Chai's additivity conjecture for base change conductors

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Let G∙:0→T→B→A→0G^\bullet:0\to T\to B\to A\to 0 be an exact sequence of semiabelian varieties over KK. Let G∙:0→T→B→A→0\mathscr{G}^\bullet:0\to\mathscr{T}\to\mathscr{B}\to\mathscr{A}\to 0 be the associated complex of Néron lft-models over OK\mathcal{O}_K, with T\mathscr{T} in degree 11. For a finite separable extension L/KL/K over which TT, BB, and AA acquire semiabelian reduction, define

c(B):=1[L:K]ℓOL ⁣(coker⁡ ⁣(Lie⁡B⊗OKOL→Lie⁡BL)),c(B):=\frac{1}{[L:K]}\ell_{\mathcal{O}_L}\!\left(\operatorname{coker}\!\left(\operatorname{Lie}\mathscr{B}\otimes_{\mathcal{O}_K}\mathcal{O}_L\to\operatorname{Lie}\mathscr{B}_L\right)\right),

and similarly define c(T)c(T) and c(A)c(A).

Chai's additivity conjecture. Assume that TT is a torus and that AA is an Abelian variety. Then

c(B)=c(T)+c(A).c(B)=c(T)+c(A).

This is the additivity property for base change conductors in an exact sequence of semiabelian varieties. The supplied text attributes the statement to Chai but gives no information about whether it has been proved or remains open.

References

Primary source

Otto Overkamp and Takashi Suzuki, “Chai's conjectures on base change conductors”, arXiv:2310.01289 (2025).

Additional references

2 papers in this index state this conjecture (2011–2023). The statement above is taken from the most recent of them; the others are arXiv:1102.5653.

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