In our previous work (arXiv:1602.06282), we established functors from the category of p-adic etale local systems on a smooth rigid analytic variety X over a finite extension of Q_p to the categories of Higgs bundles on its base change to the completed algebraic closure of Q_p, and to the categories of vector bundles with an integrable connection on its base change to B_dR respectively. These two related functors may be regarded as p-adic Simpson correspondences and p-adic Riemann-Hilbert correspondence for p-adic local systems.

Inspired by Deligne’s construction of \lambda-connections, we propose a p-adic analogue of it. Namely, we want to construct a functor from p-adic etale local systems on X to certain vector bundles with an integrable connection on the product of X with the so called Fargues-Fontaine curve. Moreover, the restrictions of the functor on the infinity point of the FF-curve and its formal neighborhood should recover p-adic Simpson correspondences and p-adic Riemann-Hilbert correspondence respectively.