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 Γ 0 (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.

In this paper, we apply ideas of Dijkgraaf and Witten [31,6] 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.

HWAJONG YOO A. For any positive integer N , we completely determine the structure of the rational cuspidal divisor class group C(N ) of X 0 (N ), which is conjecturally equal to the group of rational torsion points on J 0 (N ). More specifically, let ℓ be any given prime. For a non-trivial divisor d of N , we construct a rational cuspidal divisor Z ℓ (d) and show that the ℓ-primary subgroup of C(N ) is isomorphic to the direct sum of the cyclic groups generated by the images of the divisors Z ℓ (d). Also, we compute the order of the image of the divisor Z ℓ (d) in J 0 (N ).

Theorem 1.1 (Main theorem). Let m be an Eisenstein maximal ideal of T. Then C N [m] = 0.To prove this theorem, we classify all possible Eisenstein maximal ideals in §2. From now on, we denote by U p the p th Hecke operator T p ∈ T when p | N . Proposition 1.2. Let m be an Eisenstein maximal ideal of T. Then, it contains I M,N := (U p − 1, U q − q, I 0 (N ) : for primes p | M and q | N/M ) for some divisor M of N such that M = 1.