Rational period functions for \(\text{PSL}(2,\mathbb Z)\).

*(English)*Zbl 0790.11044
Knopp, Marvin (ed.) et al., A tribute to Emil Grosswald: number theory and related analysis. Providence, RI: American Mathematical Society. Contemp. Math. 143, 89-108 (1993).

[This is a joint review for the papers of Choie-Zagier and Parson (see Zbl 0790.11045 below).]

In 1974 the reviewer, having in mind the well-known example of the Eichler period polynomials and the rational period function (RPF) \(1/z\) appearing in Hurwitz’s transformation law for \(G_ 2\) (the Eisenstein series of weight 2 on the modular group \(\Gamma(1)\)), began the study of general RPF’s on \(\Gamma(1)\). The object originally was to prove that the two examples above are the only nontrivial ones connected with \(\Gamma(1)\), but soon infinitely many new examples emerged in each weight \(2k\), with \(k\in \mathbb Z^ +\) and \(k\) odd [Duke Math. J. 45, 47–62 (1978; Zbl 0374.10014)]. Since then the theory of RPF’s has been developed by a number of mathematicians, among them A. Ash, Y. J. Choie, E. Gethner, J. Hawkins, L. A. Parson, T. A. Schmidt and D. Zagier. The two articles under review, each of which – independently – describes completely the RPF’s on \(\Gamma(1)\), represent the culmination of (the first phase of) the theory.

An RPF in weight \(2k\) on \(\Gamma(1)\), \(k\in\mathbb Z\), is a rational function \(q\) satisfying the two relations (corresponding to the two group relations in \(\Gamma(1)\)), \[ q(z)+ z^{-2k} q(-1/z)=0,\;q(z)+ z^{-2k} q(1-1/z)+(z-1)^{-2k}q(-1/(z-1))=0.{(*)} \] That the construction of classification of RPF’s on \(\Gamma(1)\) is a highly nontrivial matter is due to the presence of two relations in \((*)\). Indeed, the classification of RPF’s on \(\Gamma_ \vartheta\), a nonnormal subgroup of index 3 in \(\Gamma(1)\), is very easy, and precisely because RPF’s on \(\Gamma_ \vartheta\) are subject only to the first of the two relations in \((*)\). John Hawkins first pointed out the connection between RPF’s on \(\Gamma(1)\) and indefinite binary quadratic forms, and he developed as well the important notion of a “minimal pole set” of an RPF [unpublished]. The role of quadratic forms is suggested by the early result [Glasg. Math. J. 22, 185–197 (1981; Zbl 0459.10017)] that an irrational pole of an RPF on \(\Gamma(1)\) is a real quadratic irrational. (The only possible rational pole of an RPF on \(\Gamma(1)\) is at 0.) A “minimal pole set” is a set \(S\) of irrational numbers, all lying in the same real quadratic extension of \(\mathbb Q\), with the property that if an RPF \(q\) on \(\Gamma(1)\) has poles in \(S\), then \(q\) has a pole at every point of \(S\). Hawkins proved that minimal pole sets are in \(1-1\) correspondence with narrow equivalence classes of primitive binary quadratic forms.

The methods of Choie-Zagier and Parson are distinct, but closely related in spirit. As Parson points out, what is crucial in both approaches is the “technique of first identifying the correct principal part at poles at a minimal pole set and then trying to adjust it to obtain an RPF by adding a rational function whose only possible pole is zero”.

For the entire collection see [Zbl 0773.00030].

In 1974 the reviewer, having in mind the well-known example of the Eichler period polynomials and the rational period function (RPF) \(1/z\) appearing in Hurwitz’s transformation law for \(G_ 2\) (the Eisenstein series of weight 2 on the modular group \(\Gamma(1)\)), began the study of general RPF’s on \(\Gamma(1)\). The object originally was to prove that the two examples above are the only nontrivial ones connected with \(\Gamma(1)\), but soon infinitely many new examples emerged in each weight \(2k\), with \(k\in \mathbb Z^ +\) and \(k\) odd [Duke Math. J. 45, 47–62 (1978; Zbl 0374.10014)]. Since then the theory of RPF’s has been developed by a number of mathematicians, among them A. Ash, Y. J. Choie, E. Gethner, J. Hawkins, L. A. Parson, T. A. Schmidt and D. Zagier. The two articles under review, each of which – independently – describes completely the RPF’s on \(\Gamma(1)\), represent the culmination of (the first phase of) the theory.

An RPF in weight \(2k\) on \(\Gamma(1)\), \(k\in\mathbb Z\), is a rational function \(q\) satisfying the two relations (corresponding to the two group relations in \(\Gamma(1)\)), \[ q(z)+ z^{-2k} q(-1/z)=0,\;q(z)+ z^{-2k} q(1-1/z)+(z-1)^{-2k}q(-1/(z-1))=0.{(*)} \] That the construction of classification of RPF’s on \(\Gamma(1)\) is a highly nontrivial matter is due to the presence of two relations in \((*)\). Indeed, the classification of RPF’s on \(\Gamma_ \vartheta\), a nonnormal subgroup of index 3 in \(\Gamma(1)\), is very easy, and precisely because RPF’s on \(\Gamma_ \vartheta\) are subject only to the first of the two relations in \((*)\). John Hawkins first pointed out the connection between RPF’s on \(\Gamma(1)\) and indefinite binary quadratic forms, and he developed as well the important notion of a “minimal pole set” of an RPF [unpublished]. The role of quadratic forms is suggested by the early result [Glasg. Math. J. 22, 185–197 (1981; Zbl 0459.10017)] that an irrational pole of an RPF on \(\Gamma(1)\) is a real quadratic irrational. (The only possible rational pole of an RPF on \(\Gamma(1)\) is at 0.) A “minimal pole set” is a set \(S\) of irrational numbers, all lying in the same real quadratic extension of \(\mathbb Q\), with the property that if an RPF \(q\) on \(\Gamma(1)\) has poles in \(S\), then \(q\) has a pole at every point of \(S\). Hawkins proved that minimal pole sets are in \(1-1\) correspondence with narrow equivalence classes of primitive binary quadratic forms.

The methods of Choie-Zagier and Parson are distinct, but closely related in spirit. As Parson points out, what is crucial in both approaches is the “technique of first identifying the correct principal part at poles at a minimal pole set and then trying to adjust it to obtain an RPF by adding a rational function whose only possible pole is zero”.

For the entire collection see [Zbl 0773.00030].

Reviewer: Marvin I. Knopp (Philadelphia)

##### MSC:

11F11 | Holomorphic modular forms of integral weight |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11E16 | General binary quadratic forms |