By the fall of 1966, I was prepared to abandon mathematics and to turn to some other life, a first step being a year or two in Turkey with my wife and children, as a prelude to an existence whose exact form was undetermined.

 Robert Langlands is a distinguished mathematician and here he is describing a mathematically low-point in his life.


I, myself, was searching for a general notion and had despaired. By the fall of 1966, I was prepared to abandon mathematics and to turn to some other life, a first step being a year or two in Turkey with my wife and children, as a prelude to an existence whose exact form was undetermined. I, who had never been anywhere outside of English-speaking North America, returned to the study of Russian and began the study of Turkish, frivolously daydreaming of a trip to Turkey --- with wife and four small children --- through the Balkans or through the Caucasus. In the end we arrived in Ankara by a more banal route. Even with the Russian and Turkish, I had time to spare and began, as an idle amusement, to calculate the constant term of the Eisenstein series for various rank-one groups. I had, curiously enough, never done this before. I discovered rather quickly a regularity of which I had been unaware. It was described in the lectures delivered at Yale some months later and included in Part 3 of this collection. The constant term, or rather the second part of the constant term, the part that expresses the functional equation was there denoted M(s) and given at the very end of §5 as a product that I write here as

∏i=1rξi(ais)ξi(ais+1)(1)

r being a small integer, often 1, and ai being a positive number. Suppose, in order not to confuse the explanations, that r is 1. The issues arising in the general case are treated in the references. It is the relation expressed by (1) that suggests and allows the passage from the theory of Eisenstein series to a general notion of automorphic L-function that can accommodate not only a non-abelian generalization of class-field theory but also, as it turned out, both functoriality and reciprocity. It was the key to the suggestions in the Weil letter. The Yale notes were written a long time ago and were hardly exemplary expositions. I have no desire at the moment to recall the details or to improve their presentation --- the reader is encouraged to consult the writings of Shahidi, for example the book Eisenstein series and automorphic L-functions --- but there are a number of points to which I would like to draw attention, and it is more convenient to refer to my own notes. I repeat, first of all, that (1) refers to rank-one parabolic subgroups, thus to Eisenstein series arising from maximal proper parabolic subgroups, so that it does not require the second or the third steps and is analytically at the level of Selberg's original arguments.