Theta function

Theta functions are complex functions defined by series whose terms involve quadratic exponents. Their transformation laws connect lattice sums, elliptic functions, and modular forms.

Jacobi theta functions: Four standard theta functions defined by series in a complex variable and a nome, with distinct parity and periodicity properties. They are the classical one-variable forms whose identities display the general family’s structure.
Complex lattice: A discrete subgroup of the complex plane generated by two linearly independent complex numbers. Its points supply the quadratic data summed over in elliptic theta functions.
Carl Gustav Jacob Jacobi: A German mathematician who developed foundational theories of elliptic functions and theta functions in the nineteenth century. Jacobi established the classical theta-function notation and many of its central identities.
Lattice Gaussian sum: A sum of Gaussian weights over the points of a lattice. Theta functions are generating functions for these sums and reveal their dependence on lattice shape.
Eisenstein series: Modular forms formed by summing reciprocal powers over integer lattice points or their modular-group counterparts. They are another lattice-based route to modular forms, with different weights and analytic behavior.
Theta characteristic: A discrete parameter specifying a theta function’s shift and phase, commonly written as a pair of vectors with entries modulo integers. Characteristics organize shifted theta series and determine their parity and transformation behavior.
Elliptic function: A meromorphic function on the complex plane that is periodic with respect to a rank-two lattice. Theta quotients construct elliptic functions with prescribed zeros and poles.
Niels Henrik Abel: A Norwegian mathematician whose work established foundational results on elliptic functions and algebraic equations. Abel’s investigations of elliptic integrals helped create the setting in which theta functions emerged.
Heat equation: A partial differential equation describing diffusion of temperature or other quantities through a medium. Theta series give fundamental solutions and periodic heat kernels.
Weierstrass elliptic function: A doubly periodic meromorphic function with a double pole, commonly used to describe elliptic curves analytically. It encodes elliptic behavior directly, while theta functions often construct it through logarithmic derivatives.

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