Modular forms, generalized Dedekind symbols and period polynomials

Fukuhara, Shinji

doi:10.1007/s002080050138

Cited by 28 publications

(54 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, sgn(x) denotes the sign of x ∈ R. It is shown ([11, Theorem 1 ′ ], [7]) that, for n odd, β + w α + w+2 maps c w,n R w,n to S w,n :…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…Furthermore, periods of cusp forms (or values of L-series associated with the forms) play important roles in number theory. A large number of papers have discussed periods and period polynomials of cusp forms ( [5,6,7,10,11,12,14,15,18,20,21]). One of the basic results is, perhaps, the result due to Eichler and Shimura ([5,11,15,18]).…”

Section: Introductionmentioning

confidence: 99%

“…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. It is known that these three spaces are isomorphic modulo trivial elements ( [7,9]). Through these isomorphisms, we can construct bases for these spaces explicitly (Theorems 2.3, 2.6 and 2.7), starting with the basis for S * w+2 determined in Theorem 2.2.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Fukuhara

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Moreover, sgn(x) denotes the sign of x ∈ R. It is shown ([11, Theorem 1 ′ ], [7]) that, for n odd, β + w α + w+2 maps c w,n R w,n to S w,n :…”

Section: Statement Of Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Fukuhara

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Self Cite

View full text Add to dashboard Cite

show abstract

“…First let us recall a few definitions in [5] which are necessary for our subsequent discussions. A Dedekind symbol is a generalization of the classical Dedekind sums ( [14]), and is defined as a complex valued function D on V := {(p, q) ∈ Z + × Z | gcd(p, q) = 1} satisfying R(p + q, q) + R(p, p + q) = R(p, q).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Hecke operators on weighted Dedekind symbols

Fukuhara

2006

Journal für die reine und angewandte Mathematik (Crelles Journa

Self Cite

View full text Add to dashboard Cite

Abstract. 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.

show abstract

“…Moreover, behaviour of period functions of modular forms with respect to modular transformations involves the action 6) which is the right action of PSL(2, Z) upon functions holomorphic in H − .…”

Section: Actions Of Modular Groupmentioning

confidence: 99%