Kato explicit reciprocity law for Siegel modular forms of weight $(3, 3)$

TL;DR

This work extends Kato's explicit reciprocity framework from modular curves to a product of modular curves realized inside the Siegel threefold, yielding an explicit reciprocity law for the unique critical twist of the $p$-adic Galois representation attached to cuspidal Siegel modular forms of weight $(3,3)$, denoted $V_\pi(2)$. The authors develop a local–global toolkit for big two-dimensional local fields, including a detailed analysis of Kähler differentials and their Galois cohomology, to connect $K_2$-regulators to de Rham period data via the Bloch–Kato dual exponential, mediated by a commutative diagram up to torsion. Core techniques combine Chern regulators from $K$-theory, the Hochschild–Serre spectral sequence, Kummer theory, and $p$-adic Hodge-theoretic filtrations, all structured to extend Scholl's modular-curve reciprocity to the Siegel setting. The resulting explicit reciprocity law strengthens the link between Euler systems and special $L$-value data for Siegel modular forms, broadening the scope of explicit reciprocity beyond rank-one modular curves and into higher-dimensional Shimura varieties.

Abstract

We extend Kato explicit reciprocity law, in the version written by Scholl, for a modular curve to a product of two modular curves. By embedding the product of two modular curves in the Siegel threefold, we deduce an explicit reciprocity law for the unique critical twist of the $p$-adic Galois representation attached to cuspidal Siegel modular forms of weight $(3,3)$.

Theorems & Definitions (43)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Theorem 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• Corollary 2.4
• Remark 2.5
• proof : Proof of Corollary \ref{["main'12"]} assuming Corollary \ref{['corollary11']}