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Introduction to Elliptic Quasi-Modular Forms via Moduli Spaces

Walter Andrés Páez Gaviria

TL;DR

The paper develops a framework for elliptic quasi-modular forms through moduli spaces and the Gauss-Manin connection, embedding Ramanujan-type differential relations into a geometric setting via enhanced elliptic curves. It defines the AMSY Gauss-Manin Lie algebra as an $\mathfrak{sl}_2(\mathbb{C})$-action on a moduli space of enhanced elliptic curves and produces explicit Ramanujan vector fields whose pullback recovers $E_2,E_4,E_6$. In the elliptic mirror-symmetry context, genus-$g$ Hurwitz-type counts are organized into partition functions, which reduce to traces of monodromy operators and are amenable to diagonalization by representation-theoretic methods; combining Frobenius theory with a half-integer product expansion yields that the resulting generating functions are quasimodular, with the $m=2$ case explicitly verified. Overall, the work unifies automorphic and topological-string-type objects for elliptic curves, linking Hodge-theoretic variation, modular forms, and Hurwitz-type counts in a coherent GMCD framework.

Abstract

In this paper we present rigorously and as succintly as possible the theory of elliptic quasi-modular forms by means of moduli spaces and the Gauss-Manin connection, and deal with one of the main historical appearances of quasi-modular forms, which was the seminal case of mirror symmetry treated by Dijkgraaf.

Introduction to Elliptic Quasi-Modular Forms via Moduli Spaces

TL;DR

The paper develops a framework for elliptic quasi-modular forms through moduli spaces and the Gauss-Manin connection, embedding Ramanujan-type differential relations into a geometric setting via enhanced elliptic curves. It defines the AMSY Gauss-Manin Lie algebra as an -action on a moduli space of enhanced elliptic curves and produces explicit Ramanujan vector fields whose pullback recovers . In the elliptic mirror-symmetry context, genus- Hurwitz-type counts are organized into partition functions, which reduce to traces of monodromy operators and are amenable to diagonalization by representation-theoretic methods; combining Frobenius theory with a half-integer product expansion yields that the resulting generating functions are quasimodular, with the case explicitly verified. Overall, the work unifies automorphic and topological-string-type objects for elliptic curves, linking Hodge-theoretic variation, modular forms, and Hurwitz-type counts in a coherent GMCD framework.

Abstract

In this paper we present rigorously and as succintly as possible the theory of elliptic quasi-modular forms by means of moduli spaces and the Gauss-Manin connection, and deal with one of the main historical appearances of quasi-modular forms, which was the seminal case of mirror symmetry treated by Dijkgraaf.
Paper Structure (8 sections, 22 theorems, 62 equations)

This paper contains 8 sections, 22 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Quasi-modular forms and Ramanujan's identities
  3. Enhanced elliptic curves and the Ramanujan vector field
  4. Mirror symmetry for elliptic curves
  5. Partition functions
  6. Reducing $\hat{N}_{g,d}$ to the trace on an operator
  7. Diagonalizing $M_d$
  8. Frobenius formula and Haneko-Zagier result

Key Result

Theorem 2.2

$\mathfrak{M}$ is equal to $\mathbb{C}[E_4,E_6]$. In other words, every modular form can be written as a polynomial in the Eisenstein series $E_4$ and $E_6$. Furthermore, $E_4$ and $E_6$ are algebraically independent.

Theorems & Definitions (51)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5
  • Example 2.6
  • Theorem 2.7
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 41 more