Rubin–Stark conjecture for function fields

Let Gamma\mathrm{Gamma} be the Galois group in the setup, let UU be the relevant group of SS-units, and let Rη\mathcal{R}_{\eta} be the regulator map associated with η=w1wn\eta=w_1^*\wedge\dots\wedge w_n^*. Write ΛS,Tn\Lambda_{S,T}^n for the subgroup of Λ0nU\Lambda_0^n U on which eχe_\chi vanishes for every character χ\chi with rχ>nr_\chi>n. Rubin–Stark conjecture. There exists an ϵ\inLambdaS,Tn\epsilon\inLambda_{S,T}^n such that

Rη(ϵ)=ΘΓ,S,T(n)(0).\mathcal{R}_{\eta}(\epsilon)=\Theta_{\Gamma,S,T}^{(n)}(0).

This is Rubin's proposed integral refinement of Stark's conjecture, relating the leading term of an equivariant LL-function to an exterior product of units. The source does not state whether the conjecture is resolved.

Sources & referencesView supporting material

Primary source

Ki-Seng Tan, “Generalized Stark formulae over function fields”, arXiv:math/0701061 (2007).

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