A prologue to functoriality and reciprocity: Part 1

Robert Langlands
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Finished in late 2012
Beyond Endoscopy

Author's comments: There is as yet no text with the title Functoriality and Reciprocity. Begun as preparations for a lecture, the text appearing here was, and remains, a first attempt to come to terms with the two topics of the title, an attempt that is perhaps doomed by its nature and by my years to remain provisional. The attempt demands not only a great deal of reflection and a good number of novel ideas, mine or those of someone else, but also a mastery of several mathematical domains --- algebraic number theory, algebraic geometry, spectral analysis, representation theory, some differential geometry, some mathematical physics --- of all of which I have acquired a smattering of knowledge over the past decades, although without mastering any. As it stands here it is only a part of the Prologue. The present text, A prologue to ``Functoriality and Reciprocity'', Part I appears, in spite of its failings, in a volume dedicated to the memory of Jonathan Rogawski, Pacific Journal of Mathematics, vol. 260, No. 2, Dec. 2012.