arXiv subjects

Sheng-Yang Kevin Ho

Publications and source records attributed to Sheng-Yang Kevin Ho.

5 recordsLinked to original sources

arXiv2026

A function field analogue of Ligozat's theorem for Drinfeld modular units

Fix a nonzero level $\mathfrak{n} \in \mathbb{F}_q[T]$. In this paper, we first establish a function field analogue of Ligozat's theorem, which serves as our main result and provides a criterion for Drinfeld modular units on the Drinfeld modular curve $X_0(\mathfrak{n})$. We further conjecture that this criterion characterizes all Drinfeld modular units; we verify the conjecture in the cases of prime power level and of level equal to the product of two primes. Second, as an application of Drinfeld modular units, we investigate the rational cuspidal divisor class group $\mathcal{C}(\mathfrak{n})$ of $X_0(\mathfrak{n})$. We construct an injective map $g$ from the group of degree $0$ rational cuspidal divisors on $X_0(\mathfrak{n})$ to the group of Drinfeld modular units on $X_0(\mathfrak{n})$ tensored with $\mathbb{Q}$ over $\mathbb{Z}$. As a result, we establish an explicit upper bound for the exponent of $\mathcal{C}(\mathfrak{n})$ for general level $\mathfrak{n}$.

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arXiv2024

Ogg's conjectures over function fields

In the early 1970s, Andrew Ogg made several conjectures about the rational torsion points of elliptic curves over $\mathbb{Q}$ and the Jacobians of modular curves. These conjectures were proved shortly after by Barry Mazur as a consequence of his fundamental study of the arithmetic properties of modular curves and Hecke algebras. In this paper, we review the function field analogues of Ogg's conjectures, their current status, and the methods that have been applied to prove some of these conjectures. The methods are based on the ideas of Mazur and Ogg, but there are interesting differences and technical complications that arise in the function field setting, as well as intriguing possible new directions for generalizations.

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arXiv2024

The Rational Torsion Subgroup of $J_0(\mathfrak{p}^r)$

Let $\mathfrak{n} = \mathfrak{p}^r$ be a prime power ideal of $\mathbb{F}_q[T]$ with $r \geq 2$. We study the rational torsion subgroup $\mathcal{T}(\mathfrak{p}^r)$ of the Drinfeld modular Jacobian $J_0(\mathfrak{p}^r)$. We prove that the prime-to-$q(q-1)$ part of $\mathcal{T}(\mathfrak{p}^r)$ is equal to that of the rational cuspidal divisor class group $\mathcal{C}(\mathfrak{p}^r)$ of the Drinfeld modular curve $X_0(\mathfrak{p}^r)$. As we completely computed the structure of $\mathcal{C}(\mathfrak{p}^r)$, it also determines the structure of the prime-to-$q(q-1)$ part of $\mathcal{T}(\mathfrak{p}^r)$.

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arXiv2024

On Drinfeld modular curves for SL(2)

We study the Drinfeld modular curves arising from the Hecke congruence subgroups of $\mathrm{SL}_2(\mathbb{F}_q[T])$. Using a combinatorial method of Gekeler and Nonnengardt, we obtain a genus formula for these curves. In cases when the genus is one, we compute the Weierstrass equation of the corresponding curve.

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arXiv2024

On the Rational Cuspidal Divisor Class Groups of Drinfeld Modular Curves $X_0(\mathfrak{p}^r)$

Let $\mathcal{C}(\mathfrak{p}^r)$ be the rational cuspidal divisor class group of the Drinfeld modular curve $X_0(\mathfrak{p}^r)$ for a prime power level $\mathfrak{p}^r\in \mathbb{F}_q[T]$. We relate the rational cuspidal divisors of degree $0$ on $X_0(\mathfrak{p}^r)$ with $Δ$-quotients, where $Δ$ is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of $\mathcal{C}(\mathfrak{p}^r)$ for arbitrary prime $\mathfrak{p}\in \mathbb{F}_q[T]$ and $r\geq 2$.

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