Motivic p-adic periods of 1-motives

TL;DR

The article delivers an explicit, functorial $p$-adic de Rham comparison isomorphism for $1$-motives, extending Beilinson’s framework to rigid-analytic settings and enabling concrete period computations for semistable curves. It develops a differential-geometric construction using rigid universal extensions to represent de Rham classes as integral forms, then proves compatibility with Beilinson and Beilinson’s Poincaré lemma in the $1$-motive setting. A new ring of motivic $p$-adic periods is introduced through rigid analytic motives, linking motivic realizations to Fontaine’s period rings and enabling the computation of periods for Kummer motives and stable hyperelliptic curves. These developments advance $p$-adic Hodge theory for $1$-motives and provide explicit motivic period data for key families of curves, with potential impact on arithmetic geometry and $p$-adic period computations.

Abstract

We give an explicit construction of the p-adic de Rham comparison isomorphism for 1-motives. In particular, we prove that our construction recovers the classical de Rham comparison isomorphism and is functorial with respect to morphisms of rigid 1-motives. In the second part, we construct a new ring of motivic p-adic periods using the formalism of rigid analytic motives. Furthermore, we explain the relation between these motivic periods and the periods of the classical p-adic de Rham comparison isomorphism. Finally, we apply our construction to explicitly compute the period pairing for several examples of 1-motives and curves.

Theorems & Definitions (123)

• Theorem : Theorem \ref{['thm_1_mot_intro']}
• Theorem : Tate, Raynaud
• Theorem
• Theorem
• Definition
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4: Beilinson's Poincaré Lemma, Beil_de_Rham
• proof