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arXiv · 1908.06411

The rational cuspidal divisor class group of $X_0(N)$

Hwajong Yoo
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Abstract

For any positive integer $N$, we completely determine the structure of the rational cuspidal divisor class group of $X_0(N)$, which is conjecturally equal to the rational torsion subgroup of $J_0(N)$. More specifically, for a given prime $\ell$, we construct a rational cuspidal divisor $Z_\ell(d)$ for any non-trivial divisor $d$ of $N$. Also, we compute the order of the linear equivalence class of the divisor $Z_\ell(d)$ and show that the $\ell$-primary subgroup of the rational cuspidal divisor class group of $X_0(N)$ is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of the divisors $Z_\ell(d)$.

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Hwajong Yoo. 2019-08-18. The rational cuspidal divisor class group of $X_0(N)$. https://arxiv.org/abs/1908.06411

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