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The Eisenstein descent on \(J_ 0(N)\). (English) Zbl 0594.14027

The paper under review deals with the general question of how to compute the rank of an abelian variety over a number field K. The standard approach uses the injection \(A(K)/n\cdot A(K)\to H^ 1(0_ k,A[n])\) (flat cohomology): One tries to find an upper bound for the righthand side, and a lower bound for the left hand side which coincides with it. Here the authors deal with A \(= Eisenstein\)-quotient of \(J_ 0(N) = Jacobian\) of the modular curve \(X_ 0(N)\), \(N=p\) or \(N=p^ 2\) (p a prime. The case \(K={\mathbb{Q}}\), \(N=p\) has been treated extensively by B. Mazur).
Under certain hypotheses (on Bernoulli-numbers) the authors can show that either the right hand side vanishes, or that the theory of Heegner points suffices to account for all of it. The details of their arguments are a little bit involved, so we prefer not to comment on them.
Reviewer: G.Faltings

MSC:

14H40 Jacobians, Prym varieties
14K15 Arithmetic ground fields for abelian varieties
11F12 Automorphic forms, one variable
14G25 Global ground fields in algebraic geometry
14K30 Picard schemes, higher Jacobians
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References:

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