Regulators and derivatives of Vologodsky functions with respect to log(p)

TL;DR

The paper investigates how p-adic regulators in bad reduction split into a continuous component computed by $p$-adic Vologodsky integration and a discrete component captured by differentiation with respect to $\\log(p)$. It develops a general framework showing that, for syntomic regulators and related non-abelian constructions, the discrete part equals the derivative of the continuous part with respect to $\\log(p)$, and provides explicit derivative formulas on curves via monodromy and dual graphs. The results connect $p$-adic heights, the unipotent Albanese map, and toric regulators, illustrating how derivatives of Vologodsky functions encode discrete arithmetic data such as local heights and monodromy, with concrete instances for zero-cycles, curves, and totally degenerate reductions. These connections offer tools to extract discrete invariants from continuous $p$-adic data and relate several regulator theories in semi-stable/bad-reduction contexts, with potential applications to Quadratic Chabauty and explicit height computations.

Abstract

We describe several instances of the following phenomenon: In bad reduction situations the \( p \)-adic regulator has a continuous and a discrete component. The continuous component is computed using Vologodsky integrals. These depend on a choice of the branch of the \( p \)-adic logarithm, determined by \( \log (p) \). They can be differentiated with respect to this parameter and the result is related to the discrete component.

Figures (2)

• Figure 1: semi-stable curve and its reduction graph.
• Figure 2: The loop determining $A(e)$.

Theorems & Definitions (63)

• Theorem 1.1
• Theorem 1.2: Bes18
• Proposition 2.1
• Corollary 2.2: BMS21[Proposition 9.14]
• Remark 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Remark 3.4