I just want to know that whether the specialization theorem holds for multi variable function fields or not? i mean when we specialize over one function field to the other , can be sure that Mordell-Weil group doesn't becomes smaller or not? Cause i only know that the theorem holds for number fields
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$\begingroup$ For a DVR $R$ with finite residue field $F\cong \mathbb{F}_{p^e}$ and with fraction field $K$, for an elliptic surface $\mathcal{E}\to \mathbb{P}^1_R$, it could happen that the rational sections of $\mathcal{E}\times_{\text{Spec}(R)}\text{Spec}(K) \to \mathbb{P}^1_K$ are only $p$-torsion points, and several distinct sections specialize over $\mathbb{P}^1_F$ to the "same" rational section. In that case, the group scheme over $\text{Spec}(R)$ of rational sections has a nonreduced fiber over $\text{Spec}(F)$. This cannot happen in characteristic $0$ because group schemes are reduced. $\endgroup$– Jason StarrCommented Apr 7, 2017 at 15:19
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