Rational Dimensions of Critical Modular $L$-Values

Definitions and mathematical statement

Let $k$ be an integer for which the space $S_k(\operatorname{SL}_2({\mathbb{Z}}))$ of holomorphic cusp forms of weight $k$ for the full modular group contains a normalized Hecke eigenform. Thus $f\in S_k(\operatorname{SL}_2({\mathbb{Z}}))$ is a simultaneous eigenvector for the Hecke operators and has Fourier expansion \[ f(z)=\sum_{n\geq 1}a_f(n)e^{2\pi i n z},\qquad a_f(1)=1. \] The existence of such an $f$ entails that $k$ is even and that the cusp-form space is nonzero.

The supplied record does not define the normalization of $\Lambda(f;s)$. Under the conventional completed modular $L$-function interpretation, set \[ L(f,s)=\sum_{n\geq 1}\frac{a_f(n)}{n^s} \quad\text{for sufficiently large }\operatorname{Re}(s), \qquad \Lambda(f;s)=(2\pi)^{-s}\Gamma(s)L(f,s), \] using analytic continuation to evaluate the completed function at the critical integers $1,\ldots,k-1$. This normalization is an explicit interpretation requiring confirmation against the source.

Define the following ${\mathbb{Q}}$-vector subspaces of ${\mathbb{C}}$: \[ \begin{aligned} L_f^+&=\operatorname{span}_{{\mathbb{Q}}}\{\Lambda(f;s):s\in{\mathbb{Z}},\ 1\leq s\leq k-1,\ s\equiv 1\pmod 2\},\\ L_f^-&=\operatorname{span}_{{\mathbb{Q}}}\{\Lambda(f;s):s\in{\mathbb{Z}},\ 1\leq s\leq k-1,\ s\equiv 0\pmod 2\}. \end{aligned} \] In particular, the endpoints in the two lists are $1,3,\ldots,k-1$ and $2,4,\ldots,k-2$, respectively.

Conjecture 1 (as supplied). For every such weight $k$ and every normalized Hecke eigenform $f\in S_k(\operatorname{SL}_2({\mathbb{Z}}))$, prove both equalities \[ \dim_{{\mathbb{Q}}}L_f^+=\dim_{{\mathbb{C}}}S_k(\operatorname{SL}_2({\mathbb{Z}})), \qquad \dim_{{\mathbb{Q}}}L_f^-=\dim_{{\mathbb{C}}}S_k(\operatorname{SL}_2({\mathbb{Z}})). \]

The supplied excerpt reports that functional relations prove the upper bounds \[ \dim_{{\mathbb{Q}}}L_f^\pm\leq\dim_{{\mathbb{C}}}S_k(\operatorname{SL}_2({\mathbb{Z}})). \] This is a report of the source's claim, not an independent verification. The supplied curation describes equality as a folklore conjecture tied to the unproved Maeda conjecture; it supplies no precise formulation of that connection.

For background, the excerpt also uses the even and odd parts of a period polynomial $P_f(X,Y)$: \[ P_f^+(X,Y)=\frac12\left(P_f(X,Y)+P_f(-X,Y)\right),\qquad P_f^-(X,Y)=\frac12\left(P_f(X,Y)-P_f(-X,Y)\right). \] It describes these as generating polynomials for critical values at odd and even integers, respectively. The supplied record does not define or normalize $P_f$, so this background does not specify an additional mathematical assertion to prove.

Short English statement

For every normalized Hecke eigenform of full modular level, the conjecture says that its odd critical completed $L$-values and its even critical completed $L$-values each span a rational vector space whose dimension equals the complex dimension of the cusp-form space of the same weight. The completion convention requires source confirmation. The supplied record labels the problem open and reports an upper bound; these are supplied assessments, not independently verified conclusions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006591$, title: Rational Dimensions of Critical Modular $L$-Values; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2026. No individual proposer is identified.
• Supplied report citation: Multiple Zeta Values and Modular Forms (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/7, pp. 433--486. Source URL and supplied literature-source link, labelled Oberwolfach Report 7/2026: https://doi.org/10.4171/owr/2026/7. The excerpt labels the assertion Conjecture 1; no precise page within the report is supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open, reports only the upper bound as proved in the report, and describes equality as a folklore conjecture tied to the unproved Maeda conjecture. This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified and states that the formulation was reconstructed and checked against the official report. That verification claim is part of the supplied record; no source visit or independent verification is claimed here.