The Hardness of Computing an Eigenform Eric Bach and Denis Charles

The Fourier coefficients of modular forms encode very interesting arithmetic data. For example, divisor sums, partition numbers, trace of Frobenius of the reduction modulo primes of an elliptic curve over Q, and more generally, trace of Frobenius of many Galois representations of dimension 2 over finite fields (this being a conjecture of Serre) are all known to be, or conjectured to be, Fourier coefficients of modular forms. A particularly important family of modular forms are the so-called Hecke eigenforms. These are modular forms that are also simultaneous eigenforms for an algebra of operators called the Hecke operators that operate on the spaces of modular forms. The Fourier coefficients of Hecke eigenforms are particularly important arithmetically. Indeed, many of the examples given above arise as Fourier coefficients of Hecke eigenforms.