TITLE: Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?
QUESTION [51 upvotes]: According to Steven Krantz's Mathematical Apocrypha (pg. 186):

As was custom, Weil often attended tea
  at [Princeton] University . Graduate
  student Steven Weintrab one day went
  about the room asking various famous
  mathematicians who was the greatest
  mathematician of the twentieth
  century. When he asked Weil, the
  answer (without hesitation) was "Carl
  Ludwig Siegel (1896-1981)."

As the title of Krantz's book suggests, the anecdote may be apocryphal. However, there are other better grounded accounts of great mathematicians expressing the highest admiration for Siegel:
(A) In The Map of My Life Shimura wrote:

I always thought that few people
  really understood my work. I knew that
  Chevalley, Eichler, Siegel, and Weil
  understood my work, and that was
  enough for me [...] Of course [Siegel]
  established himself as one of the
  giants in the history of mathematics
  long ago [...] Among his
  contemporaries, [Weil] thought highly
  of Siegel [...]

(B) In an published interview (pg. 30) Selberg said

[Siegel] was in some ways, perhaps,
  the most impressive mathematician I
  have met. I would say, in a way,
  devestatingly so. The things that
  Siegel tended to do were usually
  things that seemed impossible. Also
  after they were done, they seemed
  still almost impossible.

Why might Weil, Shimura and Selberg have been so impressed by Siegel? I should emphasize that I'm not trying to precipitate a debate about the relative standing of historical mathematicians - rather - I'm hoping to learn about aspects of Siegel's work that I might otherwise overlook. I'm also not looking for, e.g. quotations from the Wikipedia article on him, but rather, less familiar material.

REPLY [9 votes]: Recently I was digging through the meeting minutes for IAS and chanced upon a report prepared for Siegel to justify hiring him. The relevant pages are 13-18. In this report, it is stated that "we decided to propose at this juncture only one name, because it eclipses all others - that of Carl L. Siegel". 
Later on, an excerpt from a letter from Courant is given which states "the only one of his generation whose strength could be compared with that of the mathematical heroes of the preceding era". More praise is lavished on by Chevalley, where he views Siegel as "on a level with a Hilbert or a Poincare" and by Hardy where he claims that nobody questioned if "he was the equal of any mathematician in his generation, and certainly I never doubted it myself". 
With this in mind, it seems that it was in fact the consensus of the mathematical community then that Siegel was the greatest of his time, not just of Weil. Details of his accomplishments are given in the report for those interested. One thing I personally find puzzling is the lack of his influence in current areas of modern mathematics, as suppose to say Weil or Kolmogorov.