For Jacobians of genus-2 curves on a K3 surface over number field k, there is a finite extension l/k with infinitely many fibers where the rank jumps, and under extra geometric conditions this holds on a non-thin set.
Large rank jumps on elliptic surfaces and the Hilbert prop- erty
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Rank jumps for Jacobians of Hyperelliptic curves on K3 surfaces
For Jacobians of genus-2 curves on a K3 surface over number field k, there is a finite extension l/k with infinitely many fibers where the rank jumps, and under extra geometric conditions this holds on a non-thin set.