Arithmetic Dijkgraaf-Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas

2023/08/10 by Yuqi Deng, Deng, Yuqi, Riku Kurimaru +3
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2308.05467

Abstract

We compute Hirano's formula for the mod 2 arithmetic Dijkgraaf-Witten invariant Z_k for the ring of integers of the quadratic field k=ℚ(√(p₁⋯ p_r)), where p_i's are distinct prime numbers with p_i ≡ 1 \pmod4, and give a simple formula for Z_k in terms of the graph obtained from quadratic residues among p₁,⋯, p_r. Our result answers the question posed by Ken Ono. We also give a density formula for mod 2 arithmetic Dijkgraaf-Witten invariants.

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