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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.

##### MSC:
 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
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