Log24

Research Contributions

Raussen and Skau: This brings us to the next
topic: Your Ph.D. thesis from 1950, when you were
twenty-five years old. It has been extensively cited
in the literature under the sobriquet “Tate’s thesis”.
Several mathematicians have described your thesis
as unsurpassable in conciseness and lucidity and as
representing a watershed in the study of number
fields. Could you tell us what was so novel and fruitful
in your thesis?

Tate: Well, first of all, it was not a new result, except
perhaps for some local aspects. The big global
theorem had been proved around 1920 by the
great German mathematician Erich Hecke, namely
​the fact that all L -functions of number fields,
abelian L -functions, generalizations of Dirichlet’s
L -functions, have an analytic continuation
throughout the plane with a functional equation
of the expected type. In the course of proving
it Hecke saw that his proof even applied to a new
kind of L -function, the so-called L -functions with
Grössencharacter. Artin suggested to me that one
might prove Hecke’s theorem using abstract
harmonic analysis on what is now called the adele
ring, treating all places of the field equally, instead
of using classical Fourier analysis at the archimedian 
places and finite Fourier analysis with congruences 
at the p -adic places as Hecke had done. I think I did
a good job —it might even have been lucid and
concise!—but in a way it was just a wonderful 
exercise to carry out this idea. And it was also in the
air. So often there is a time in mathematics for 
something to be done. My thesis is an example. 
Iwasawa would have done it had I not.

[For a different perspective on the highlighted areas of
mathematics, see recent remarks by Edward Frenkel.]