arXiv:math/0308028v4 [math.HO] 17 Jul 2005 Ramanujan’s Most Singular Modulus MARK B. VILLARINO August 4, 2013 Abstract We present an elementary self-contained detailed computation of Ramanujan’s most famous singular modulus, k 210 , based on the Kronecker Limit Formula. Contents 1 The Singular Modulus α 2 1.1 Introduction .................................... 2 1.2 Singular moduli and units in quadratic fields .................. 3 1.3 Binary quadratic forms. The “numeri idonei” of Euler ............. 5 1.4 Elliptic modular functions and abelian extensions ............... 5 1.5 Prospectus ..................................... 7 2 Ramanujan’s Function F (α) 7 2.1 Complete elliptic integrals ............................ 7 2.2 The modular equation .............................. 8 2.3 Ramanujan’s theorem for α 2 , α 3 , and α 7 ................... 9 2.4 Ramanujan’s two-step algorithm ......................... 12 2.4.1 Ramanujan’s invariant g n ........................ 12 2.4.2 “a very curious algebraical lemma” ................... 13 2.4.3 The computation of α 30 ......................... 15 2.4.4 The computation of k 210 ......................... 16 3 Kronecker’s Limit Formula and the Computation of g 210 18 3.1 Binary quadratic forms .............................. 19 3.2 Kronecker’s limit formula ............................. 22 3.3 The relation with Ramanujan’s function g n ................... 24 3.4 The Kronecker–Weber computation of the weighted sum for m = 210 ... 26 3.5 The Dirichlet computation of the weighted sum ................ 29 3.5.1 The Dirichlet class number formulas .................. 33 3.5.2 The final computation of g 210 ...................... 33 1