Belyi Function for Wiman Sextic and Cusp Forms of a Noncongruence Subgroup

arXiv Math · · 1 min read · Natural Sciences

Read research and analysis on Belyi Function for Wiman Sextic and Cusp Forms of a Noncongruence Subgroup published by ICANEWS, a global research journal for emerging researchers.

Key Takeaways

  • Explicit determination of an algebraic Belyi function on a unique smooth projective model ($\widetilde{W}$) of the Wiman sextic curve ($W$).
  • Description of the complex uniformization of $\widetilde{W}$ using modular functions associated with a noncongruence subgroup ($\Gamma_{\widetilde{W}} \subset \rm SL_2(\mathbb{Z})$).
  • Direct proof of the unbounded denominators conjecture in weight 2 for the subgroup $\Gamma_{\widetilde{W}}$.

Why This Matters

The research contributes a distinct methodological approach to proving the unbounded denominators conjecture. This involves specific techniques like degeneration over $\mathbb{F}_{5}$ and explicit Puiseux series expansions, providing an alternative perspective to existing proofs.

Overview

This research focuses on the explicit determination of an algebraic Belyi function on $\widetilde{W}$, which is identified as a unique smooth projective model of the Wiman sextic curve $W$. The study also details the complex uniformization of this model using modular functions. These modular functions are specifically associated with a noncongruence subgroup, denoted as $\Gamma_{\widetilde{W}} \subset \rm SL_2(\mathbb{Z})$.

A secondary objective and application of this work is to present a direct proof concerning the unbounded denominators conjecture. This proof is specifically developed for weight 2 and applies to the noncongruence subgroup $\Gamma_{\widetilde{W}}$.

Research Context

The unbounded denominators conjecture is a known topic within its field. Prior progress on this conjecture has been made by various researchers, including work by Dong, Lin, and Ng. The conjecture is currently understood to be known in full generality due to the work of Calegari, Dimitrov, and Tang.

Approach

The primary approach involves explicitly determining the algebraic Belyi function for $\widetilde{W}$. Following this, the complex uniformization of $\widetilde{W}$ is described. This description is framed in terms of modular functions that are linked to the specific noncongruence subgroup $\Gamma_{\widetilde{W}}$.

For the proof of the unbounded denominators conjecture, the methodology employed differs from previous work. It utilizes a degeneration of $\widetilde{W}$ over the finite field $\mathbb{F}_{5}$. In conjunction with this degeneration, the method incorporates explicit Puiseux series expansions.

Findings

  • An algebraic Belyi function was explicitly determined for $\widetilde{W}$, the unique smooth projective model of the Wiman sextic curve $W$.
  • The complex uniformization of $\widetilde{W}$ was described through modular functions associated with the noncongruence subgroup $\Gamma_{\widetilde{W}} \subset \rm SL_2(\mathbb{Z})$.
  • A direct proof for the unbounded denominators conjecture in weight 2 was provided for the subgroup $\Gamma_{\widetilde{W}}$.

Research Information

Institution
arXiv
Original Study
View Publication
Source
arXiv Math

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