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

TL;DR

This paper constructs a $p$-adic analogue of Saito's higher residue pairing using crystalline cohomology and the twisted de Rham--Witt complex, connecting it with the $p$-adic Riemann--Hilbert correspondence.

Contribution

It develops a canonical $p$-adic higher residue pairing on relative crystalline cohomology, linking crystalline geometry with residue theory and extending primitive forms to the $p$-adic setting.

Findings

Defines a filtered $F$-isocrystal with a natural bilinear form.

Recovers classical Grothendieck residue at the special fiber.

Identifies the pairing with the flat extension of the residue form via $p$-adic Riemann--Hilbert.

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.