{"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/a-constructive-approach-to-zauner-s","title":"A Constructive Approach to Zauner's Conjecture via the Stark Conjectures","arxiv_id":"2501.03970","date":"2025-01-07","proceeding":null,"authors":["Marcus Appleby","Steven T Flammia","Gene S Kopp"],"abstract":"We propose a construction of $d^2$ complex equiangular lines in $\\mathbb{C}^d$, also known as SICPOVMs, conjectured by Zauner to exist for all d. The construction gives a putatively complete list of SICs with Weyl-Heisenberg symmetry in all dimensions d > 3. Specifically, we give an explicit expression for an object we call a ghost SIC, which is constructed from the real multiplication values of a special function and which is Galois conjugate to a SIC. The special function, the Shintani-Faddeev modular cocycle, is more precisely a tuple of meromorphic functions indexed by a congruence subgroup of ${\\rm SL}_2(\\mathbb{Z})$. We prove our construction gives a valid SIC in every case assuming two conjectures: the order 1 abelian Stark conjecture for real quadratic fields and a special value identity for the Shintani-Faddeev modular cocycle. The former allows us to prove that the ghost and the SIC are Galois conjugate over an extension of $\\mathbb{Q}(\\sqrt{\\Delta})$ where $\\Delta = (d+1)(d-3)$, while the latter allows us to prove idempotency of the presumptive fiducial projector. We provide computational tests of our SIC construction by cross-validating it with known solutions, particularly the extensive work of Scott and Grassl, and by constructing four numerical examples of nonequivalent SICs in d=100, three of which are new. We further consider rank-r generalizations called r-SICs given by maximal equichordal configurations of r-dimensional complex subspaces. We give similar conditional constructions for r-SICs for all r, d such that r(d-r) divides $(d^2-1)$. Finally, we study the structure of the field extensions conjecturally generated by the r-SICs. If K is any real quadratic field, then either every abelian Galois extension of K, or else every abelian extension for which 2 is unramified, is generated by our construction; the former holds for a positive density of field discriminants.","url_abs":"https://arxiv.org/abs/2501.03970v2","url_pdf":"https://arxiv.org/pdf/2501.03970v2.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":"a-constructive-approach-to-zauner-s","repo_url":"https://github.com/sflammia/zauner.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}