It's fairly easy to write down families of elliptic curves over $\mathbb{Q}(t)$ such that almost every (i.e. when the "height" of $t$ is sufficiently large) curve in the family has positive rank over $\mathbb{Q}$. One can do this by constructing the fibration so it has sections /$\mathbb{Q}$ a priori, or by fiddling around with Gauss sums; see papers of Fermigier, Mestre, Arms/Miller/Lozano-Robledo, etc. Are there any known examples of families of genus *two* curves over $\mathbb{Q}(t)$ such that the Jacobian of almost every curve $C_t$ in the family has provably positive rank? Can one do this while requiring that $\mathrm{Jac}(C_t)$ be $\overline{\mathbb{Q}}$-simple for almost all $t$?

## 1 Answer

My guess for some examples is the family of (genus 2) hyperelliptic curves y^{2}=degree 6 poly in x passing through n "randomly chosen" rational points (for n=2,3,4,5, or 6). The family of such curves has dimension 7-n, and I would guess that if a hyperelliptic curve has n "random" rational points on it then the Q rank of its Jacobian is usually at least n-1.

An obvious place to look for explicit examples is

MR1406090 Cassels, J. W. S.; Flynn, E. V. Prolegomena to a middlebrow arithmetic of curves of genus 2. London Mathematical Society Lecture Note Series, 230. Cambridge University Press, Cambridge, 1996. ISBN: 0-521-48370-0

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