Abstract
Let N and M be distinct prime numbers and let J o (NM) be the Jacobian of the modular curve X o (NM). The old part of J o (NM) is defined as the span of the images of certain natural maps from J o (N) and J o (M) to J o (NM). Ribet [16] and Ling [9] determined the old part of J o (NM) for most pairs (N, M). In this paper, we use the structure of the kernel of the Eisenstein ideal in J o (N) (and J o (M)) to complete the task of determining the old subvariety of J o (NM). Our results were previously known in the special case where neither N nor M are congruent to 1 modulo 16.
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Csirik, J.A. (2004). The old Subvariety of J0 (NM). In: Cremona, J.E., Lario, JC., Quer, J., Ribet, K.A. (eds) Modular Curves and Abelian Varieties. Progress in Mathematics, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7919-4_4
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DOI: https://doi.org/10.1007/978-3-0348-7919-4_4
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