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Title: Compatible families of elliptic type

Abstract: In axiomatizing their study of Frobenius distributions 
in GL(2)-extensions of the rationals, Lang and Trotter introduce 
the notion of an adelic Galois representation of "elliptic type," 
and they ask in passing whether every such representation arises 
from an elliptic curve. Roughly speaking, the question is whether 
certain conditions that are necessary for a strictly compatible 
family of l-adic representations to come from an elliptic curve 
are also sufficient. In spite of the extraordinary advances that 
have been made in Galois representation theory during the intervening 
decades, the problem of Lang and Trotter may not yet be ripe for 
a solution, but the obstacles that remain appear to be interesting 
questions in their own right.