What does this research mean for the field?

The Steinitz-conductor identity h * Nm(A) = f holds for the self-intersection degree h of the exotic Weil class on CM abelian fourfolds across all tested pairs of imaginary quadratic fields and cyclic cubic conductors, with zero exceptions. Novelty: ClaimNovelty.CONFIRMATORY. Consensus alignment: ConsensusAlignment.SUPPORTS_CONSENSUS.

What question did this study set out to answer?

The primary aim is to verify the Steinitz-conductor identity for self-intersection degrees of Weil classes on CM abelian fourfolds across various pairs of quadratic fields and conductors.

February 25, 2026Open Access

Self-Intersection Patterns Beyond Cyclic Cubics (Foundations of Constructive Relativity, Paper 65)

Key Points

The primary aim is to verify the Steinitz-conductor identity for self-intersection degrees of Weil classes on CM abelian fourfolds across various pairs of quadratic fields and conductors.
Utilized integer arithmetic for calculations of class numbers and eigenforms.
Examined 1,220 pairs of imaginary quadratic fields and cyclic cubic conductors.
Determined conditions for free lattices versus those requiring Steinitz twists.
Tested properties of non-cyclic cubics to identify symmetry requirements.
Confirmed the identity h * Nm(A) = f across all tested pairs without exceptions.
Identified 738 pairs with free lattices and 482 requiring a Steinitz twist.
Established the importance of prime factors' inertia in determining lattice behavior.
Verified that non-cyclic cubics do not follow the scalar identity in the tested cases.

Abstract

Description: Paper 65 of the Foundations of Constructive Relativity series. MAIN RESULT: We verify the Steinitz-conductor identity h * Nm (A) = f for the self-intersection degree h of the exotic Weil class on CM abelian fourfolds A₊, ₅, where K is an imaginary quadratic field, F is a totally real cyclic cubic of conductor f, and A is the Steinitz ideal class of the rank-1 OK-Hermitian Weil lattice Wᵢnt. The identity is confirmed across all 1, 220 pairs (K, F) with K = Q (sqrt (-d) ), d 1). The Steinitz twist is forced if and only if f is not represented by the principal binary quadratic form of K. When all prime factors of f are inert in K, the lattice is necessarily free. For non-cyclic (S₃) cubics, the scalar identity breaks: h² = disc (F) never holds among 216 tested pairs, confirming that Z/3Z Galois symmetry is essential. This extends the h = f identity of Papers 56-57 (Heegner fields) and the Steinitz correction of Paper 58 to all imaginary quadratic fields with class number up to 20. The computation uses purely integer arithmetic (class numbers via reduced binary quadratic forms, cyclic cubics via algebraic discriminant solving). CRM level: BISH + WLPO. Package contains: LaTeX source and PDF (10 pages), self-contained Python computation script, all generated CSV/JSON data files, and 4 PNG plots. Access right: Open Access License: Creative Commons Attribution 4. 0 International (CC BY 4. 0) Language: English Keywords (one per line): CM abelian varietiesSteinitz classself-intersection degreecyclic cubicsconductorimaginary quadratic fieldsclass numberbinary quadratic formsconstructive mathematicsWLPO Subjects: Algebraic geometryMathematical logic and foundations Related identifiers: DOI 10. 5281/zenodo. 17054050 | relation: isPartOf | type: publication (parent series DOI) DOI 10. 5281/zenodo. 18737090 | relation: isNewVersionOf | type: publication (preceded by Paper 64) Additional notes: AI-assisted writing using Claude (Anthropic, Opus 4. 6) under human direction. All mathematical content specified by the author. Computation is self-contained Python with purely integer arithmetic (no external APIs). No Lean formalization for this paper.

Self-Intersection Patterns Beyond Cyclic Cubics (Foundations of Constructive Relativity, Paper 65)

Key Points

Abstract

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