Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modular Arrangements

A. Levin, N. Sakharova

TL;DR

This work develops a framework to study periods of mixed Hodge structures on the square of modular curves by exploiting Hecke arrangements and a Modular Cauchy kernel expressed via Zagier series. By constructing bimodular differential forms with prescribed residues along Hecke divisors and using level maps and the Fricke involution, it derives regulator formulas that connect geometric data to Bloch–Wigner dilogarithm values, culminating in a Hauptmodul-based Aomoto dilogarithm on genus-zero arrangements. The approach integrates CM-point intersection theory, spectral-filtered cohomology of arrangement complements, and Rudenko’s reciprocity to reduce complex periods to algebraic dilogarithm values. The results offer explicit descriptions of residues and poles for differential forms on $X^2$, and provide a pathway to compute dilogarithmic regulators on modular surfaces with arithmetic significance. This framework advances the arithmetic-geometric understanding of modular curves, their square, and associated period integrals with potential implications for Néron–Severi theory and split abelian surface moduli. $X^2$ serves as a natural laboratory where modular forms, Hecke correspondences, and Aomoto-like dilogarithms intertwine in a concrete, computable setting.

Abstract

The modular curves serve as excellent objects for testing conjectures in arithmetic geometry. They possess a natural geometric definition in contrast with rather nontrivial structure. On the other hand, they are well-studied from the perspective of number theory. Furthermore, there is a well-developed and powerful analytic technique available. We will use the square of the modular curve as the experimental object to investigate the arithmetic properties of the periods of mixed Hodge structures. There is an additional reason for this study: this square is naturally associated with a family of (Hecke) curves. These curves form components of the Neron-Severi locus, allowing for the interpretation of the square of the moduli curve as the moduli space of split (i.e., the product of two elliptic curves) abelian surfaces.

Modular Arrangements

TL;DR

This work develops a framework to study periods of mixed Hodge structures on the square of modular curves by exploiting Hecke arrangements and a Modular Cauchy kernel expressed via Zagier series. By constructing bimodular differential forms with prescribed residues along Hecke divisors and using level maps and the Fricke involution, it derives regulator formulas that connect geometric data to Bloch–Wigner dilogarithm values, culminating in a Hauptmodul-based Aomoto dilogarithm on genus-zero arrangements. The approach integrates CM-point intersection theory, spectral-filtered cohomology of arrangement complements, and Rudenko’s reciprocity to reduce complex periods to algebraic dilogarithm values. The results offer explicit descriptions of residues and poles for differential forms on , and provide a pathway to compute dilogarithmic regulators on modular surfaces with arithmetic significance. This framework advances the arithmetic-geometric understanding of modular curves, their square, and associated period integrals with potential implications for Néron–Severi theory and split abelian surface moduli. serves as a natural laboratory where modular forms, Hecke correspondences, and Aomoto-like dilogarithms intertwine in a concrete, computable setting.

Abstract

The modular curves serve as excellent objects for testing conjectures in arithmetic geometry. They possess a natural geometric definition in contrast with rather nontrivial structure. On the other hand, they are well-studied from the perspective of number theory. Furthermore, there is a well-developed and powerful analytic technique available. We will use the square of the modular curve as the experimental object to investigate the arithmetic properties of the periods of mixed Hodge structures. There is an additional reason for this study: this square is naturally associated with a family of (Hecke) curves. These curves form components of the Neron-Severi locus, allowing for the interpretation of the square of the moduli curve as the moduli space of split (i.e., the product of two elliptic curves) abelian surfaces.

Paper Structure

This paper contains 16 sections, 3 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. 1-2-0 of the Elliptic Modularity
  3. The Generalities of the Modular Curves.
  4. The Level Structure and the Modular Correspondences.
  5. The CM-Points.
  6. The Arrangements
  7. General Setup.
  8. Regular Functions and Differential Forms.
  9. The Modular Cauchy kernel.
  10. Bimodular differential forms whose residues are the cusps forms of weight 2.
  11. Differential 2-forms on the complement of the "triangle" on the $X\times X$.
  12. The Hauptmodule Modular Aomoto Dilogarithm
  13. The Regulator Formulas.
  14. Compact Curves.
  15. The Square of the Modular Curve. Modular Aomoto Dilogarithm.
  16. ...and 1 more sections

Key Result

Lemma 1

The form $\mathrm{CoRes}^2(n, a, b)$ has the following singularities: the simple pole at $w=0$$($with the asymptotic $\left(-dq/q +O(1) dq \right) \wedge \left( \Xi^{\ast}(z, a)d z-\Xi^{\ast}(z, b)dz \right))$, the simple pole at the image of the points $z=a, b$ with the residue $\Xi^{\ast}(w, a)dw$

Figures (1)

  • Figure 1: Left: Hecke arrangement on $X \times X$. Right: $X_0(n) \times X_0(n)$. The diagonal $\Delta$ on $X_0(n) \times X_0(n)$ maps to the Hecke curve $T_n$ under the mapping $\lambda \times \rho$.

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Definition 4
  • Theorem 1
  • proof
  • Lemma 2