Cuspidal Q-rational Torsion Subgroups of Jacobians of Modular Curves

Abstract

對於p是一個大於3的質數並且r是一個正整數，讓Γ是一個介於Γ_1(p^r) 和 Γ_0(p^r)之間的congruence subgroup。在這篇論文中，我們給予 the group of modular units on X(Γ) that have divisors defined over Q 一個明確的basis。然後應用此結果 去決定 the order of the cuspidal Q-rational torsion subgroup of J(Γ) generated by the divisor classes of cuspidal divisors of degree 0 defined over Q 當 Γ 是 Γ_0(p^r)或 Γ_1(p^r) 。

For p > 3 an odd prime and r > 0, let Γ be a congruence subgroup between Γ_1(p^r) and Γ_0(p^r). In this dissertation, we give an explicit basis for the group of modular units on X(Γ) that have divisors defined over Q. As an application, we determine the order of the cuspidal Q-rational torsion subgroup of J(Γ) generated by the divisor classes of cuspidal divisors of degree 0 defined over Q when Γ = Γ_0(p^r), Γ_1(p^r).