Dedekind Symbols Research Articles

Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series

Let E(z, s) be the non-holomorphic Eisenstein series for the modular group $${\mathrm{SL}}(2,{{\mathbb {Z}}})$$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $$s=1$$ of E(z, s) is essentially the logarithm of the Dedekind eta function. This eta function is a weight 1/2 modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $${\Gamma }$$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map $$S:{\Gamma }\rightarrow {{\mathbb {R}}}$$ which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from $${\Gamma }$$ to $${{\mathbb {C}}}$$ produces higher-order Eisenstein series. These series share many of the properties of E(z, s) though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher-order Dedekind symbols $$S^*:{\Gamma }\rightarrow {{\mathbb {R}}}$$. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups $${\Gamma }_0(N)$$ are rational.

Quantum modular forms and Hecke operators

It is known that there are one-to-one correspondences among the space of cusp forms, the space of homogeneous period polynomials and the space of Dedekind symbols with polynomial reciprocity laws. We add one more space, the space of quantum modular forms with polynomial period functions, to extend results from Fukuhara. Also, we consider Hecke operators on the space of quantum modular forms and construct new quantum modular forms.

Modular forms of real weights and generalised Dedekind symbols

In a previous paper, I have defined non-commutative generalised Dedekind symbols for classical PSL(2,mathbf {Z})-cusp forms using iterated period polynomials. Here I generalise this construction to forms of real weights using their iterated period functions introduced and studied in a recent article by R. Bruggeman and Y. Choie.

Modular cocycles and linking numbers

It is known that the 3-manifold SL(2,Z)\SL(2,R) is diffeomorphic to the complement of the trefoil knot in S3. E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind’s eta function under SL(2,Z). In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.

Generalized Dedekind sums and equidistribution mod 1

Dedekind sums are well-studied arithmetic sums, with values uniformly distributed mod 1. Based on their relation to certain modular forms, Dedekind sums may be defined as functions on the cusp set of SL(2,Z). We present a compatible notion of Dedekind sums, which we name Dedekind symbols, for any non-cocompact lattice Γ<SL(2,R), and prove the corresponding equidistribution mod 1 result. The latter part builds up on a paper of Vardi, who first connected exponential sums of Dedekind sums to Kloosterman sums.

Non-commutative generalized Dedekind symbols

We define and study generalized Dedekind symbols with values in non--necessarily commutative groups, generalizing constructions of Sh. Fukuhara in [Fu1], [Fu2]. Basic examples of such symbols are obtained by replacing period integrals of modular forms (cf. [Ma1], [Ma2], [Kn1], [Kn2], [ChZ]) by iterated period integrals introduced and studied in [Ma3], [Ma4].

The elliptic Apostol–Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol–Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide’s result (J Number Theory 128:1060–1073, 2008) on his elliptic Dedekind–Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain formulas by Ramanujan (Collected Papers of Srinivasa Ramanujan, pp. 136–162. AMS Chelsea Publishing, Providence, 2000), van der Pol (Indag Math 13:261–271, 272–284, 1951), Rankin (Proc R Soc Edinburgh Sect A 76:107–117, 1976) and Skoruppa (J Number Theory 43:68–73, 1993).

Dedekind symbols with plus reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols), and the symbols are determined uniquely by their reciprocity laws, up to an additive constant. For Dedekind symbols D and F, we can consider two kinds of reciprocity laws: D ( p , q ) − D ( q , − p ) = R ( p , q ) and F ( p , q ) + F ( q , − p ) = T ( p , q ) . The first type, which we call minus reciprocity laws, have been studied extensively. On the contrary, the second type, which we call plus reciprocity laws, have not yet been investigated. In this note we study fundamental properties of Dedekind symbols with plus reciprocity law F ( p , q ) + F ( q , − p ) = T ( p , q ) . We will see that there is a fundamental difference between Dedekind symbols with minus and plus reciprocity laws.

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

Let be the vector space of cusp forms of weight w + 2 on the full modular group, and let denote its dual space. Periods of cusp forms can be regarded as elements of . The Eichler-Shimura isomorphism theorem asserts that odd (or even) periods span . 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 .

Hecke operators on weighted Dedekind symbols

Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind symbols. We prove that these newly introduced operators are compatible with Hecke operators on the space of modular forms. As an application, we present formulae to give Fourier coefficients of Hecke eigenforms. In particular we give explicit formulae for generalized Ramanujan's tau functions.

The Dedekind symbol associated with the Eisenstein series of weight two

Dedekind symbols generalize the classical Dedekind sums (symbols). These symbols are determined uniquely, up to additive constants, by their reciprocity laws. For k ≧ 2, there is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws of degree 2k − 2 and the space of modular forms of weight 2k for the full modular group \(PSL_2 (\mathbb{Z}).\) However, this is not the case when k = 1 as there is no modular form of weight two; nevertheless, there exists a unique (up to a scalar multiple) quasi-modular form (Eisenstein series) of weight two. The purpose of this note is to define the Dedekind symbol associated with this quasi-modular form, and to prove its reciprocity law. Furthermore we show that the odd part of this Dedekind symbol is nothing but a scalar multiple of the classical Dedekind sum. This gives yet another proof of the reciprocity law for the classical Dedekind sum in terms of the quasi-modular form.

Dedekind symbols with polynomial reciprocity laws

A Dedekind symbol is a generalization of the classical Dedekind symbol (sum). A Dedekind symbol is characterized by its reciprocity law. Dedekind symbols with polynomial reciprocity laws are of special interest and importance, as such symbols are known to correspond bijectively to cusp forms for the full modular group, and to period polynomials. However, explicit forms of such Dedekind symbols are not yet known. In this article we construct Dedekind symbols explicitly by means of ‘‘Poincare series’’, and then show that these symbols satisfy polynomial reciprocity laws and that they form a spanning set for the space of Dedekind symbols with polynomial reciprocity laws. That is, we show that any Dedekind symbol with polynomial reciprocity law can be expressed as a linear combination of these symbols.

Dedekind symbols associated with J-forms and their reciprocity law

We define a Dedekind symbol associated with a J-form (J-forms generalize the usual Jacobi forms). Then we prove the reciprocity law for Dedekind symbols. As an example, we give an explicit description for the Dedekind symbol associated with Weierstrass ℘-function. The reciprocity law in this case then yields new trigonometric identities. This in turn gives rise to a “generating function” of Apostol reciprocity law for the generalized Dedekind sums.

Generalized Dedekind symbols associated with the Eisenstein series

We have shown recently that the space of modular forms, the space of generalized Dedekind sums, and the space of period polynomials are all isomorphic. In this paper, we will prove, under these isomorphisms, that the Eisenstein series correspond to the Apostol generalized Dedekind sums, and that the period polynomials are expressed in terms of Bernoulli numbers. This gives us a new more natural proof of the reciprocity law for the Apostol generalized Dedekind sums. Our proof yields as a by-product new polylogarithm identities.

Modular forms, generalized Dedekind symbols and period polynomials

Modular forms, generalized Dedekind symbols and period polynomials

Exceptional values of the Dedekind symbol

Five new exceptional values of the Dedekind symbol are presented, and a conjecture is proposed on the necessary and sufficient conditions for integers to be exceptional values.