This article is devoted to the elliptic Stark conjecture formulated by Darmon (Forum Math Pi 3:e8, 2015), which proposes a formula for the transcendental part of a p-adic avatar of the leading term at s=1\documentclass[12pt]{minimal}
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\begin{document}$$s=1$$\end{document} of the Hasse–Weil–Artin L-series L(E,ϱ1⊗ϱ2,s)\documentclass[12pt]{minimal}
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\begin{document}$$L(E,\varrho _1\otimes \varrho _2,s)$$\end{document} of an elliptic curve E/Q\documentclass[12pt]{minimal}
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\begin{document}$$E/\mathbb {Q}$$\end{document} twisted by the tensor product ϱ1⊗ϱ2\documentclass[12pt]{minimal}
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\begin{document}$$\varrho _1\otimes \varrho _2$$\end{document} of two odd 2-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a 2×2\documentclass[12pt]{minimal}
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\begin{document}$$2\times 2$$\end{document}p-adic regulator involving the p-adic formal group logarithm of suitable Stark points on E. This conjecture was proved by Darmon (Forum Math Pi 3:e8, 2015) in the setting where ϱ1\documentclass[12pt]{minimal}
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\begin{document}$$\varrho _1$$\end{document} and ϱ2\documentclass[12pt]{minimal}
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\begin{document}$$\varrho _2$$\end{document} are induced from characters of the same imaginary quadratic field K. In this note, we prove a refinement of this result that was discovered experimentally by Darmon (Forum Math Pi 3:e8, 2015, [Remark 3.4]) in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of Darmon (Forum Math Pi 3:e8, 2015) holds in a particular setting where the Hida–Rankin p-adic L-function associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both E and K.
