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here is my question : Let $K$ be any field, $ E \to spec(K)$. Let $ v \in M_K $ an archimedean place. We know that $ \overline{K_v} \simeq \mathbb{C}$ and there exists $\tau_v \in \mathbb{H}$ such that : $$ E(\overline{K_v}) \simeq \mathbb{C} / \mathbb{Z} + \tau_v \mathbb{Z} $$ My question is : 1) can we have an upper bound for $Im(\tau_v)$ by a function depending on $v$ ?

2) if we fix an ample invertible sheaf $L$ which give an embedding $E \to P^n(K)$ can we compute $\tau$ in function of the data of $L$ ? Or $L^{\otimes n}$ ?

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  • $\begingroup$ $\tau$ can be computed by integrating a $1$-form over two different loops in $E(\mathbb C)$ (i.e. elliptic integrals). $\endgroup$
    – Will Sawin
    Commented Mar 13, 2020 at 16:17
  • $\begingroup$ I know that if I define this norm (Falting Norm) on the pull back of $\Omega^1_{\mathcal{E}}$ of the Néron model $\mathcal{E}$ over valuation ring of E : $$ \|s\|^2 = \dfrac{i}{2} \int_{E(\overline{K_v})} s \wedge \overline{s} $$ And if we take $\mathrm{d} z$, we have : $$ \|dz\|^2 = Im(\tau_v) $$ Do you have an other example ? Or maybe an ohter choice of section $s$ which give more informations ? $\endgroup$
    – MoinsUnPuissanceN
    Commented Mar 16, 2020 at 17:42

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