Derived heights and generalized Mazur-Tate regulators (Q1345226)

Let \(E\) be an elliptic curve defined over a number field \(K\), and let \(L/K\) be an abelian extension with Galois group \(G\). \textit{B. Mazur} and \textit{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.