Topology in Arithmetic

Question

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.

Answer 1

Dear user02138, you may be interested in the Bombieri-Lang conjecture. It has a weak form and a strong form. The latter is:

If $X$ s a smooth variety over a number field $K$ which is of general type over a number field, then there are only finitely many curves of (geometric) genus $\leq 1$ contained in $X$, and outside those curves $X$ has only finitely many $K$-rational points.

The key word here is "general type" which is a geometric condition (defined for a proper smooth variety as the canonical divisor $K$ being big, that is the order magnitude of the dimensions of the space of global section of the bundle $L(nK)$ is $n^d$ as $n$ goes to infinity), which for proper smooth curves is equivalent to the Mordell's condition $g \geq 2$. Under this condition (which some experts in algebraic geometry may discuss in more detail that I can -- for example is it a condition that can be read on $X(\mathbb C)$ as topological space like for curves?), you get strong results on the number of rational points. Of course this conjecture is wide-open, except in the cases of curves or more generally subvarieties of abelian varieties, proved by Faltings.