Strong deflection limit-analysis using Picard-Fuchs equation in Einstein-Maxwell-Dilaton spacetime

We consider the deflection of light by a spherically symmetric and electrically charged black hole solution in the Einstein-Maxwell-Dilaton theory with specific values of the dilaton coupling constant where the deflection angle can be represented as elliptic integrals. We show that the deflection angle as a function of two dimensionless variables $(s,z)$, which are related to the background charge and the impact parameter, respectively, satisfies a system of 2nd-order linear partial differential equations called the Picard-Fuchs (PF) equations. For each case of the dilaton coupling, the PF equations lead to 1st-order ordinary differential equations with respect to the variable $s$ for the constants $\bar{a}$ and $\bar{b}$ in the log-formula for the strong deflection limit. Using the Hamiltonian system associated with Painlevé VI equations, which holds as a result of the integrability of the PF equations, we solve the equations for $\bar{a}$ and $\bar{b}$. By requiring consistency with the Schwarzschild case in the zero-charge limit, $\bar{a}$ and $\bar{b}$ for nonzero charge are uniquely determined.