Theta functions

Theta functions are a special class of functions that arise in various areas of mathematics, including complex analysis, number theory, and algebraic geometry. They are particularly significant in the study of elliptic functions and modular forms.

Jacobi forms are a class of functions that arise in the context of several areas in mathematics, including number theory, algebraic geometry, and the theory of modular forms. They are particular kinds of quasi-modular forms that exhibit specific transformation properties under the action of certain groups.

The metaplectic group is a significant concept in the fields of mathematics, particularly in representation theory and the theory of symplectic geometry. It is a double cover of the symplectic group, which means that it serves as a sort of "two-fold" representation of the symplectic group, capturing additional structure that cannot be represented by the symplectic group alone.

The Schottky problem, often referred to in the context of number theory and algebraic geometry, is named after the mathematician Friedrich Schottky. It addresses questions related to the representation of certain algebraic structures, particularly in connection with the theory of abelian varieties and modular forms. In more specific terms, the Schottky problem can be framed as follows: it concerns the characterization of Jacobians of algebraic curves.

In the context of mathematics and specifically in the field of number theory, the term "Theta characteristic" often refers to a certain type of characteristic of a Riemann surface or algebraic curve that arises in the study of Abelian functions, Jacobi varieties, and the theory of divisors. 1. **Theta Functions**: Theta characteristics are closely related to theta functions, which are special functions used in various areas of mathematics, including complex analysis and algebraic geometry.

In mathematics, particularly in the theory of abelian varieties and algebraic geometry, a *Theta divisor* is a specific kind of divisor associated with a principally polarized abelian variety (PPAV). More formally, if \( A \) is an abelian variety and \( \Theta \) is a quasi-projective variety corresponding to a certain polarization, then the theta divisor \( \theta \) is defined as the zero locus of a section of a line bundle on \( A \).

The theta function of a lattice is a special type of mathematical function that arises in the context of complex analysis, number theory, and mathematical physics. Specifically, it is related to the theory of elliptic functions, modular forms, and can be used in various applications including statistical mechanics and string theory. A lattice in this context is typically defined as a discrete subgroup of the complex plane generated by two linearly independent complex numbers \( \omega_1 \) and \( \omega_2 \).