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Zbl 1013.11019
Culp-Ressler, Wendell
Rational period functions on the Hecke groups. (English)
[J] Ramanujan J. 5, No.3, 281-294 (2001). ISSN 1382-4090; ISSN 1572-9303

Rational period functions (RPFs) have received the attention of a number of mathematicians since the reviewer first discussed them in [Duke Math. J. 45, 47-62 (1978; Zbl 0374.10014)]. The notion of RPF arises in a natural generalization of modular form: Suppose $f$ is holomorphic in $\text{Im }z>0$ and has the transformation properties $$f(z+1)=f(z),\quad z^{-2k} f(-1/z)=f(z)+q(z),\tag*$$ with $k\in \bbfZ$ and $g(z)$ a rational function. If, further, $f(z)$ has a left-finite expansion in the variable $\text{exp}(2\pi iz)$ (at the cusp $i\infty$), then we call $f$ a modular integral of weight $2k$, with RPF $q(z)$, on the modular group $\text{SL}(2,\bbfZ)$. {\it A. Ash} [Am. J. Math. 111, 35-51 (1989; Zbl 0664.10014)] gave an abstract characterization of RPFs on $\text{SL}(2,\bbfZ)$, and this was followed by an explicit characterization given, independently, by {\it Y. J. Choie} and {\it D. Zagier} [Contemp. Math. 143, 89-109 (1993; Zbl 0790.11044)] and {\it L. A. Parson} [Contemp. Math. 143, 109-116 (1993; Zbl 0790.11045)]. The author points out that others (Schmidt, Sheingorn) ``have taken steps toward an explicit characterization of RPFs on the Hecke groups''.\par In Theorem 2 of the article under under review Culp-Ressler gives ``an explicit characterization of a class of RPFs on the Hecke groups. [This] characterization is for RPFs for which $k$ (half of the weight) is odd and whose irreducible pole sets'' have a certain symmetry property. The author's Theorem 1 gives an explicit expression for any RPF on a Hecke group, but as yet he has not determined when this expression yields a bona fide RPF.
[Marvin I.Knopp (Philadelphia)]

MSC 2000:: *11F67 Special values of automorphic L-series, etc
11F11 Modular forms, one variable

Keywords: rational period functions; modular integral; Hecke groups

Citations: Zbl 0374.10014; Zbl 0664.10014; Zbl 0790.11044; Zbl 0790.11045

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