Let $T\!: y^2 = x^3 + a(t, s)x + b(t, s)$ be an elliptic threefold over a finite field $\mathbb{F}_q$ of characteristic $p > 3$. In other words, we have an elliptic curve over the function field $\mathbb{F}_q(t, s)$ in two independent variables $t$, $s$. Fixing $t \in \mathbb{F}_{q}$, except for some $t$ we obtain elliptic surfaces $T_{t}$, i.e., elliptic curves over $\mathbb{F}_q(s)$. Each surface $T_{t}$ generates the Mordell--Weil lattice, i.e., the lattice $T_t(\mathbb{F}_q(s))$. Help me please. How are these lattices related to each other ? Are they isomorphic in some sense ?
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2$\begingroup$ They will all but finitely many contain a fixed sublattice coming from divisors on the threefold, but each one can have rank larger than this fixed lattice, and in a varying way. I don't think much can be said. $\endgroup$– Will SawinCommented Oct 27, 2022 at 18:14
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