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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.

More generally any function unaltered by all the substitutions of a group of linear substitutions of its variable is called an Automorphic Function.