Akashi series, characteristic elements and congruence of Galois representations (Q2800144)

Let \(F\) be a number field and \(F_\infty/F\) a \(p\)-adic Lie extension containing the cyclotomic \(\mathbb{Z}_p\)-extension \(F^{\mathrm{cyc}}\) of \(F\) and unramified outside a finite set of primes. Put \(G:=\mathrm{Gal}(F_\infty/F)\), \(H:=\mathrm{Gal}(F_\infty/F^{\mathrm{cyc}})\) and assume \(G\) has no \(p\)-torsion. The paper deals with Akashi series, Euler characteristics and characteristic elements associated to the Pontrjagin dual of Selmer groups \(\mathrm{Sel}(A/F_\infty)\) for certain Galois representations \(A\) (a classical example is \(A=\mathcal{A}[p^\infty]\) for some abelian variety \(\mathcal{A}\)). Two Galois representations \(A\) and \(B\) are called \textit{congruent} if \(A[\pi]\simeq B[\pi]\) and \(A_v[\pi]\simeq B_v[\pi]\) for any \(v|p\) (where \(\pi\) is a local parameter for the ring of integers \(\mathcal{O}\) of a finite extension of \(\mathbb{Q}_p\), \(M[\pi]\) is the \(\pi\)-torsion of an \(\mathcal{O}\)-module \(M\) and \(A_v\) is the local realization of \(A\)). Under several hypotheses, using fine results on the structure of Selmer groups, the author shows that when \(F\) is not totally real, the Akashi series of two congruent representations \(A\) and \(B\) are both units (or both non-units) in the Iwasawa algebra \(\mathcal{O}[[G/H]]\). Similar results are provided for the Euler characteristics and the characteristic elements, the main tool for the proof always being a comparison theorem between the \(\mathcal{O}[[H]]\)-ranks of the duals of \(\mathrm{Sel}(A/F_\infty)\) and of \(\mathrm{Sel}(B/F_\infty)\).