Local heights on elliptic curves and intersection multiplicities (Q2907094)

The main result of the paper under review is a formula which relates the local heights on elliptic curves over number fields to intersection theory on a regular model over the ring of integers. Precisely, let \(E\) be an elliptic curve defined over a number field \(K\). It is well known that the canonical height \(\widehat{h}\) on \(E\) has a decomposition NEWLINE\[NEWLINE\widehat{h}(P)=\frac{1}{[K:\mathbb{Q}]}\sum_{v\in M_K}[K_v:\mathbb{Q}_v]\lambda_v(P)NEWLINE\]NEWLINE where \(M_K\) denotes the set of places of \(K\). These functions \(\lambda_v: E(K_v)\to \mathbb{R}\) are called the local heights. On the other side, let \(R\) be the ring of integers in \(K\) and let \(\mathcal{C}\) be the minimal regular model of \(E\) over \(R\). To every \(Q\in E(K)\), \(\mathrm{Q}\) is denoted by the closure of \((Q)\in \text{Div}(E)(K)\), this approach can be extended to the group \(\text{Div}(E)(K)\) of \(K\)-rational divisors on \(E\) by linearity. For any finite place \(v\) and any \(D\in \text{Div}(E)(K)\) of degree zero, Hriljac's lemma confirms that there exists a \(v\)-vertical \(\mathbb{Q}\)-divisor \(\Phi_v(D)\)on \(\mathcal{C}\) such that \((\mathrm{D}+\Phi_v(D)\centerdot F)_v=0\) for any \(v\)-vertical \(\mathbb{Q}\)-divisor \(F\) on \(\mathcal{C}\), where \((\cdot\centerdot\cdot)_v\) denotes the intersection multiplicity on \(\mathcal{C}\) above \(v\). Then the authors show that if \(v\) is finite and \(P\in E(K)\backslash\{O\}\) and \(E\) is given by a Weierstrass equation that is minimal at \(v\), one has \(\lambda_v(P)=2(\mathrm{P}\centerdot \mathrm{O})_v-(\Phi_v((P)-(O))\centerdot \mathrm{P}-\mathrm{O})_v\). This result can be regarded as a local analog of Faltings and Hriljac's theorem which relates the Arakelov intersection to the canonical heights.