On rational Eisenstein primes and the rational cuspidal groups of modular Jacobian varieties

Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for $\Gamma_0(N)$, and its rational Eisenstein primes of $\mathbb{T}(N)$ containing $\ell$, defined in Section 3. If $\mathfrak{m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak{m}$ is of the form $(\ell, ~\mathcal{I}^D_{M, N})$, where the ideal $\mathcal{I}^D_{M, N}$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}] \neq 0$, where $\mathcal{C}(N)$ is the rational cuspidal group of $J_0(N)$. To do this, we compute the precise order of the cuspidal divisor $\mathcal{C}^D_{M, N}$, defined in Section 4, and the index of $\mathcal{I}^D_{M, N}$ in $\mathbb{T}(N)\otimes \mathbb{Z}_\ell$.Comment: Many arguments are clarified, and many details are filled i

Rational torsion points on Jacobians of modular curves

Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb{Q}$ and its Jacobian variety $J_0(3p)$ over $\mathbb{Q}$. Let $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ be the group of rational torsion points on $J_0(3p)$ and the cuspidal group of $J_0(3p)$, respectively. We prove that the $3$-primary subgroups of $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ coincide unless $p\equiv 1 \pmod 9$ and $3^{\frac{p-1}{3}} \equiv 1 \!\pmod {p}$

The rational cuspidal divisor class group of X0(N)X_0(N)

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 $Z_\ell(d)$.Comment: Comments are welcom

Abelian arithmetic Chern-Simons theory and arithmetic linking numbers

Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of n-th power residue symbols. This formalism leads to a precise arithmetic analogue of a 'path-integral formula' for linking numbers

Arithmetic Chern-Simons theory II

In this paper, we apply ideas of Dijkgraaf and Witten [6, 32] on 3 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern–Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of cases and point towards some arithmetic applications

NON-OPTIMAL LEVELS OF A REDUCIBLE MOD l MODULAR REPRESENTATION

Let l >= 5 be a prime and let N be a square-free integer prime to l. For each prime p dividing N, let ap be either 1 or -1. We give sufficient criteria for the existence of a newform f of weight 2 for G0( N) such that the mod l Galois representation attached to f is reducible and Upf = apf for primes p dividing N. The main techniques used are level raising methods based on an exact sequence due to Ribet. c.2018 American Mathematical Societ