Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)

For and an even integer , let be the space of period polynomials of weight on with eigenvalue under the Fricke involution. We determine the dimension formula for and construct an explicit basis for it using period functions for weakly holomorphic modular forms. Furthermore, for a quadratic form , we...

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Bibliographic Details
Published in:Open mathematics (Warsaw, Poland) Poland), 2024-12, Vol.22 (1)
Main Authors: Choi, SoYoung, Kim, Chang Heon, Lee, Kyung Seung
Format: Article
Language:English
Subjects:
11F11
Analytic functions
cycle integrals
Cycle ratio
Eigenvalues
Half planes
Integrals
period polynomials
Polynomials
Quadratic forms
rational period functions
weakly holomorphic modular forms
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Summary:For and an even integer , let be the space of period polynomials of weight on with eigenvalue under the Fricke involution. We determine the dimension formula for and construct an explicit basis for it using period functions for weakly holomorphic modular forms. Furthermore, for a quadratic form , we define the function on the complex upper half-plane as a generating function of the cycle integrals of the canonical basis elements for the space of weakly holomorphic modular forms of weight and eigenvalue under the Fricke involution on . We also show that is a modular integral on . Our approach focuses on calculating cycle integrals within rather than , which allows us to overcome certain technical challenges. This study extends earlier work by Choi and Kim ( , J. Math. Anal. Appl. (2015), no. 2, 741–758) which focused on eigenvalue +1, providing new insights by examining eigenvalue cases in the theory of rational period functions and cycle integrals in this setting.
ISSN:2391-5455
DOI:10.1515/math-2024-0102