On the Arf invariant in historical perspective

Originally this manuscript was prepared for my talk at the workshop on Sequences, Curves and Codes in Antalya, 25–29 September 2009. Later I had given a talk with the same title on October 5, 2009 at the conference on Positivity, Valuations, and Quadratic Forms in Konstanz. In the discussion after the talk I learned that there may be an error in Arf’s paper and perhaps his main theorem has to be modified. I am indebted to Karim Johannes Becher for this comment. Indeed, after another check I found the error in the proof of one of Arf’s lemmas. Accordingly the manuscript had to be corrected, taking care of the situation and clarifying the scope of Arf’s theorems after the correction. For the convenience of the reader I have included in an appendix a counterexample and also a proof of the corrected version of Arf’s theorem. This appendix has been prepared jointly with Falko Lorenz.