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Spectral Curves with Complex Multiplication in Hermitian Matrix Models
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DOI：10.1016/j.nuclphysb.2026.117582.png)
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Abstract
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We show that elliptic curves with complex multiplication (CM) naturally emerge in the spectral geometry of Hermitian one-matrix models in the two-cut phase. Focusing on a symmetric quartic potential, we derive the corresponding genus-one spectral curve and compute its modular j-invariant in closed form as a function of the quartic coupling g. We identify specific values of g for which the elliptic curve exhibits CM, i.e., its endomorphism ring is larger than Z . This establishes a direct connection between number-theoretic structures and the spectral data of random matrix ensembles.
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