Spectral Curves with Complex Multiplication in Hermitian Matrix Models

Abstract: 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 $\mathbb{Z}$. This establishes a direct connection between number-theoretic structures and the spectral data of random matrix ensembles.