# Automorphic form

A meromorphic function on a bounded domain $D$ of the complex space $\mathbf C ^ {n}$ that, for some discrete group of transformations $\Gamma$ operating on this domain, satisfies an equation:

$$f ( \gamma ( x ) ) = j _ \gamma ^ {-m} ( x ) f ( x ) , \ x \in D , \gamma \in \Gamma ,$$

Here $j _ \gamma (x)$ is the Jacobian of the mapping $x \rightarrow \gamma (x)$ and $m$ is an integer known as the weight of the automorphic form. If the group $\Gamma$ acts fixed-point free, then automorphic forms define differential forms on the quotient space $D / \Gamma$ and vice versa. Automorphic forms may be used in the construction of non-trivial automorphic functions (cf. Automorphic function). It has been shown that if $g(x)$ is a function that is holomorphic and bounded on a domain $D$, then the series

$$\sum _ {\gamma \in \Gamma } g( \gamma (x)) j _ \gamma ^ {m} (x)$$

converges for large values of $m$, thus representing a non-trivial automorphic function of weight $m$. These series are called Poincaré theta-series.

The classical definition of automorphic forms, given above, has recently served as the starting point of a far-reaching generalization in the theory of discrete subgroups of Lie groups and adèle groups.

#### References

 [1] H. Poincaré, , Oeuvres de H. Poincaré , Gauthier-Villars (1916–1965) [2] C.L. Siegel, "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen (1955)