Actually, we know all Diophantine equations are not decidable.
Does Faltings's theorem imply the sets of Diophantine equations are decidable? That is , there is an algorithm that decide whether those equations are decidable.
"Let $C$ be a non-singular algebraic curve of genus (mathematics)|genus $g$ over $\mathbb{Q}$. Then the set of rational points on $C$ may be determined as follows:
Case $g$ = 0: no points or infinitely many; $C$ is handled as a conic section. Case $g$ = 1: no points, or $C$ is an elliptic curve and its rational points form a finitely generated abelian group (''Mordell's Theorem'', later generalized to the Mordell–Weil theorem). Moreover Mazur's torsion theorem restricts the structure of the torsion subgroup. Case $g$ > 1: according to the Mordell conjecture, now Faltings's Theorem, $C$ has only a finite number of rational points."
For example, in case $g = 0$, is there an algorithm that decide whether the equations have integral solutions to them?
Could any one give a list of classes of Diophantines which are decidable?