# The old subvariety of Jo(pq)

• Kenneth A. Ribet
Chapter
Part of the Progress in Mathematics book series (PM, volume 89)

## Abstract

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
$$A = {\rm Im} age({J_0}{(p)^2}\xrightarrow{\alpha }{J_0}(pq)),B = {\rm Im} age({J_0}({q^2})\xrightarrow{\beta }{J_0}(pq))$$
of the two indicated degeneracy maps. Here, J o (N) denotes the Jacobian Pic°(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 [6], §2a.

Tate Summing

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