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The present paper discusses various mathematical aspects about the rapiditysymmetry in chiral Potts model (CPM) in the context of algebraic geometry andgroup theory . We re-analyze the symmetry group of a rapidity curve in$N$-state CPM, explore the universal group structure for all $N$, and furtherenlarge it to modular symmetries of the complete rapidity family in CPM. Aswill be shown in the article that all rapidity curves in $N$-state CPMconstitute a Fermat hypersurface in $\PZ^3$ of degree 2N as the naturalgeneralization of the Fermat K3 elliptic surface $(N=2)$, we conduct a thoroughalgebraic geometry study about the rapidity fibration of Fermat surface and itsreduced hyperelliptic fibration via techniques in algebraic surface theory.Symmetries of rapidity family in CPM and hyperelliptic family in$\tau^{(2)}$-model are exhibited through the geometrical representation of theuniversal structural group in mathematics.

Fermat Surface and Group Theory in Symmetry of Rapidity Family in Chiral Potts Model