**G. Lomadze**

## On Entire Modular Forms of Half-Integral weight on
the Congruence Subgroup $\Gm_0(4N)$

**Abstract:**

Some entire modular forms of weight $\frac{s}{2}$ on the congruence subgroup
$\Gm_0(4N)$ are constructed if $s$ is odd $>11$. The constructed type of entire
modular forms is useful for revealing the arithmetical meaning of additional
terms in formulas for the number of representations of positive integers by
positive quadratic forms with integral coefficients if the number $s$ of
variables is odd $>11$.

**Keywords:**

Entire modular form, multiplier system, congruence subgroup, theta-function with
characteristics.

**MSC 2000:** 11E20, 11E25.