History[edit]
In 1926, Siegel proved the theorem effectively in the special case , so that he proved this theorem conditionally, provided the Mordell's conjecture is true.
In 1929, Siegel proved the theorem unconditionally by combining a version of the Thue–Siegel–Roth theorem, from diophantine approximation, with the Mordell–Weil theorem from diophantine geometry (required in Weil's version, to apply to the Jacobian variety of C).
In 2002, Umberto Zannier and Pietro Corvaja gave a new proof by using a new method based on the subspace theorem.[1]
Effective versions[edit]
Siegel's result was ineffective for (see effective results in number theory), since Thue's method in diophantine approximation also is ineffective in describing possible very good rational approximations to almost all algebraic numbers of degree . Siegel proved it effectively only in the special case in 1926. Effective results in some cases derive from Baker's method.
