Filtration along Modular Theta Cycles

Definitions and mathematical statement

Let $f$ denote the modular form in the supplied question, and let $\tau$ denote its theta operator. For the indices occurring in the source, write \[ f_0=f,\qquad f_j=\tau(f_{j-1})=\tau^j(f)\quad(1\leq j\leq p). \] The source calls $(f_1,\ldots,f_p)$ the theta cycle of $f$. Write $\operatorname{Fil}(g)$ for the filtration of a form $g$, provided that this filtration is defined in the intended modular-form theory. This notation names the source's filtration; its definition and value space are not supplied.

The requested determination is the behavior of filtration under successive applications of $\tau$, namely the sequence \[ \operatorname{Fil}(f),\quad \operatorname{Fil}(\tau(f)),\quad\ldots,\quad \operatorname{Fil}(\tau^p(f)), \] under the hypotheses defining the intended theta cycle. The source does not specify those hypotheses, the modular-form space, the characteristic, the meaning of $p$, the theta operator, or the filtration. It also does not state a periodicity identity or the treatment of vanishing iterates. Consequently, this is an unresolved formulation of a determination problem, rather than a fully specified universally quantified assertion. No particular filtration convention, numerical weight-change rule, or cycle-closing identity is imposed here.

The supplied background places the question in the context of unitary Shimura varieties. Its dated assessment distinguishes a signature $(n,1)$ case, for which it attributes an explicit theta-cycle filtration formula to de Shalit and Goren, from a broader unitary setting whose general filtration behavior it labels open. The original question does not specify how these settings delimit its intended scope.

Short English statement

Determine how the filtration of a modular form changes through the source's theta cycle $\tau(f),\ldots,\tau^p(f)$, including its relation to the filtration of $f$. The underlying modular-form setting, theta operator, filtration, and cycle conditions require clarification before a precise general statement can be given.

Sources

• Supplied local problem registry, permanent UnsolvedMath ID $30004058$, title "Filtration along Modular Theta Cycles," exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The registry records proposed year 2019 and supplies no individual proposer.
• Supplied report citation: Arithmetic of Shimura Varieties (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/2, pp. 65--131. https://doi.org/10.4171/owr/2019/2. This is the supplied report-level provenance; no exact problem page or passage is supplied.
• Supplied literature reference: de Shalit--Goren, theta cycles. https://doi.org/10.2140/ant.2019.13.1829.
• Supplied literature reference: La Porta, generalized theta operators. https://arxiv.org/abs/2209.03717.
• Supplied literature reference: Eischen--Mantovan, entire theta operators. https://academic.oup.com/imrn/article/2022/21/16405/6326546.
• Literature assessment checked at 2026-08-22: the supplied dated curation labels the problem partially solved, attributes an explicit theta-cycle filtration formula in signature $(n,1)$ to de Shalit and Goren, and describes later work as constructing further weight-shifting operators while general filtration behavior across unitary Shimura varieties remains open. This assessment is supplied dated curation derived from the OWR material, not independently verified here; its time-sensitive status and cited claims require independent verification.