Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials

Abstract

Abstract. Let S w+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 S w+2 , we will determine a new basis for S w+2 . The even period polynomials of this basis elements are expressed explicitly by means of Bernoulli polynomials.Next we consider three spaces-S w+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 S w+2 in terms of Bernoulli numbers B k and divisor functions σ k (n), which are quite different from the Eichler-Selberg trace formula.

Twisted Hecke L-values and period polynomials

“…Then in [8,12] it is shown that the values of r m (R n ) can be expressed in terms of the Bernoulli numbers for integers m and n with opposite parity satisfying 0 m 2k − 2 and 1 n 2k − 3. In [7], by considering the natural correspondence of S 2k (SL(2, Z)), its dual, and the space of Dedekind symbols, the first author of the present article found bases for S 2k (SL(2, Z)) in terms of R n , which in turn give explicit expression for Hecke operators in terms of Bernoulli numbers and sum-of-divisor functions. For the case Γ 0 (2), this is done in [8] with a different approach.…”

Quantum modular forms and Hecke operators

“…Fukuhara showed that there is a one-to-one correspondence among the spaces S w+2 , E ± w and U ± w (see [6,8,9]). …”

A basis for the space of modular forms

“…To state the result in [3] we need the following notation and convention: form a basis for S * 2k . Next we will display a basis for S 2k .…”