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Winding quotients for virtual period maps of rank 1

Kyoji Saito

TL;DR

The paper develops a rank-1 model of virtual period maps that exhibits a winding phenomenon, producing a winding quotient $E_{\tau}$ that reframes the inversion problem for period maps. It introduces $q$-multiplicatively periodic objects $\mathcal{U},\mathcal{V},\mathcal{W}$ on the winding quotient, whose pull-backs recover the Weierstrass data up to Eisenstein corrections, linking to Borcherds' expansions. The main result provides an explicit morphism from the winding quotient $\mathbb{C}_w^\times$ to the elliptic curve $E_{\tau}$ in terms of $\mathcal{W}$ and its derivative, thereby solving the new inversion problem in this non-geometric setting. The work connects classical elliptic function theory with $q$-shifted factorial frameworks and suggests intriguing links to mathematical physics via propagator-type constructions on elliptic curves.

Abstract

We illustrate a rank 1 model of virtual period maps and their associated winding quotient, where the winding quotient is a new phenomenon appeared in a recent study of virtual period maps and it requires a reformulation of the classical Jacobi inversion problem for the period maps. We answer to the new inversion problem by introducing the q-multiplicatively periodic function, whose pull-back to the winding covering space is the Weierstrass p-function up to a correction by Eisenstein series E2. The function appears also in the study of mathematical physics as the propagator on elliptic curves.

Winding quotients for virtual period maps of rank 1

TL;DR

The paper develops a rank-1 model of virtual period maps that exhibits a winding phenomenon, producing a winding quotient that reframes the inversion problem for period maps. It introduces -multiplicatively periodic objects on the winding quotient, whose pull-backs recover the Weierstrass data up to Eisenstein corrections, linking to Borcherds' expansions. The main result provides an explicit morphism from the winding quotient to the elliptic curve in terms of and its derivative, thereby solving the new inversion problem in this non-geometric setting. The work connects classical elliptic function theory with -shifted factorial frameworks and suggests intriguing links to mathematical physics via propagator-type constructions on elliptic curves.

Abstract

We illustrate a rank 1 model of virtual period maps and their associated winding quotient, where the winding quotient is a new phenomenon appeared in a recent study of virtual period maps and it requires a reformulation of the classical Jacobi inversion problem for the period maps. We answer to the new inversion problem by introducing the q-multiplicatively periodic function, whose pull-back to the winding covering space is the Weierstrass p-function up to a correction by Eisenstein series E2. The function appears also in the study of mathematical physics as the propagator on elliptic curves.

Paper Structure

This paper contains 8 sections, 1 theorem, 37 equations.

Table of Contents

  1. Introduction
  2. The virtual period map $z^{\tau}$ on $\mathbb{C}_z^\times$
  3. Lifting of the map $z^{\tau}$ to the monodromy covering space $\mathbb{C}_{\mathrm{Log}z}$
  4. Winding of the monodromy covering space $\mathbb{C}_{\mathrm{Log}z}$ to the range $\mathbb{C}_w^\times$
  5. Winding of the domain $\mathbb{C}_z^\times$ of $z^{\tau}$ to the elliptic curve $E_\tau$
  6. $q$-multiplicatively quasi-periodic function $\mathcal{V}(w,q)$ and periodic function $\mathcal{W}(w,q)$
  7. Solution to the new inversion problem
  8. Appendix

Key Result

Theorem 7.1

For each fixed $q=\exp(2\pi\sqrt{-1}\tau)$, the pair of functions: gives a morphism: $\mathbb{C}_w^\times \to E_\tau \subset \mathbb{P}^2$ making the diagram $(*)$ in section 5 commutative.

Theorems & Definitions (10)

  • Remark 1.1
  • proof
  • proof
  • Remark 6.1
  • proof
  • Remark 6.2
  • Remark 6.3
  • Theorem 7.1
  • proof
  • proof