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Zbl 0869.11037
Culp-Ressler, Wendell
A Hecke correspondence theorem for modular integrals with rational period functions. (English)
[J] Ill. J. Math. 40, No.4, 586-605 (1996). ISSN 0019-2082

In the 1930's Erich Hecke used the Mellin transform and its inverse to demonstrate a systematic relationship between infinite exponential sums and Dirichlet series. In particular, entire modular forms on $\Gamma(1)= SL(2,\bbfZ)$ (which satisfy a modular relation) correspond to Dirichlet series which satisfy a functional equation.\par In the 1970's Marvin Knopp defined generalizations of modular forms which he called modular integrals. The modular relation for a modular integral involves a period function $q(z)$. An important class of period functions is the class of rational period functions.\par In 1992 John Hawkins and Marvin Knopp proved a Hecke correspondence theorem for modular integrals of weight $2k\in\bbfZ$ with rational period functions on $\Gamma_\theta= \langle S^2,T\rangle$, the subgroup of index 3 in $\Gamma(1) =\langle S, T\rangle$. Here we are using $S=\left (\smallmatrix 1 & 1 \\ 0 & 1 \endsmallmatrix \right)$ and $T= \left(\smallmatrix 0 & -1\\ 1 & 0\endsmallmatrix \right)$. The group $\Gamma_\theta$ has a single relation $T^2=I$, which gives rise to a single relation for rational period functions on $\Gamma_\theta$, $q+q |T=0$. Hawkins and Knopp proved that a modular integral on $\Gamma_\theta$ corresponds to a Dirichlet series with a functional equation which involves a remainder term $R(s)$. The relation $q+q|T=0$ gives rise to a relation for the remainder term, $R(2k-s)+ i^{2k}R(s)=0$.\par This paper presents a Hecke correspondence theorem for modular integrals on the full modular group $\Gamma (1)$. $\Gamma(1)$ has a second group relation, $(ST)^3=I$, which imposes a second condition on rational period functions on $\Gamma(1)$, $q+q |ST+ q|(ST)^2=0$. We define a mapping $\rho$, which acts on remainder terms in a way analogous to the action of $|ST$ on rational period functions. We show that the second relation for a rational period function imposes a second relation on the corresponding remainder term, $R+\rho(R) +\rho^2 (R)=0$. Our results depend on the classification of rational period functions on $\Gamma(1)$, a problem which was solved in 1993 in papers by L. Alayne Parson and, independently, by Young Ju Choie and Don Zagier. We describe their results and modify them in order to write rational period functions in a way that emphasizes the second relation.
[W.Culp-Ressler (Lancaster, PA)]

MSC 2000:: *11F11 Modular forms, one variable
11F66 Dirichlet series and functional equations related to modular forms
11F67 Special values of automorphic L-series, etc

Keywords: rational period functions; Hecke correspondence; modular integrals; full modular group

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