{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/iwasawa-theory-for-p-torsion-class-group","title":"Iwasawa Theory for $p$-torsion Class Group Schemes in Characteristic $p$","arxiv_id":"2107.12555","date":"2021-07-27","proceeding":null,"authors":["Jeremy Booher","Bryden Cais"],"abstract":"We investigate a novel geometric Iwasawa theory for $\\mathbf{Z}_p$-extensions of function fields over a perfect field $k$ of characteristic $p>0$ by replacing the usual study of $p$-torsion in class groups with the study of $p$-torsion class group schemes. That is, if $\\cdots \\to X_2 \\to X_1 \\to X_0$ is the tower of curves over $k$ associated to a $\\mathbf{Z}_p$-extension of function fields totally ramified over a finite non-empty set of places, we investigate the growth of the $p$-torsion group scheme in the Jacobian of $X_n$ as $n\\rightarrow \\infty$. By Dieudonn\\'e theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator $V$. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$-module structure of the space $M_n:=H^0(X_n, \\Omega^1_{X_n/k})$ of global regular differential forms as $n\\rightarrow \\infty.$ For example, for each tower in a basic class of $\\mathbf{Z}_p$-towers we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + \\lambda_r n + c_r(n)$ for all $n$ sufficiently large, where $a_r, \\lambda_r$ are rational constants and $c_r : \\mathbf{Z}/m_r \\mathbf{Z} \\to \\mathbf{Q}$ is a periodic function, depending on $r$ and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on $\\mathbf{Z}_p$-towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$.","url_abs":"https://arxiv.org/abs/2107.12555v2","url_pdf":"https://arxiv.org/pdf/2107.12555v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"iwasawa-theory-for-p-torsion-class-group","repo_url":"https://github.com/jeremybooher/MAGMA-Towers","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}