Class 2 Galois representations of Kummer type (Q1770307)

Let \(k\) be a field of characteristic \(0\), let \(\bar k\) be an algebraic closure of \(k\) and let \(G_k= \text{Gal}(\bar k/k)\) be the absolute Galois group of \(k\). A projective Galois representation of \(G_k\), i.e., a continuous homomorphism \(P\colon G_k \to \text{PGL}_n(\bar k)\), is said to be of Kummer type if: (i) it is absolutely irreducible of degree greater than 1, (ii) its image \(P(G_k)\) is abelian and (iii) \(k\) contains a primitive \(e\)th of unity, where \(e\) is the exponent of \(P(G_k)\). Thus the fixed field of the kernel of a projective Galois representation of \(G_k\) is a Kummer extension of \(k\). A central pair over \(k\) is a pair \((A,f)\), where \(A\) is a finite abelian group \(A\) and \(f\) is a 2-cocycle \(f\) with values in \(k^*\). Under suitable hypotheses (that the central pair is ``nondegenerate'' and ``full''), one can attach to a central pair over \(k\) a Kummer extension \(k_f\) of \(k\) with \([k_f:k]=| A| \). One of the statements of the paper says that there exists a one-to-one correspondence between the set of isomorphism classes of nondegenerate and full central pairs \((A,f)\) over \(k\) and the set of isomorphism classes of irreducible projective Galois representations \(P\) of \(G_k\) of Kummer type and with kernel \(k_f\). Secondly, the author considers absolutely irreducible linear representations of \(G_k\) of Kummer type, i.e., absolutely irreducible and continuous representations \(L: G_k \to GL_n(\bar k)\) such that the associated projective representation is of Kummer type. Thus the images \(L(G_k)\) of such representations are groups of nilpotent class at most 2, and a representation is said to be of class 2 if its image is in fact a group of class 2. Two linear representations \(L_1\) and \(L_2\) of \(G_k\) are said to belong to the same genus if there exists a linear character \(\lambda\) of \(G_k\) such that \(L_2\) is isomorphic to \(\lambda L_1\). The main result of the paper is the following: there exists a one-to-one correspondence between the set of isomorphism classes of nondegenerate full central pairs \((A,f)\) over \(k\) satisfying a certain property (``regularly cyclic'') and the set of genera of linear class 2 representations of \(G_k\) of Kummer type.