Related papers: On rational Eisenstein primes and the rational cus…

Let $\mathcal{C}_N$ be the cuspidal subgroup of the Jacobian $J_0(N)$ for a square-free integer $N>6$. For any Eisenstein maximal ideal $\mathfrak{m}$ of the Hecke ring of level $N$, we show that $\mathcal{C}_N[\mathfrak{m}]\neq 0$. To…

Let $\ell \geq 5$ be a prime and let $N$ be a non-squarefree integer not divisible by $\ell$. For a rational Eisenstein prime $\mathfrak{m}$ of the Hecke ring $\mathbb{T}(N)$ of level $N$ acting on $J_0(N)$, we precisely compute the…

Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup of the Jacobian variety $J_0(\frak{n})$ of the Drinfeld modular curve $X_0(\frak{n})$. We prove that for any prime number $\ell$ not dividing…

In this paper, we study the torsion subgroup of $J_0(N)$ over the field generated by those points in the cuspidal group, where $N$ is an odd positive integer. We prove that, considered as Hecke modules, this group and the cuspidal subgroup…

We study the rational torsion subgroup of the modular Jacobian $J_0(N)$ for $N$ a square-free integer. We give a new proof of a result of Ohta on a generalization of Ogg's conjecture: for a prime number $p \nmid 6N$, the $p$-primary part of…

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…

For a positive integer $N$, let $J_0(N)$ be the Jacobian of the modular curve $X_0(N)$. In this paper we completely determine the structure of the rational cuspidal subgroup of $J_0(N)$ when the largest perfect square dividing $N$ is either…

For a positive integer $N$, let $\mathscr{C}_N(\mathbb{Q})$ be the rational cuspidal subgroup of $J_0(N)$ and $\mathscr{C}(N)$ be the rational cuspidal divisor class group of $X_0(N)$, which are both subgroups of the rational torsion…

Let N be a square-free positive integer and let f be a newform of weight 2 on \Gamma_0(N). Let A denote the abelian subvariety of J_0(N) associated to f and let m be a maximal ideal of the Hecke algebra T that contains Ann_T(f) and has…

Let $N$ be a positive integer and let $J_0(N)$ be the Jacobian variety of the modular curve $X_0(N)$. For any prime $p\ge 5$ whose square does not divide $N$, we prove that the $p$-primary subgroup of the rational torsion subgroup of…

Let $\frak{p}$ and $\frak{q}$ be two distinct prime ideals of $\mathbb{F}_q[T]$. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve $X_0(\frak{p}\frak{q})$ to compare the rational torsion subgroup of the Jacobian…

We use pseudodeformation theory to study the analogue of Mazur's Eisenstein ideal with certain squarefree levels. Given a prime number $p>3$ and a squarefree number $N$ satisfying certain conditions, we study the Eisenstein part of the…

Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of…

Let $\mathbf{G}=\mathrm{U}(2,n)$ be the unitary group associated to a Hermitian space over a quadratic imaginary number field $E$. We assume that 2 is unramified in $E$, and the Hermitian space splits at all finite places and has signature…

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…

For a prime $\mathfrak{p} \subseteq \mathbb{F}_{q}[T]$ and a positive integer $r$, we consider the generalised Jacobian $J_{0}(\mathfrak{n})_{\mathbf{m}}$ of the Drinfeld modular curve $X_{0}(\mathfrak{n})$ of level…

We consider the generalised Jacobian $J_{0}(N)_{\mathbf{m}}$ of the modular curve $X_{0}(N)$ of level $N$, with respect to the modulus $\mathbf{m}$ consisting of all cusps on the modular curve. When $N$ is odd, we determine the group…

Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps…

For a prime number $q\geq 5$ and a positive integer $N$ prime to $q$, Ribet proved the action of the Hecke algebra on the component group of the Jacobian variety of the modular curve of level $Nq$ at $q$ is "Eisenstein", which means the…

Let $F/\mathbb{Q}$ be any totally real number field and $\frak{N}$ an ideal of its ring of integers of norm $N$ and define, for every even $n$, the $[F:\mathbb{Q}]$-dimensional multiweight $\textbf{n}=(n,...,n)$. We prove that for a non CM…