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The reciprocity conjecture supposes these motives come from a different type of analytical mathematical object discovered by Langlands called automorphic representations, Arthur notes.

The conception which any one frames of another's mind is more or less after the pattern of his own mind, Ð is automorphic.

In their analytical form, as groups of linear transformations of a single variable, the groups are those on which the theory of automorphic functions depends.

Automorphic, aw-to-mor′fik, adj. marked by automorphism, the ascription to others of one's own characteristics.

A similar utility, of a more extended kind, belongs to automorphic functions in general; but it can be shown that such functions necessarily have an infinite number of essential singularities except for the simplest cases.