Higher Residue Pairing for $p$-adic Isocrystals and the $p$-adic Riemann--Hilbert Correspondence

Abstract

We construct a canonical sesquilinear pairing on the relative crystalline cohomology of a smooth proper family of varieties over a complete discretely valued $p$-adic field. Motivated by the role of Saito's higher residue pairing in the theory of primitive forms and complex variations of Hodge structure, we develop a $p$-adic analogue based on the twisted relative de~Rham--Witt complex. We show that this twisted complex defines a filtered $F$-isocrystal whose cohomology carries a natural flat, Frobenius-compatible, and non-degenerate bilinear form. Its specialization at the uniformizer recovers the classical Grothendieck residue on the special fiber, providing a direct bridge between crystalline geometry and residue theory. Using the $p$-adic Riemann--Hilbert correspondence of Faltings and Liu--Zhu, we further identify the resulting pairing with the unique flat extension of this residue form to the corresponding $p$-adic local system. The construction is functorial in the family and compatible with base change and $p$-adic comparison isomorphisms. This yields a genuine $p$-adic analogue of Saito's higher residue pairing and supplies foundational ingredients for a prospective theory of $p$-adic primitive forms, $p$-adic TERP structures, and $p$-adic Frobenius manifolds.