The ℚ-rational cuspidal group of J₁(2p)

Abstract

Let p be a prime not equal to 2 or 3. In this paper we study the ℚ-rational cuspidal group 𝒞_ℚ of the jacobian J₁(2p) of the modular curve X₁(2p). We prove that the group 𝒞_ℚ is generated by the ℚ-rational cusps. We determine the order of 𝒞_ℚ, and give numerical tables for all p ≤ 127. These tables give also other cuspidal class numbers for the modular curves X₁(2p) and X₁(p). We give a basis of the group of the principal divisors supported on the ℚ-rational cusps, and using this we determine the explicit structure of 𝒞_ℚ for all p ≤ 127. We determine the structure of the Sylow p-subgroup of 𝒞_ℚ, and the explicit structure for all p ≤ 4001.