The characteristic-polynomial conjecture for $y^2=x^p-1$

Let $C_p$ be the genus $g$ curve

$$y^2=x^p-1,$$

where $p=2g+1$ is prime. For the characteristic polynomial $P_{\gamma^d}(T)$ associated with the indicated component representative, the characteristic-polynomial conjecture.

$$P_{\gamma^d}(T)=T^{2g}+1$$

for any $d$ relatively prime to $2g$. This conjecture is an observation-based prediction about the characteristic polynomials of components of the Sato–Tate group and is used to motivate corresponding moment predictions.