  A "modified" variant of the Weierstrass sigma, zeta, and elliptic functions
is proposed whereby the zeta function is redefined by $\zeta(z)$ $\mapsto$
$\tilde \zeta(z)$ $\equiv$ $\zeta(z) - \gamma_2z$, where $\gamma_2$ is a
lattice invariant related to the almost-holomorphic modular invariant of the
quasi-modular-invariant weight-2 Eisenstein series. If $\omega_i$ is a
primitive 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 the
modified sigma function is much simpler than that of the original, and it
becomes the building block for the modular-invariant formulation of
lowest-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 the
high-symmetry (square and hexagonal) lattices, the modified and original sigma
functions coincide.
