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### The old subvariety of J o (pq) [chapter]

Kenneth A. Ribet
1991 Arithmetic Algebraic Geometry
Let p and q be distinct primes. The old part of J o (pq) is the abelian subvariety A + B of J o (pq) generated by the images of the two indicated degeneracy maps. Here, J o (N ) denotes the Jacobian Pic o (X o (N )) of the standard modular curve X o (N ), for each integer N ≥ 1. Also, we have written J o (p) 2 for the product J o (p) × J o (p), and have used analogous notation for J o (q) 2 . The definitions of α and β will be given below; see also  , §2a. The structure of A was determined
more » ... A was determined in  . Namely, the kernel of α is the Shimura subgroup Σ p of J o (p), viewed as a subgroup of J o (p) 2 via the antidiagonal embedding x → (x, −x). Thus we have A = J o (p) 2 /Σ p and, analogously, B = J o (q) 2 /Σ q . Since A and B are known, we consider that to understand A + B is to understand A ∩ B, which is a finite abelian group. The main purpose of this note is to identify A ∩ B, up to groups of 2-power order. In other words, we identify the -primary part of A ∩ B for each odd prime . Let C p be the cuspidal subgroup of J o (p). This group is cyclic of order num( p−1 12 ), and appears frequently in  . (The symbol "num" denotes the