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Frobenius lift for the cyclotomic map on spectral symplectic cohomology

Ascertain whether, for a Weinstein domain M in the setting of Theorem 3 (the cyclotomic structure on SH^•(M; S)), the composed map SH^•(M; S) → (SH^•(M; S))^{Φ C_p} → (SH^•(M; S))^{t C_p} admits a Frobenius lift to the homotopy fixed points (SH^•(M; S)_{^p})^{h C_p}. Determine the same question after p-completing SH^•(M; S).

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Background

In the paper the author constructs a cyclotomic structure on the spectral version of symplectic cohomology SH•(M; S) for suitable symplectic manifolds, yielding compatible genuine C_{pk}-actions and geometric fixed-point equivalences along iterations. This opens a pathway toward arithmetic structures analogous to those in p-adic Hodge theory.

A central step in such arithmetic frameworks is the existence of a Frobenius lift from geometric fixed points to Tate fixed points via homotopy fixed points, mirroring cyclotomic Frobenius constructions in topological cyclic homology. Establishing the existence of this lift in the present Floer-theoretic setting would concretely connect the constructed cyclotomic structure to standard cyclotomic frameworks and enable further arithmetic applications.

References

We will end the introductory discussion with a precise foundational conjecture, which we do not know how to resolve, and has a significant impact on any future theory of ``arithmetic aspects of symplectic topology'':

Question: Let M be a Weinstein domain and suppose that we are in the setting of Theorem \ref{thm:sh-is-cyclotomic}. Does the composed map \begin{equation} SH\bullet (M, S) \to (SH\bullet(M, S)){\Phi C_p} \to (SH\bullet(M, S)){tC_p}, \end{equation} where the second map in the composition is the canonical map from geometric fixed points to tate fixed points admit a Frobenius lift, i.e. a lift to the homotopy fixed points (SH\bullet(M, S)_{\hat{p}){hC_p}$? The same question may be asked for the corresponding map obtained by $p$-completing $SH\bullet(M, S)$.

— Cyclotomic Structures in Symplectic Topology (2405.18370 - Rezchikov, 28 May 2024) in Introduction