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# Why did André Weil name Carl Ludwig Siegel the greatest mathematician of the 20th century?

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.