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The genus $g$ of a nonsingular curve $C$ of degree $n$ is defined as $g = \frac{1}{2}(n-1)(n-2)$. Let $C(Q)$ denote the set of rational points on $C$. By Faltings, we know that $C (Q) < \infty$ for all $g\geq 2$.

I'm curious whether there exists another result (apart from Faltings') that relates the size of $C(Q)$ and $g$ ? More strongly, does the size of $C(Q)$ vary directly as $g$ ?

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  • $\begingroup$ I didn't understand. You need to know whether there is a rational solution of the equation? $$g=\frac{1}{2}(n-1)(n-2)$$ $\endgroup$
    – individ
    Commented Jan 13, 2016 at 17:31
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    $\begingroup$ @individ: no, that expression is a definition of $g$. $\endgroup$ Commented Jan 13, 2016 at 17:31
  • $\begingroup$ What do you mean by vary directly as $g$? For example, Fermat's last theorem implies there are infinitely many $g$s for which $C(\mathbb{Q})$ is a small constant. $\endgroup$
    – Mohan
    Commented Jan 13, 2016 at 18:38
  • $\begingroup$ How can you not knowing the shape of the curve. Using only the degree of $n$ to say, what about solutions? Even to the extent $n=2$ without knowing what kind of curve has nothing to say about solutions. $\endgroup$
    – individ
    Commented Jan 14, 2016 at 5:00

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It is still an open question whether, for every $g \geq 2$ and any number field $K$, there exists a uniform bound $B_g(K)$ such that every curve $C$ of genus $g$ over $K$ has at most $B_g(K)$ rational points. This is known as the uniform Mordell conjecture.

Thus, not only do we not know the size $C(K)$ in terms of $g$, we don't even know whether there is a uniform bound.

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