6 problems

Let $\Gamma\subseteq\operatorname{SL}_2(\mathbb{Z})$ be a congruence subgroup of prime level, and let $X(\Gamma)$ be the associated modular curve with Jacobian $J(\Gamma)$. Let…

Let $N$ be a positive integer, let $L$ be the level-dependent integer used for the Galois action, and let $\mathcal{D}_N$ and $F_{m,h}$ be the indexing set and modular-unit basis i…

Let $N$ be a positive integer. Let $C_N(\mathbb{Q})$ be the subgroup of $J_0(N)(\mathbb{Q})$ represented by degree-zero cuspidal divisors, and let $C(N)$ be the subgroup formed by…

Let $N$ be a positive integer. Write $\mathscr C_\mathbb Q(N)$ for the subgroup of rational cuspidal divisor classes on $X_0(N)$ generated by divisors supported on rational cusps,…

Let $A$ be the ring of functions regular away from a fixed place at infinity, and let $\mathfrak n\lhd A$ be a non-zero square-free ideal. Let $X_0(\mathfrak n)$ be the smooth proj…

Let $p$ be a prime and $n$ a positive integer. For the cuspidal divisor class group $\mathscr C_1^\infty(p^n)$, let $[p]$ denote multiplication by $p$: … Define its $p$-rank to be…