Modular forms

Modular functions played a prominent role in the mathematics of the 19th century, where they appear in the theory of elliptic functions, i.e., elements of the function field of an elliptic curve, but also in the theory of binary quadratic forms. The term seems to stem from Dirichlet, but the functions are clearly present in the works of Gauss, Abel and Jacobi. They play an important role in the work of Kronecker, Eisenstein and Weierstrass, and later in that century they appear as central themes in the work of Poincar´e and Klein. The theory of Riemann surfaces developed by Riemann became an important tool, and Klein and Fricke studied and popularized the Riemann surfaces defined by congruence subgroups of the modular group SL(2,ℤ). Modular forms appear as theta functions in the work of Jacobi in the 1820’s, and, up to a factor q^1/24, already in Euler’s identity They show up in a natural way in the expansions of elliptic functions and as such they were studied by Eisenstein, but the concept of modular forms was formalized only later. Apparently, it was Klein who introduced the term “Modulform”, cf. page 144 of Klein-Fricke [12].