Ie, let $E/S$ be an elliptic curve over some scheme $S$. Is it possible to have an automorphism $\alpha$ of $E$ over $S$ such that for some geometric point $s\in S$ its pullback to $E_s$ is the identity?
2026-03-29 12:26:37.1774787197
Can you have a nontrivial automorphism of an elliptic curve $E/S$ which when restricted to a geometric fiber is the identity?
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Obviously, the answer is yes if $S$ is not connected.
If $S$ is connected, the answer is NO because Grothedieck's Rigidity Lemma (GRL) implies that $$ \text{End}_S(E)\longrightarrow\text{End}_s(E) $$ is injective for every geometric point $s\in S$. Unfortunately, I could find no link to a GRL page. The reference I know is to the book of Mumford, Fogarty and Kirwan on Geometric Invariant Theory (Proposition 6.1 there)