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arXiv · 2605.17973

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

Kohei Aoyama·Youhei Morita·Yasuhiro Wakabayashi
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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.

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Kohei Aoyama, Youhei Morita, Yasuhiro Wakabayashi. 2026-05-18. Genus formulas for dormant modular curves and asymptotic behavior of their function fields. https://arxiv.org/abs/2605.17973

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