Lower Bounds for Function-Field Quaternionic Periods

Definitions and mathematical statement

Put $k=\mathbb F_q(t)$, where $q$ is a fixed prime power, and fix a quaternion algebra $D$ over $k$. For a quadratic extension $F/k$, write $D_F=D\otimes_k F$, and let $g$ denote the genus of $F$. For a global function field $L$, let $\mathbb A_L$ denote its adele ring. Write $G_k=D^\times$, $G_F=D_F^\times$, and let $Z_F$ be the center of $G_F$, so that $Z_F(\mathbb A_F)=\mathbb A_F^\times$. The extension $F/k$ induces the inclusion of adelic groups used below.

Let $K\subseteq G_F(\mathbb A_F)$ be a compact subgroup, and put \[ H_K=K\cap G_k(\mathbb A_k). \] The supplied record defines \[ m=[M:H_K], \] where $M$ is a maximal compact subgroup of $G_k(\mathbb A_k)$ containing $H_K$. The choice of $M$ and the finiteness of this index are not specified. In particular, compactness of $K$ alone does not supply the finite-index hypothesis needed to use $m$ as a positive integer parameter.

Consider a normalized Hecke eigenfunction $f$ on \[ X_{F,K}=G_F(F)\backslash G_F(\mathbb A_F)\big/\left(KZ_F(\mathbb A_F)\right). \] The source does not specify the normalization, the Hecke operators, or the automorphic function space from which $f$ is taken. Its proposed period is \[ P_F(f)=\int_{G_k(k)\backslash G_k(\mathbb A_k)}f(x)\,\mathrm d\mu(x), \] where $f$ is restricted along the adelic inclusion. The measure $\mu$, its normalization, and convergence of this integral are not specified. The displayed integration domain has not been divided by the center; it is retained exactly as supplied.

Supplied lower-bound question. With $q$ and $D$ fixed, does there exist a constant $C$, depending only on $q$ and $D$, such that, as $F/k$ varies over quadratic extensions of genus $g$, $K$ varies over the stated compact subgroups, and $f$ varies over every normalized Hecke eigenfunction on $X_{F,K}$, one has \[ P_F(f)\stackrel{?}{=}\Omega\!\left(m^{-C}q^{g/2}\right)? \] This is the supplied target, with unresolved qualifications. In particular, a complex-valued period has no specified ordered lower bound, and the source does not state whether an absolute value is intended. Under an absolute-value interpretation with fully uniform implied constants, the proposed inequality would take the form \[ |P_F(f)|\ge c\,m^{-C}q^{g/2}, \] with $c>0$ independent of $F,K,f$. That interpretation is not adopted as a correction: the intended phase convention or absolute value, the dependence of $c$, and whether the bound applies for every genus or only beyond a uniform genus threshold remain unspecified. The source also does not impose a nonvanishing or distinction condition on $f$. These issues prevent an unqualified precise inequality from being extracted.

The supplied dated curation describes this as open and says that the source passage checks only the split standard-level case, while the proposed universal bound over varying extensions and compact subgroups remains unproved. The record does not specify that special case sufficiently to formulate a separate verified assertion.

Short English statement

For a fixed quaternion algebra over $\mathbb F_q(t)$, the supplied question asks for a uniform lower bound of size $m^{-C}q^{g/2}$ for periods of every normalized Hecke eigenfunction over varying quadratic extensions. A precise formulation still requires the intended absolute value or phase convention, normalization of functions and measures, convergence of the period, a finite level index, and the scope of the asymptotic quantifiers. The supplied open status and split standard-level qualification are dated curation requiring independent review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json: permanent ID $30004500$, title "Lower Bounds for Function-Field Quaternionic Periods," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. This draft uses the supplied complete registry record.
• Supplied report citation: Automorphic Forms and Arithmetic (2021), Oberwolfach Reports, pp. 1317--1383, DOI 10.4171/owr/2020/26. Supplied source URL and literature-source link, labeled "Oberwolfach Report 26/2020": https://doi.org/10.4171/owr/2020/26. The record gives proposed year 2020 and does not identify an individual proposer.
• Literature assessment dated exactly 2026-08-22: the supplied status is "open." The supplied assessment says that the function-field lower-bound question is checked only in the split standard-level case in the source passage, whereas the proposed universal bound over varying extensions and compact subgroups remains unproved. This is supplied dated curation, not an independently verified literature assessment; its mathematical claims and time-sensitive status require independent verification.
• The registry labels its statement corrected_verified and reports that the official report was checked to restore the quaternion algebra, varying quadratic extension, genus, compact subgroup, index, automorphic quotient, and parameter dependence. This is a supplied verification claim, not verification performed for this draft. The supplied citation uses the year 2021, while the report label and DOI use 2020; both are retained.