“My Life’s Good Fortune”

A Lie Symmetry Connection between Jacobi's Modular Differential Equation and Schwarzian Differential Equation

“…Thus Abel and Jacobi are recognized jointly for developing the theory of the elliptic functions in their current form [21]. A "noble competition" between these two young men may have developed [21], but subsequent polemics about the unquestionable priority of Abel and the alleged speculation that Jacobi may have been unfair and not cognizant of Abel's original contribution ( [28], [29]) may be countered if one recalls that on March 14, 1829, nearly a month before Abel's death, Jacobi wrote to Legendre [16] "Quelle découverte de M. Abel que cette généralisation de l'intégrale d'Euler! A-t-on jamais vu pareille chose!…”

“…It is well-known that Sophus Lie (1842-1899) was influenced by Abel's and especially Galois' work on algebraic equations as well as Jacobi's work on partial differential equations [13], [14]. Less well-known 2 may be his involvement with the republication of Abel's work [2] and unfruitful search for Abel's original manuscript [29] that Jacobi prized so much and went unnoticed and was subsequently lost at the Paris Academy 3 . In this paper we apply Lie group analysis to Jacobi's modular differential equation, which he derived for linking two different moduli of an elliptic integral [18], and obtain a six-dimensional Lie symmetry algebra isomorphic to sl(2, IR) × sl(2, IR) which transforms it into the well-known Schwarzian differential equation [15].…”