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Using some of BCnrd's ideas together with a different construction, I'll give a positive answer to Kevin Buzzard's stronger question; i.e., there is a counterexample for any non-etale smooth morphism. …

Let $Y=\operatorname{Spec} k$ and let $X=\mathbb{P}^1_k$, viewed as $\operatorname{Spec} k[t]$ glued to $\operatorname{Spec} k[t^{-1}]$. Let $T = \operatorname{\bf Spec}(\mathcal{O}_X + \mathcal{O}_X( …

For instance, one can find examples over $\mathbf{Q}$ by using the fact that the endomorphism ring injects into the endomorphism ring of the reduction modulo any prime of good reduction, and combining …

Since you asked for it, here is a little bit about the role of valuations in the Lang-Nishimura theorem, one version of which is as follows (my version implies yours): Theorem (Lang-Nishimura): Let $ …

Computer experiments (in the early 1960s!) led Birch and Swinnerton-Dyer to the formulation of their conjecture, which stimulated the development of much of arithmetic geometry.

Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication "If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true." for ev …