On the 𝑝-adic Second Main Theorem

Levin, Aaron

doi:10.1090/S0002-9939-2014-12530-5

On the $p$-adic Second Main Theorem
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Proc. Amer. Math. Soc. 143 (2015), 633-640 Request permission

Abstract:

We study the Second Main Theorem in non-archimedean Nevanlinna theory, giving an improvement to the non-archimedean Second Main Theorems of Ru and An in the case where all the hypersurfaces have degree greater than one and all intersections are transverse. In particular, under a transversality assumption, if $f$ is a nonconstant non-archimedean analytic map to $\mathbb {P}^n$ and $D_1,\ldots , D_q$ are hypersurfaces of degree $d$, we prove the defect relation \begin{equation*} \sum _{i=1}^q\delta _f(D_i)\leq n-1+\frac {1}{d}, \end{equation*} which is sharp for all positive integers $n$ and $d$.

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