Abstract
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 \(s=1\) of the Hasse–Weil–Artin L-series \(L(E,\varrho _1\otimes \varrho _2,s)\) of an elliptic curve \(E/\mathbb {Q}\) twisted by the tensor product \(\varrho _1\otimes \varrho _2\) of two odd 2-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a \(2\times 2\) 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 \(\varrho _1\) and \(\varrho _2\) 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.
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Notes
Indeed, when g is cuspidal this hypothesis not only asks that \(\alpha \ne \beta \) but also that there should exist no real quadratic field F in which p splits such that \(\varrho _g \simeq {{\mathrm{Ind}}}_F^\mathbb {Q}(\xi )\) for some character \(\xi \) of F. However, in our CM setting, the existence of a character \(\xi \) of a real quadratic field F such that \(\mathrm {Ind}_\mathbb {Q}^K(\psi )\simeq \mathrm {Ind}_\mathbb {Q}^F(\xi )\) implies that \(\mathrm {Gal\,}(H/K)\simeq C_4\). Then F is the single real quadratic field contained in the quadratic extension of K cut out by \(\psi ^2\), and the condition \(\alpha \ne \beta \) implies that p cannot split in F.
Indeed, when \(\theta (\psi )\) is Eisenstein, it is shown in [10, Sect. 1] that a necessary condition for hypotheses C–C\({^\prime }\) to hold is that \(\alpha = \beta \), but this is automatically satisfied because \(\psi ^2=1\). As explained in loc. cit., this is also expected to be a sufficient condition.
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Casazza, D., Rotger, V. Stark points and the Hida–Rankin p-adic L-function. Ramanujan J 45, 451–473 (2018). https://doi.org/10.1007/s11139-016-9824-y
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DOI: https://doi.org/10.1007/s11139-016-9824-y