The Cartan conjecture for \(p\)-adic holomorphic curves (Q1278924)

The main result is formulated as: Let \(V\) be a vector space of dimension \(n+1\) over \({\mathbb{C}}_p\). Let \({\mathcal G}=\{g_j\}_{j=0}^q\) be a finite family of \(p\)-adic curves \(g_j:{\mathbb{C}}_p\rightarrow {\mathbb{P}}(V^*)\) in general position with \(q\geq n\). Take an integer \(k\) with \(1\leq k\leq n\). Let \(f:{\mathbb{C}}_p\rightarrow {\mathbb{P}}(V)\) be a \(p\)-adic holomorphic curve which is \(k\)-flat over \(\mathcal R\) such that each pair \((f,g_j)\) is free for \(j=0,\dots ,q\). Assume that \(g_j\) grows slower than \(f\) for each \(j\). Then we have \(\sum _{j=0}^q\delta _f(g_j)\leq 2n-k+1\). The first page of the paper compares this theorem in the complex case and the \(p\)-adic case with theorems on Nevanlinna theory by many authors, e.g., R.~Nevanlinna, H.~Cartan, P.~Vojta, H. H.~Khóai. It seems to the reviewer that the above theorem is an important result which encompasses the previous results in the \(p\)-adic case. However he has been unable to trace the definitions and their meaning of all the terms in the formulation. It might be worthwhile to present the theorem and its proof in a readable form.