Conjectured by Mordell and later proved by Faltings, a non-singular algebraic curve of genus $g$ over $\mathbb{Q}$ has finitely many rational points if $g > 1$. Since the genus of the Fermat curve $x^{n} + y^{n} = 1$ is $\frac{n(n-1)}{2}$ by the degree formula, Faltings' Theorem implies that it can only have finitely many rational solutions for $n > 2$, which proves a weak form of FLT.
Are there other topological invariants of curves or surfaces which bound the genus or (more directly) the number of rational points that can exist on them, e.g., arithmetic/geometric genus, (Zariski) multiplicity, Milnor number, Hirzebruch signature, Casson Invariant, Rohklin Invariant, etc.?
For example, if $f = \sum_{i=0}^{n} z_{i}^{a_{i}}$ with positive integers $a_0, \dots, a_n$, then one defines a Brieskorn-Pham manifold $\Sigma(a_0, \dots, a_n)$ as the intersection of the corresponding hypersurface of $f$ with a sufficiently small sphere, namely, $f^{-1}(0) \cap S^{2n+1}_{\epsilon}$. Brieskorn-Pham manifolds give examples of exotic spheres in certain odd dimensions for certain exponents. Is there a connection between certain topological attributes of $\Sigma(a_0, \dots, a_n)$, including topological invariants, and the number of rational points on the curve $f = 1$?
NB: This question was first asked on Math.SE, but went unanswered.