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A "modified" variant of the Weierstrass sigma, zeta, and elliptic functionsis proposed whereby the zeta function is redefined by $\zeta(z)$ $\mapsto$$\tilde \zeta(z)$ $\equiv$ $\zeta(z) - \gamma_2z$, where $\gamma_2$ is alattice invariant related to the almost-holomorphic modular invariant of thequasi-modular-invariant weight-2 Eisenstein series. If $\omega_i$ is aprimitive half-period, $\tilde\zeta(\omega_i)$ = $\pi \omega_i^*/A$, where $A$is the area of the primitive cell of the lattice. The quasiperiodicity of themodified sigma function is much simpler than that of the original, and itbecomes the building block for the modular-invariant formulation oflowest-Landau-level wavefunctions on the torus. It is suggested that the"modified" sigma function is more natural than the original Weierstrass form,which was formulated before quasi-modular forms were understood. For thehigh-symmetry (square and hexagonal) lattices, the modified and original sigmafunctions coincide.

A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus