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Automorphic loops and metabelian groups

Mark Greer, Lee Raney (2020)

Commentationes Mathematicae Universitatis Carolinae

Given a uniquely 2-divisible group G , we study a commutative loop ( G , ) which arises as a result of a construction in “Engelsche elemente noetherscher gruppen” (1957) by R. Baer. We investigate some general properties and applications of “ ” and determine a necessary and sufficient condition on G in order for ( G , ) to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that G is metabelian if and only if ( G , ) is an automorphic loop. We...

Automorphic realization of residual Galois representations

Robert Guralnick, Michael Harris, Nicholas M. Katz (2010)

Journal of the European Mathematical Society

We show that it is possible in rather general situations to obtain a finite-dimensional modular representation ρ of the Galois group of a number field F as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over F , provided one works “potentially.” The proof is based on a close study of the monodromy of the Dwork family of Calabi–Yau hypersurfaces; this in turn makes use of properties of rigid local systems and the classification...

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