Riemann hypothesis for period polynomials attached to the derivatives of \(L\)-functions of cusp forms for \(\Gamma_0(N)\)
Period polynomials play a critical role in the theory of modular forms. They encode the critical \(L\)-values of modular forms, and they form spaces which determine modular forms via the Eichler-Shimura isomorphism. There has been a great deal of work developing their arithmetic structure, for example, Manin's Theorem on the transcendence of their coefficients, and the seminal work of \textit{W. Kohnen} and \textit{D. Zagier} [in: Modular forms, Symp. Durham/Engl. 1983, 197--249 (1984; Zbl 0618.10019)] on the structure of rational period polynomials. Recently, their analytic structure, as polynomials, has received much attention. Specifically, their roots have been shown to be tightly constrained in various situations. A few particular examples include the work of Conrey-Farmer-Imamoglu [\textit{J. B. Conrey} et al., Int. Math. Res. Not. 2013, No. 20, 4758--4771 (2013; Zbl 1305.11030)] showing that all but 5 of the roots of the odd parts of all level \(1\) cusp forms lie on the unit circle and of \textit{A. El-Guindy} and \textit{W. Raji} [Bull. Lond. Math. Soc. 46, No. 3, 528--536 (2014; Zbl 1311.11027)] showing that all roots of the full period polynomials lie on the unit circle when the level is \(1\). This was greatly generalized by Jin-Ma-Ono-Soundararajan [\textit{S. Jin} et al., Proc. Natl. Acad. Sci. USA 113, No. 10, 2603--2608 (2016; Zbl 1412.11075)], who showed that the same is true in all levels for newforms of any level, where unit circle is replaced by a circle of appropriate radius dictated by the Fricke involution. This has been termed the Riemann Hypothesis for Period Polynomials (RHPP). Afterwards, several papers have considered RHPPs in other contexts. For instance, \textit{N. Diamantis} and \textit{L. Rolen} [Res. Math. Sci. 5, No. 1, Paper No. 9, 15 p. (2018; Zbl 1457.11032); J. Reine Angew. Math. 770, 1--25 (2021; Zbl 1472.11152)] considered analogues of period polynomials which are built instead out of higher derivatives of modular form \(L\)-functions. This paper considers the Diamantis-Rolen conjecture in the case of first derivatives. The authors prove this is almost all cases. Specifically, for forms of weight \(4\) and \(6\) of any level, the roots of the first \(L\)-derivative ``period polynomial'' lie on the appropriate circle, and that for weights at least \(8\) and sufficiently large levels, that the zeros lie on the same circles. In fact, they prove the conjecture for all but finitely many cusp forms, which suggests that a computational verification may be able to prove the full conjecture (at least for first derivatives). This builds on the story of these period polynomials, adding a nice set of results and analytic techniques to the literature that hint at the existence of structure which warrants further investigation.
scientific article; zbMATH DE number 7472992
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