The following theorem is a very deep application of Roth’s theorem. Let be a number field and let be a finite set of places of . Let be the http://planetmath.org/node/RingOfSIntegersring of -integers in . Let be a smooth projective curve of genus defined over and let be a non-constant function in the function field of , i.e. .
Theorem (Siegel’s Theorem).
Assume that has genus . Then the set is finite.
In particular, when is the coordinate functions and , Siegel’s theorem implies that a curve of genus has only finitely many integral points. For example, this shows that an elliptic curve defined over can only have finitely many points defined over .
|Date of creation||2013-03-22 15:57:24|
|Last modified on||2013-03-22 15:57:24|
|Last modified by||alozano (2414)|