = 0)\). We formulate a conjecture predicting the precise order of vanishing of the element \(\theta\), and expressing the value of its leading coefficient in terms of our generalized regulator \(\Lambda\). In certain cases, we show that our refinement of the Mazur-Tate conjecture follows from the classical conjecture of Birch and Swinnerton-Dyer.
A particularly interesting special case (which partly motivated the present study) arises when \(K\) is a quadratic field and \(L/K\) is an extension of \(K\) of dihedral type. In this case, degeneracies in the Mazur-Tate height seem to be the rule rather than the exception. This case is discussed in section 4.3.

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11R54 Other algebras and orders, and their zeta and \(L\)-functions
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14H52 Elliptic curves
Full Text: DOI
[1] M. Bertolini, Iwasawa theory, \(L\)-functions and Heegner points , Ph.D. thesis, Columbia University, 1992.
[2] M. Bertolini and H. Darmon, Derived \(p\)-adic heights ,
[3] J. W. S. Cassels, Arithmetic on curves of genus \(1\). III. The Tate-Šafarevič and Selmer groups , Proc. London Math. Soc. (3) 12 (1962), 259-296. · Zbl 0106.03705 · doi:10.1112/plms/s3-12.1.259
[4] J. W. S. Cassels, Arithmetic on curves of genus \(1\). IV. Proof of the Hauptvermutung , J. Reine Angew. Math. 211 (1962), 95-112. · Zbl 0106.03706 · doi:10.1515/crll.1962.211.95 · crelle:GDZPPN002179873 · eudml:150551
[5] Cassels, J. W. S. and Frölich, A., eds., Algebraic number theory , Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the Inter national Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London, 1967. · Zbl 0153.07403
[6] H. Darmon, A refined conjecture of Mazur-Tate type for Heegner points , Invent. Math. 110 (1992), no. 1, 123-146. · Zbl 0781.11023 · doi:10.1007/BF01231327 · eudml:144045
[7] H. Darmon, Heegner points, Heegner cycles, and congruences , proceedings of a conference on elliptic curves and related topics, Ste-Adèle, Québec, 1992.
[8] H. Darmon, Euler systems and refined conjectures of Birch Swinnerton-Dyer type , proceedings of a conference on \(p\)-adic monodromy and the Birch-Swinnerton-Dyer conjecture, Boston University, 1991, · Zbl 0823.11036
[9] A. Fröhlich, Galois module structure of algebraic integers , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 1, Springer-Verlag, Berlin, 1983. · Zbl 0501.12012
[10] B. H. Gross, On the values of abelian \(L\)-functions at \(s=0\) , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), no. 1, 177-197. · Zbl 0681.12005
[11] B. H. Gross, Heegner points on \(X_ 0(N)\) , Modular forms (Durham, 1983) ed. R. A. Rankin, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 87-105. · Zbl 0559.14011
[12] B. Mazur, Rational points of abelian varieties with values in towers of number fields , Invent. Math. 18 (1972), 183-266. · Zbl 0245.14015 · doi:10.1007/BF01389815 · eudml:142180
[13] B. Mazur and J. Tate, Canonical height pairings via biextensions , Arithmetic and geometry, Vol. I eds. Artin Micheal and Tate John, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 195-237. · Zbl 0574.14036
[14] B. Mazur and J. Tate, Refined conjectures of the “Birch and Swinnerton-Dyer type” , Duke Math. J. 54 (1987), no. 2, 711-750. · Zbl 0636.14004 · doi:10.1215/S0012-7094-87-05431-7
[15] J. S. Milne, Arithmetic duality theorems , Perspectives in Mathematics, vol. 1, Academic Press Inc., Boston, MA, 1986. · Zbl 0613.14019
[16] B. Perrin-Riou, Fonctions \(L p\)-adiques, Théorie d’Iwasawa et points de Heegner , Bull. Soc. Math. France 115 (1987), 455-510. · Zbl 0645.14010 · doi:10.1007/BF01388982 · eudml:143491
[17] P. Schneider, Iwasawa \(L\)-functions of varieties over algebraic number fields. A first approach , Invent. Math. 71 (1983), no. 2, 251-293. · Zbl 0511.14010 · doi:10.1007/BF01389099 · eudml:142991
[18] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques , Invent. Math. 15 (1972), no. 4, 259-331. · Zbl 0235.14012 · doi:10.1007/BF01405086 · eudml:142133
[19] T. Shintani, On construction of holomorphic cusp forms of half integral weight , Nagoya Math. J. 58 (1975), 83-126. · Zbl 0316.10016
[20] K.-S. Tan Harvard Ph.D. Thesis, 1991.
[21] K.-S. Tan, \(p\)-adic pairings , proceedings of a conference on \(p\)-adic monodromy and the Birch Swinnerton-Dyer conjecture, Boston University, 1991, · Zbl 0840.14030
[22] J. Tate, Duality theorems in Galois cohomology over number fields , Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 288-295. · Zbl 0126.07002
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