Consequences
Faltings' 1983 paper had as consequences a number of statements which had previously been conjectured:
- The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points;
- The Shafarevich conjecture that there are only finitely many isomorphism classes of abelian varieties of fixed dimension and fixed polarization degree over a fixed number field with good reduction outside a given finite set of places; and
- The Isogeny theorem that abelian varieties with isomorphic Tate modules (as Ql-modules with Galois action) are isogenous.
The reduction of the Mordell conjecture to the Shafarevich conjecture was due to Parshin (1971). A sample application of Faltings' theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to an + bn = cn, since for such n the curve xn + yn = 1 has genus greater than 1.
Read more about this topic: Faltings' Theorem
Famous quotes containing the word consequences:
“The consequences of our actions grab us by the scruff of our necks, quite indifferent to our claim that we have “gotten better” in the meantime.”
—Friedrich Nietzsche (1844–1900)
“[As teenager], the trauma of near-misses and almost- consequences usually brings us to our senses. We finally come down someplace between our parents’ safety advice, which underestimates our ability, and our own unreasonable disregard for safety, which is our childlike wish for invulnerability. Our definition of acceptable risk becomes a product of our own experience.”
—Roger Gould (20th century)
“The horror of Gandhi’s murder lies not in the political motives behind it or in its consequences for Indian policy or for the future of non-violence; the horror lies simply in the fact that any man could look into the face of this extraordinary person and deliberately pull a trigger.”
—Mary McCarthy (1912–1989)