Analogies in Number Theory
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Title: Analogies in Number Theory
Abstract: In his 1940 letter from prison, Andre Weil described a
"Rosetta stone" of mathematics, which provides a unifying, tripartite analogy
between the work of Riemann on his eponymous surfaces and their function theory,
the work of Dedekind, Kummer, Weber and others on algebraic number fields,
and the work of Artin and Weil himself on function fields over finite fields.
This analogy is a touchstone in number theory, with each of these
three "pillars" influencing the development and trajectory of the other two.
I will survey Weil's famous analogy, emphasizing the profound impact
it has had on the genesis and development of number theory through the
particular lens of Iwasawa theory. I will also describe an exciting new trajectory
in the Iwasawa theory of function fields over finite fields, and recent developments therein.