## Abstract

A "modified" variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by ζ(z) ζ(z) ≡ ζ(z)-γ2z, where γ_{2} is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ω_{i} is a primitive half-period, ζ (ωi) = πωi/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.

Original language | English (US) |
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Article number | 071901 |

Journal | Journal of Mathematical Physics |

Volume | 59 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2018 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics