SOME MODULAR EQUATIONS IN THE FORM OF SCHLÄFLI

Abstract

On page 90 of his first notebook, S. Ramanujan records Schl¨afli-type modular equations for degrees 3, 5, 7, 11, 13, 17 and 19.  In this paper, we establish Schl¨afli-type modular equations for degrees 11, 13, 17 and 19 which are recorded by Ramanujan in his first notebook.  We also establish several new Schl¨afli-type modular equations of degrees 2, 4, 9, 15, 23, 25, 29, 31, 47 and 71.  As an application, we deduce some explicit evaluations of Ramanujan-Weber class invariants.