Strong Birch–Swinnerton-Dyer formula in the everywhere-good-reduction case

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Let AA be an abelian variety of dimension gg over a number field FF, assume that AA has good reduction everywhere, and let PA,∞(η)P_{A,\infty}(\eta) and aη\mathfrak a_\eta be the archimedean period and associated fractional ideal used to define the global volume. Let AtA^t be the dual abelian variety, let ΘNT(A)\Theta_{NT}(A) be the Néron–Tate regulator, and let \Sha(A/F)\Sha(A/F) be the Tate–Shafarevich group. Everywhere-good-reduction strong BSD formula. One has

L∗(A,1)=PA,∞(η) NF/Q(aη) ΘNT(A) [\Sha(A/F)]∣dF∣g/2[A(F)tor][At(F)tor].L^*(A,1)=\frac{P_{A,\infty}(\eta)\,\mathbb N_{F/\mathbb Q}(\mathfrak a_\eta)\,\Theta_{NT}(A)\,[\Sha(A/F)]}{|d_F|^{g/2}[A(F)_{\mathrm{tor}}][A^t(F)_{\mathrm{tor}}]}.

This is the strong BSD formula after the finite period factor becomes 11 under everywhere good reduction. It is a specialization of the preceding conjecture, and the source gives no resolution beyond that of the general BSD conjecture.

References

Primary source

S. Lichtenbaum and N. Ramachandran, “Values of zeta functions of arithmetic surfaces at s=1”, arXiv:2101.06530 (2022).

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