arXiv subjects

Kenji Terao

Publications and source records attributed to Kenji Terao.

3 recordsLinked to original sources

arXiv2026

Computing class groups and gonalities of algebraic curves over finite fields

We give practical algorithms for computing the divisor class group and the gonality of a curve over a finite field, achieving several orders of magnitude speedup over existing methods for sufficiently large genus or residue field. The approach relies on introducing a precomputation step involving power series-expansions, which allows for an efficient amortized computation of large numbers of Riemann-Roch spaces.

math.NT↗

arXiv2025

Degrees of points with rational $j$-invariant on $X_{0}(n)$ and $X_{1}(n)$

We give a classification of the degrees of the points with rational $j$-invariant on the modular curves $X_{0}(n)$ and $X_{1}(n)$. The degrees which occur infinitely often are computed unconditionally, while those which occur finitely often are determined assuming a conjecture of Zywina. To achieve this, we define the notion of $\mathcal{H}$-closures of subgroups of $\operatorname{GL}_{2}(\widehat{\mathbb{Z}})$, and compute the $\mathcal{B}_{0}(n)$- and $\mathcal{B}_{1}(n)$-closures of images of Galois representations of elliptic curves defined over $\mathbb{Q}$. An application to computing the set of isolated $j$-invariants in $\mathbb{Q}$ is also given.

math.NT↗

arXiv2024

Isolated points on modular curves

We study isolated points on the modular curves $X_{H}$, for $H$ a subgroup of $\operatorname{GL}_{2}(\mathbb{Z}/n \mathbb{Z})$ for some $n \geq 1$. In particular, we prove a single-sink theorem for such isolated points, which traces the existence of all such isolated points with the same $j$-invariant back to an isolated point on a single curve. Building on this result, we also present a uniform strategy for determining the isolated points on any family of modular curves. As an example, we use this strategy to classify the isolated points with rational $j$-invariant on all modular curves of level 7, as well as the modular curves $X_{0}(n)$, the latter assuming a conjecture on images of Galois representations of elliptic curves over $\mathbb{Q}$. Underpinning all of this, we develop a theory of isolated divisors on geometrically disconnected varieties, which may be of independent interest.

math.NT↗