Timeline for How does the number of connected components of the Néron model change in a family of abelian varieties?
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| Sep 12, 2023 at 13:23 | comment | added | Nathan Lowry | @AriyanJavanpeykar Isn't the family in the post a counterexample to this claim? This family contains (up to isomorphism) every elliptic curve over $\mathbb{Q}_p$ so, in particular, it contains elliptic curves whose Neron polygon has arbitrarily many sides. Perhaps the claim will be true with some additional hypotheses on the base (proper, projective, etc.)? | |
| Sep 12, 2023 at 4:24 | comment | added | Ariyan Javanpeykar | Isn't the number of components constant over a dense open $U$ of the base $S$? If so, restrict the abelian scheme $A\to S$ to $S\setminus U = S_2$. Now you have an abelian scheme $A\to S_2$ whose component group is constant over a dense open $U_2\subset S_2$. This process terminates (in about $\dim S+1$ steps). | |
| Sep 11, 2023 at 14:54 | history | asked | Nathan Lowry | CC BY-SA 4.0 |