Explicit Formulas for Hecke Operators on Cusp Forms, Dedekind Symbols and Period Polynomials

Let Sw+2 be the vector space of cusp forms of weight w + 2 on the full modular group, and let S∗ w+2 denote its dual space. Periods of cusp forms can be regarded as elements of S∗ w+2. The Eichler-Shimura isomorphism theorem asserts that odd (or even) periods span S w+2 . However, periods are not linearly independent; in fact, they satisfy the Eichler-Shimura relations. This leads to a natural question: which periods would form a basis of S∗ w+2. First we give an answer to this question. Passing to the dual space Sw+2, we will determine a new basis for Sw+2. The even period polynomials of this basis elements are expressed explicitly by means of Bernoulli polynomials. Next we consider three spaces—Sw+2, the space of even Dedekind symbols of weight w with polynomial reciprocity laws, and the space of even period polynomials of degree w. There are natural correspondences among these three spaces. All these spaces are equipped with compatible action of Hecke operators. We will find explicit form of period polynomials and the actions of Hecke operators on the period polynomials. Finally we will obtain explicit formulas for Hecke operators on Sw+2 in terms of Bernoulli numbers Bk and divisor functions σk(n), which are quite different from the Eichler-Selberg trace formula.