1.Elliptic functions. 2.Hormomorphisms. 3.The modular function. 4.Fourier expansions. 5.The modular equation. 6.Higher levels. 7.Automorphisms of the modular functions field. 8.Results from algebraic number theory. 9.Reduction of elliptic curves. 10.Complex multiplication. 11.Shimura’s reciprocity law. 12.The function. 13.The l-adict and p-adic representation of deuring. 14.Ihara’s theory. 15.The tate parametrization. 16.The Isogeny theorems. 17.Division points over number fields.. 18.Product expansions. 19.The fundamental theta function. 20.The Kronecker limit formulas. 21.The first limit formula and L-series. 22.The second limit formula and L-series.