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 $L^{\otimes n}$ ?