Deficient points on extensions of abelian varieties by \(G_ m\) (Q1117284)

Let V be a commutative algebraic group over a number field k which is an extension of an abelian variety A by the multiplicative group \(G_ m\). For each prime number \(\ell\), denote by \(V_{\ell}\) the group of \(\ell\)- torsion points of V. It generates a finite Galois extension \(k(V_{\ell})/k\). The authors analyze the set \(D=D(V/k)\) of all those k- rational points x of V such that \(\ell^{-1}x\) are \(k(V_{\ell})\)- rational for all sufficiently large \(\ell\) (called ``deficient points''). Obviously, D contains the group T of all k-rational torsion points of V, and (by elementary number theory) \(D=T\) if \(A=(0).\) A surprising fact shown here is that, in general, \(D\neq T\). Motivated by a question raised by D. Bertrand, and using some constructions of L. Breen, this work gives a precise (and clear) accout of what D is (a constructive answer) and when \(D\neq T\) (cohomological interpretations). \textit{K. A. Ribet} gives a motivic description of D in another paper [J. Number Theory 25, 152-161 (1987; Zbl 0666.14001)], together with a proof of a conjectural statement in remark 4.4.