Explicit local heights (Q1299560)

Let \(K\) be a finite extension of \(\mathbb{Q}_p\), and denote by \(|\cdot |\) the absolute value on \(K\), normalized such that \(|p|=1/p\). Further, let \(E\) denote an elliptic curve defined over \(K\), and for a \(K\)-rational point \(Q\in E(K)\), write \((x_Q,y_Q)\) for the coordinates of \(Q\) with respect to a minimal defining equation. In a previous paper, \textit{G. Everest} and \textit{B. N. Fhlathuin} [Math. Proc. Camb. Philos. Soc. 120, 13-25 (1996; Zbl 0865.11068)] showed that if \(\lambda (Q)\) denotes the local canonical height of \(Q\), then \(\int_G \log |x-x_Q|d\mu_G =2\lambda (Q)\), where \(G\) is any compact group containing \(Q\), and \(\mu_G\) is the Haar measure on \(G\), normalized such that \(\mu (G)=1\). In the present paper, the author proves some analogues of Jensen's formula \[ \int_0^1 \log |e^{2\pi it} -a|dt =\log \max (1,|a|). \] For instance, if \(p\not= 2,3\), \(Q\) has non-singular reduction and \(G\) is the topological closure of \(Q\), he shows that \(\int_G \log |x-x_Q|d\mu_G =\log\max (1,|x_Q|)\). He proves a similar, but more complicated formula in case that \(Q\) has split multiplicative reduction.