Kramer–Tunnell local root-number conjecture for hyperelliptic Jacobians

Let $K$ be a local field of characteristic zero, let $L/K$ be a quadratic extension, and let $C/K$ be a hyperelliptic curve with Jacobian $J$. Write $C^L$ and $J^L$ for the quadratic twists by $L/K$, let $g$ be the genus of $C$, let $\Delta_C$ be its discriminant, let $i_d(C)$ and $i_d(C^L)$ denote the relevant deficiency indicators, and let $( (-1)^g\Delta_C,L/K)$ denote the quadratic Hilbert symbol. Kramer–Tunnell local root-number conjecture. One has

$$w(J/K)w(J^L/K)=((-1)^g\Delta_C,L/K)i_d(C)i_d(C^L)(-1)^{\dim_{\mathbb{F}_2}J(K)/N_{L/K} J(L)}.$$

This conjectural relation generalises the Kramer–Tunnell formula for elliptic curves and is intended to compare local root-number terms with local 2-Selmer contributions. The supplied text says that the authors prove the formula in many instances and that in all cases it follows from standard global conjectures, but does not establish it unconditionally in full generality.