Jing Yu in his paper "On Arithmetic Of Hyperelliptic Curves" on page 5 asserts the following
The most interesting case is certainly the case $k = \mathbb Q$ and $D \in \mathbb Z[t]$.
To decide whether there are non-constant units in $\mathbb Q[t, \sqrt{D}]$, one reduces the equation $y^2 = D(t)$ modulo various odd primes $p$. For all but finitely many $p$ (namely those dividing the discriminant of the polynomial $D(t)$), one obtains curve $C_p$ over the finite field $\mathbb F_p$. Given such a prime $p$ of "good" reduction. If the divisor class of $\infty_{+} - \infty_{-}$ is of order $n$ in $Jac\ C$, then unless $p | n$, the t is also of order $n$ in $Jac\ C_p$. This fact follows from the theory of reduction of abelian varieties.
I wonder where is it possible to read the proof of mentioned fact or some similar theory. Could you please provide some links to papers or books in which I can read that this fact follows from the theory of reduction of abelian varieties. Or it would be great if it is possible to explain it right here.
P.S. $C$ is a hyperelliptic curve over the base field $k = \mathbb Q$, $D$ is a square free polynomial with $\deg D$ is even, positive and the leading coefficient of $D$ is a square in $k^{\times}$.
