Integral Parahoric Comparison and Diagonal Bockstein Obstruction Cancellation at p = 41
Year: 2026
Type: preprint
Abstract: Let $p=41$ and consider the integral chamber-glued parahoric source $J_{\mathrm{geom}}$ and hyperspecial target lattice $L_K$ associated with a specified principal series of $GSp_4(Q_{41})$. We determine the $H$-fixed subspace of the residual kernel of the summation map $J_{\mathrm{geom}}/41J_{\mathrm{geom}}\to L_K/41L_K$, where $H$ is the elementary abelian Klingen unipotent quotient. We establish an integral comparison of the original 42-leaf source with its normalized presentation, a restricted Bockstein criterion, and an integral identification of the target relation lattice with a three-step intertwiner image. The source class in the 40-dimensional Levi constituent has two diagonal components. Their target contributions combine as $g_Y^{\mathrm{diag}}=g_Y^{(v=5)}+\Delta$ with $\Delta=-41e_2+41e_{1054}$ modulo $41^2$; although the isolated single-channel target is obstructed, the diagonal target has an explicit preimage under the relation matrix. Together with the corresponding $20$-dimensional constituent, this yields $V_{41}^{H}\cong X_{20}\oplus Y_{40},\qquad \dim_{F_{41}}V_{41}^{H}=60.$ The argument is confined to $p=41$. The integral source and Levi-module statements needed for this conclusion are proved within the appendices. Reproducible finite-ring computations and falsification tests are supplied separately as supporting evidence.
Source: Preprints.org
Author: Lin Tao
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225783605
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