2021/06/13 by Saber, Hicham, Sebbar, Abdellah · 1 citation
#11F03 #11F11 #34M05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2106.06903
For every positive integer r, we solve the modular Schwarzian differential equation \h,τ\=2π2r2E4, where E4 is the weight 4 Eisenstein series, by means of equivariant functions on the upper half-plane. This paper supplements previous works \citeforum, ramanujan, where the same equation has been solved for infinite families of rational values of r. This also leads to the solutions to the modular differential equation y''+r2π2E4 y=0 for every positive integer r. These solutions are quasi-modular forms for SL2(\mathbb Z) if r is even or for the subgroup of index 2, SL2(\mathbb Z)2, if r is odd.