The old subvariety of J 0(pq) and the Eisenstein kernel in Jacobians

Abstract

When p, q are distinct odd primes, and γ:J 0(p)2×J 0(q)2→J 0(pq) is the natural map defined by the degeneracy maps, Ribet [10] determined the odd part of the kernel of γ. We study the 2-primary part of this kernel through its intersection with the Eisenstein kernel J 0(p)[I p )2×J 0(q)[I q ]2. We determine this intersection for p≢1 mod 16, q≢1 mod 16, and also produce new elements of ker γ whenever p≡9 mod 16 or q≡9 mod 16. These sharpen Ribet's results in [10]. © 1993 The Magnes Press.