Orthogonal low-lying zeros and forced-zero deflation

Definitions and mathematical statement

The AIM question is "Local statistics for zeros for $L$-functions of orthogonal type," literally attributed in the source to "Chantal Davis." Its verification identifies workshop organizer Chantal David and her March 2014 Arizona Winter School notes as the likely intended attribution, explicitly calling this a source-level typo rather than an extraction error. No family, scaling, or limiting regime is fixed in the original.

The attached report interprets the question as low-lying zero statistics in orthogonal function-field families, for example quadratic twists of a fixed elliptic curve over ${\mathbb{F}}_q(t)$. It says existing fixed-$q$ one-level results with Fourier support in $(-1,1)$, including later trace/lower-order refinements, do not distinguish $O$, $SO(\mathrm{even})$, and $SO(\mathrm{odd})$ when the forced central zero is counted. Precise family hypotheses and references are not supplied.

Its candidate finite-rank formula concerns Haar orthogonal matrices. Choose with probability one half a matrix from $SO(2N)$, and with probability one half from $SO(2N+1)$, deleting the odd matrix's forced $+1$ eigenvalue. Both resulting multisets have $2N$ eigenangles. For an even trigonometric polynomial \[ F(\theta)=f_0+2\sum_{k=1}^Kf_k\cos(k\theta),\qquad0\leq K\leq2N-2, \] the report asserts \[ {\mathbb{E}}\sum_{\theta_j}F(\theta_j) =2Nf_0+\sum_{k=1}^K(-1)^kf_k. \] The stable Fourier range and equal mixture weights are part of the statement.

For an even test function $\varphi$ with Fourier convention $\widehat\varphi(u)=\int_{{\mathbb{R}}}\varphi(x)e^{-2\pi iux}\,dx$, take the canonical periodization \[ F_N(\theta)=\sum_{m\in{\mathbb{Z}}}\varphi\!\left(\frac{2N}{2\pi}(\theta+2\pi m)\right),\qquad f_k=\frac1{2N}\widehat\varphi\!\left(\frac{k}{2N}\right). \] Under regularity and Fourier-support assumptions making this a polynomial in the preceding range, the claimed expectation is \[ \widehat\varphi(0)+\mathcal L_N,\qquad \mathcal L_N=\frac1{2N}\sum_{k=1}^K(-1)^k\widehat\varphi\!\left(\frac{k}{2N}\right),\qquad |\mathcal L_N|\leq\frac{\|\widehat\varphi'\|_1}{2N}. \] Thus the deflated balanced ensemble has the unitary leading term with a controlled alternating leakage. Adding back the forced eigenvalue contributes $F_N(0)/2$ at finite rank; in the usual decaying-test large-$N$ regime this recovers the reported orthogonal law $\widehat\varphi(0)+\varphi(0)/2$. The limiting law is not substituted for an exact finite-rank identity. Nor does the matrix identity alone establish a zero-statistics theorem for an unspecified $L$-function family.

Short English statement

Study orthogonal low-lying zeros with explicit treatment of forced central zeros. The report proposes an exact even/deflated-odd matrix formula and a controlled Fourier correction explaining why balanced deflated data can have a unitary leading term.

Sources

• AIM workshop "Arithmetic statistics over finite fields and function fields," Question 29: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf. The PDF literally says "Chantal Davis," according to the supplied verification; no PDF was accessed here.
• Local canonical registry aim-algebraic-number-theory-notes.json, notes[58], tag question; permanent UnsolvedMath ID $20000407$, source code catalogue entry. Original remarks and literature field are empty, and the clean statement is absent.
• Supplied verification cites the official workshop page/summary identifying Chantal David and her March 2014 Arizona Winter School notes on orthogonal one-level densities; no URLs or full bibliographic entries for those sources are supplied.
• Supplied research attempt 1 proposes the finite-rank formula and leakage bound. It is machine generated, not peer reviewed, and requires independent verification before citation; full citations for the existing function-field density results are absent.