Genus formulas for dormant modular curves and asymptotic behavior of their function fields

TL;DR

This paper constructs and analyzes towers of dormant modular curves derived from moduli spaces of higher-level dormant PGL2-opers, providing explicit genus formulas and studying their asymptotic properties in relation to classical modular curves.

Contribution

It introduces a new class of dormant modular curves from higher-level PGL2-opers and derives explicit genus formulas to analyze their asymptotic behavior.

Findings

Established explicit genus formulas for dormant modular curves.

Compared asymptotic properties of these towers with classical modular and Drinfeld cases.

Demonstrated asymptotically good behavior of the constructed towers.

Abstract

Towers of algebraic function fields over finite fields play a fundamental role in arithmetic geometry and coding theory. Classical examples arising from modular and Drinfeld modular curves exhibit asymptotically good behavior. In this paper, we introduce an analogous construction derived from the moduli spaces of higher-level dormant $\mathrm{PGL}_2$-opers of prescribed radii on $4$-pointed stable curves of genus $0$. These spaces, which we refer to as dormant modular curves, form projective systems under level reduction. Building on previous results in the moduli theory of dormant opers, we establish an explicit formula for computing the genera of these curves. This formula allows us to study the asymptotic behavior of the corresponding towers of function fields and to compare them with the classical modular and Drinfeld modular cases.