<div dir="ltr"><div class="gmail_default" style="font-family:comic sans ms,sans-serif;font-size:large">"... <i style="color:rgb(0,0,0);font-family:Helvetica,Arial,"Nimbus Sans L",sans-serif;font-size:15px;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial">My office in Ankara was next to that of Cahit Arf, and when I mentioned the question to him, he drew my attention to a paper of Hasse that had appeared in a journal not widely read, the</i><span style="color:rgb(0,0,0);font-family:Helvetica,Arial,"Nimbus Sans L",sans-serif;font-size:15px;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial;float:none;display:inline"><span> </span>Acta Salmanticensia<span> </span></span><i style="color:rgb(0,0,0);font-family:Helvetica,Arial,"Nimbus Sans L",sans-serif;font-size:15px;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial">of 1954. He fortunately had a reprint. So I could begin to think seriously about the matter. The critical idea came in April 1968 in a hotel room in Izmir, where I had gone to deliver a lecture. It was the understanding that all identities needed were consequences of four basic ones, formulated in the notes as the four main lemmas. Once this is understood and basic facts about Gauss sums are understood, as in the papers of Lamprecht and Davenport-Hasse, three of these four identities are not so difficult to establish. The second main lemma turned out, on the other hand, to be a major obstacle. Fortunately while leafing idly through journals in the library, either in Ankara or later in New Haven (I no longer remember), I came across Dwork's paper in which the first and the second main lemmas were proved. Dwork had indeed tried to establish a product formula for what has come to be called the<span> </span><span class="gmail-MathJax" id="gmail-MathJax-Element-14-Frame" tabindex="0" style="display:inline;font-style:normal;font-weight:normal;line-height:normal;font-size:15px;text-indent:0px;text-align:left;text-transform:none;letter-spacing:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;padding:0px;margin:0px"><span style="border:0px;padding:0px;margin:0px;max-width:none;max-height:none;min-width:0px;min-height:0px;vertical-align:0px;line-height:normal;text-decoration:none;white-space:nowrap"><span class="gmail-math" id="gmail-MathJax-Span-76" style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;text-decoration:none;width:0.549em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;text-decoration:none;width:0.44em;height:0px;font-size:18.3px"><span style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;text-decoration:none"><span class="gmail-mrow" id="gmail-MathJax-Span-77" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;text-decoration:none"><span class="gmail-mi" id="gmail-MathJax-Span-78" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;text-decoration:none;font-family:STIXGeneral;font-style:italic">ϵ</span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;text-decoration:none;width:0px;height:2.516em"></span></span></span><span style="display:inline-block;border-width:0px;border-top-style:initial;border-right-style:initial;border-bottom-style:initial;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.063em;line-height:normal;text-decoration:none;overflow:hidden;width:0px;height:0.67em"></span></span></span><span class="gmail-MJX_Assistive_MathML" style="padding:0px;border:0px;display:inline;margin:0px;vertical-align:0px;line-height:normal;text-decoration:none;height:1px;width:1px;overflow:hidden"><span><span>ϵ</span></span></span></span>-factor..."</i></div><div class="gmail_default" style="font-family:comic sans ms,sans-serif;font-size:large"><i style="color:rgb(0,0,0);font-family:Helvetica,Arial,"Nimbus Sans L",sans-serif;font-size:15px;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial">Robert Langlands</i></div><div class="gmail_default" style="font-family:comic sans ms,sans-serif;font-size:large"><i style="color:rgb(0,0,0);font-family:Helvetica,Arial,"Nimbus Sans L",sans-serif;font-size:15px;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial"><a href="http://publications.ias.edu/rpl/section/22">http://publications.ias.edu/rpl/section/22</a><br></i></div></div>