Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the p-adic integration over Igusa towers of Siegel modular varieties

Marco Adamo Seveso

TL;DR

This work develops a concrete p-adic integration framework for Igusa towers on Siegel modular varieties, extending genus 1 methods to higher genus via integral truncated dual BGG complexes and unit-root splittings. It builds a robust representation-theoretic and geometric foundation (via automorphic sheaves, q-expansions, and de Rham data) to define and analyze p-depleted de Rham complexes. The central result is the acyclicity of these p-depleted complexes in degrees 1 through d_g (under P(0)=0), enabling explicit primitive constructions that enable explicit reciprocity laws in higher genus. The approach blends deep representation theory with p-adic and Igusa-tower geometry to produce tools of broad arithmetic significance.

Abstract

We develop an explicit $p$-adic integration theory for Igusa towers of modular Siegel manifolds, which finds applications to explicit reciprocity laws.

On the p-adic integration over Igusa towers of Siegel modular varieties

TL;DR

This work develops a concrete p-adic integration framework for Igusa towers on Siegel modular varieties, extending genus 1 methods to higher genus via integral truncated dual BGG complexes and unit-root splittings. It builds a robust representation-theoretic and geometric foundation (via automorphic sheaves, q-expansions, and de Rham data) to define and analyze p-depleted de Rham complexes. The central result is the acyclicity of these p-depleted complexes in degrees 1 through d_g (under P(0)=0), enabling explicit primitive constructions that enable explicit reciprocity laws in higher genus. The approach blends deep representation theory with p-adic and Igusa-tower geometry to produce tools of broad arithmetic significance.

Abstract

We develop an explicit -adic integration theory for Igusa towers of modular Siegel manifolds, which finds applications to explicit reciprocity laws.

Paper Structure

This paper contains 13 sections, 34 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Integral truncated dual BGG complexes and representations of symplectic groups
  3. Preliminary results and notations
  4. Induced modules and the dual BGG complex
  5. The degree filtration on the induced modules in the symplectic case
  6. The isomorphism $\mathrm{Sym}_{W}^{2,\vee }\left( 1\right) \overset{\sim }{\longrightarrow }\mathfrak{u}^{-}$
  7. The isomorphism $\mathrm{J}^{\prime }\overset{\sim }{\longrightarrow }\mathrm{Ind}_{\mathbf{Q}^{-}}^{\mathbf{G}}\left( \mathrm{1}\right) \left[ \mathbf{Y}\right] _{\leq 1}^{\vee }$
  8. Automorphic sheaves attached to representations of symplectic groups
  9. $q$-expansion and degeneration at the $0$-dimensional cusps
  10. $p$-depletions and $p$-adic integration over the Igusa tower
  11. Description of the de Rham complex
  12. Acyclicity of $p$-depleted de Rham complexes
  13. Proof of Proposition \ref{['Primitives PKey']}

Key Result

Theorem 1.1

For every $P$ such that $P\left( 0\right) =0$, the complex $H^{0}\left( X_{0},\mathcal{L}_{{ \if@compatibility \mathchar"0115 {} \mathchar"0115 } }\otimes _{\mathcal{O}_{X_{0}}}\Omega _{X_{0}/\Bbbk }^{\cdot }\right) ^{\left[ P\right] }$ is acyclic in degree $p=1,...,d_{g}$.

Theorems & Definitions (87)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 77 more