Twist-minimal trace formulas for Hecke operators : Modular forms

Holomorphic modular forms have been studied actively in both algebraic number theory and analytic number theory, related with the class field theory, Galois representations and elliptic curves. They are providing both new methods and a source of problems in analytic number theory. In particular modular forms are important examples of automorphic L-functions. The theory of modular forms have been well-established, but still, there are unsolved conjectures.

The main object of this project is to provide a method to compute the space of holomorphic modular forms for arbitrary weight, level and Nebentypus characters. There are several ways, including the method of modular symbols, but our tool is based on the Eichler-Selberg trace formula. In the recently published book by H. Cohen and F. Strömberg, the newform trace formulas (based on the newform theory of modular forms) and their applications (e.g. Pari/GP package, which can evaluate L-functions for modular forms and other numerical values associated with modular forms) are explained.

When the level is non-square free, the corresponding space of modular forms becomes more complicated. The newform spaces can be further sieved down to the twist-minimal forms, i.e., those newforms whose conductor cannot be reduced by twisting with Dirichlet characters. Our aim is to derive twist-minimal Eichler-Selberg trace formulas and to find applications including computing the space, evaluating associated L-functions, etc.