Higher Rademacher Symbols

Abstract

Certain higher Rademacher symbols are defined that give class functions on the modular group. Their basic properties are derived via a two-variable reformulation of Eichler-Shimura cohomology. This reformulation better explains the role of cycle integrals and also yields new results, about the integrality of the values of the symbols. The Rademacher symbols determine the values at non-positive integers of the zeta function for a narrow ideal class of a real quadratic field. This result is equivalent to one of Siegel, but is proven in a new way by using an identity for the value of such a zeta function at a positive integer greater than one as a sum of certain double zeta values defined for the quadratic field.

Keywords:

• Rademacher symbol
• real quadratic field
• zeta values

Acknowledgments

I thank the referee for suggesting several improvements of the paper’s exposition.

Notes

1 Study of the “minus” continued fraction goes back at least to Möbius [25].

2 This freedom is a result of the fact that the dimension of the space of polynomials available to construct ${\Psi }_n^{\prime }$ is in general larger than the dimension of a certain cohomology group determined by these polynomials.

3 E.g. the coefficient of x₁ in $G_{24}(x)$ is 5932154033364156392062962058217938594635552972840.

4 Examples show that the related formulas given in Theorem 8 of [21] and in the example that follows it need to be corrected.

Additional information

Funding

Research supported by NSF grant DMS 1701638.