Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Siegel modular forms arising from higher Chow cycles

Shouhei Ma

TL;DR

The paper builds a bridge between higher Chow cycles on low-dimensional abelian varieties and Siegel modular forms by interpreting the primitive infinitesimal invariant as a holomorphic section of a Koszul-constructed automorphic bundle, yielding meromorphic Siegel modular forms of weight $\mathrm{Sym}^4\det^{-1}$. It develops an automorphic Koszul complex framework, proves functoriality under rank-1 degeneration via a K-theory elevator that corresponds to the Siegel operator, and establishes rigidity which implies countability of obtainable modular forms. The approach simultaneously handles the boundary behavior via partial toroidal compactifications and extends the Siegel operator to meromorphic forms, culminating in a limit-formula analysis that verifies the elevator–Siegel-operator correspondence in genus $g=2,3$. The results illuminate a precise, low-dimensional link between cycles in algebraic geometry and vector-valued Siegel modular forms, with potential arithmetic applications and questions about explicit expressions and regularity at cusps.

Abstract

We prove that the infinitesimal invariant of a higher Chow cycle of type (2,3-g) on a generic abelian variety of dimension g<4 gives rise to a meromorphic Siegel modular form of (virtual) weight Sym^{4}det^{-1} with bounded singularity, and that this construction is functorial with respect to rank 1 degeneration, namely the K-theory elevator for the cycle corresponds to the Siegel operator for the modular form.

Siegel modular forms arising from higher Chow cycles

TL;DR

The paper builds a bridge between higher Chow cycles on low-dimensional abelian varieties and Siegel modular forms by interpreting the primitive infinitesimal invariant as a holomorphic section of a Koszul-constructed automorphic bundle, yielding meromorphic Siegel modular forms of weight . It develops an automorphic Koszul complex framework, proves functoriality under rank-1 degeneration via a K-theory elevator that corresponds to the Siegel operator, and establishes rigidity which implies countability of obtainable modular forms. The approach simultaneously handles the boundary behavior via partial toroidal compactifications and extends the Siegel operator to meromorphic forms, culminating in a limit-formula analysis that verifies the elevator–Siegel-operator correspondence in genus . The results illuminate a precise, low-dimensional link between cycles in algebraic geometry and vector-valued Siegel modular forms, with potential arithmetic applications and questions about explicit expressions and regularity at cusps.

Abstract

We prove that the infinitesimal invariant of a higher Chow cycle of type (2,3-g) on a generic abelian variety of dimension g<4 gives rise to a meromorphic Siegel modular form of (virtual) weight Sym^{4}det^{-1} with bounded singularity, and that this construction is functorial with respect to rank 1 degeneration, namely the K-theory elevator for the cycle corresponds to the Siegel operator for the modular form.
Paper Structure (30 sections, 21 theorems, 74 equations)

This paper contains 30 sections, 21 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Siegel modular forms
  3. Siegel modular varieties
  4. Automorphic vector bundles
  5. Siegel modular forms
  6. Higher Chow cycles
  7. Higher Chow cycles
  8. Normal functions
  9. The main construction
  10. Automorphic Koszul complexes
  11. The main construction
  12. On the primitive part
  13. Rigidity
  14. Meromorphic Siegel operator
  15. Partial toroidal compactification
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $U$ be a Zariski open set of ${\mathcal{A}_{\Gamma}}$ and $Z$ be a family of higher Chow cycles of type $(2, 3-g)$ on $\pi^{-1}(U)\to U$, nullhomologous when $g=3$. Then the primitive infinitesimal invariant of $Z$ gives a meromorphic Siegel modular form $f_{Z}$ of weight ${{\rm Sym}}^4\otimes \

Theorems & Definitions (51)

  • Theorem 1.1: Theorem \ref{['thm: main construction']}
  • Theorem 1.2: Theorem \ref{['thm: elevator Siegel']}
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Proposition 4.3
  • ...and 41 more