On the representation of the Picard modular function by $θ$ constants I-II

Spectral Curves of Operators with Elliptic Coefficients

“…In this section we follow a similar approach to that of Matsumoto [20] in a genus 4 problem, who developed earlier results of Shiga [27] and Picard [22]. Write the equation of C in the form…”

“…It is a rare event, that a period matrix can be calculated from an algebraic equation, since the two in general are transcendental functions of each other. The method consists in the traditional one [2,26,27] of choosing a suitable homology basis and calculating the covering action on it. We believe our result on the period matrix of the Halphen curve and its reduction to be new.…”

Thomae Type Formulae For Singular Z N Curves

“…The modular properties of certain family of curves (1.2) were investigated by Burhardt [6], Hutchinson [19], more recently by Shiga [34] and Koike [26], Diez [7] and others. This curve (1.2) appeared in Zverovich [40] as the curve related to a solvable Riemann-Hilbert problem with quasipermutation monodromy matrices.…”

Construction of CM Picard curves

“…This formula goes back to Picard [25] and has already been worked out for a special symplectic basis in [30]. But since the period matrices constructed in Section 3 are in general not of a special form, we have to review the arguments in [25,30] and give a more general formulation using geometric considerations.…”