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Joe Silverman
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Here are three.

Wiles theorem: Every elliptic curve over Q is parameterized by modular functions.

Faltings Theorem/Mordell Conjecture: A curve of genus at least 2 has only finitely many rational points.

Faltings Theorem/Shafarevich Conj: There are only finitely many abelian varieties with good reduction outside a given finite set of primes.

Joe Silverman
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