Beilinson's conjecture for the special value of an elliptic curve over a number field

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Let A/FA/F be an elliptic curve, let ΣFR\Sigma_F^{\mathbb R} and ΣFC\Sigma_F^{\mathbb C} denote the real and complex archimedean places, and choose periods Ωσ,1,Ωσ,2\Omega_{\sigma,1},\Omega_{\sigma,2} for the complex tori Aν(C)A_\nu(\mathbb{C}). Let dFd_F be the discriminant of FF.

Beilinson's conjecture. If L(1,A)≠0L(1,A)\neq0, then

L(1,A)∈∣dF∣1/2∏σ∈ΣFRΩσ,1∏σ∈ΣFCIm⁡(Ωσ,1Ωσ,2‾)Q×.L(1,A)\in |d_F|^{1/2}\prod_{\sigma\in\Sigma_F^{\mathbb R}}\Omega_{\sigma,1}\prod_{\sigma\in\Sigma_F^{\mathbb C}}\operatorname{Im}(\Omega_{\sigma,1}\overline{\Omega_{\sigma,2}})\mathbb{Q}^\times.

This is described as an unrefined rank-zero version of the leading-term formula related to the Birch--Swinnerton-Dyer conjecture. The paper says it is deduced from Beilinson's conjectures, but the displayed formula itself is used as a conjectural input.

References

Primary source

Xavier Guitart and Santiago Molina, “Periods of modular forms and applications to the conjectures of Oda and of Prasanna-Venkatesh”, arXiv:2507.05021 (2025).

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