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Is there any relationship between the number of the rational points on a curve over $\mathbb{Q}$ of genus $\geq 2$ and its Jacobian?

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    $\begingroup$ Are you working over a number field? Are you working over a finite field? $\endgroup$
    – Jason Starr
    Commented Jan 25, 2019 at 10:49
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    $\begingroup$ The number of rational points on a curve defined over $\mathbb{Q}$ (and more generally, a number field) can be bounded in terms of the genus and the Mordell-Weil rank of the Jacobian, provided said rank is sufficiently small compared to the genus. More recent work in non-abelian Chabauty can extend this further to the case when the rank of the Neron-Severi group is large. See for example projecteuclid.org/euclid.dmj/1476450482 $\endgroup$
    – Stanley Yao Xiao
    Commented Jan 25, 2019 at 11:51
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    $\begingroup$ There are conjectures, and for certain families theorems, saying that $$\#C(\mathbb Q)\le \kappa(g)^{1+\operatorname{rank} J(\mathbb Q)}.$$ Here $g$ is the genus of $C$. However, the Bombieri-Lang conjecture (rational points are not Zariski dense on varieties of general type) implies a stronger uniform bound $\#C(\mathbb Q)\le\kappa'(g)$. Is this what you had in mind? $\endgroup$
    – Joe Silverman
    Commented Jan 25, 2019 at 12:13

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