A $\tau$-function solution of the sixth Painlevé transcendent

We represent and analyze the general solution of the sixth Painlevé transcendent $\mathcal P_6$ in the Picard–Hitchin–Okamoto class in the Painlevé form as the logarithmic derivative of the ratio of $\tau$-functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the $\mathcal P_6$ equation and the uniformization of algebraic curves and present examples.