Representations Functions of Definite Binary Quadratic Forms over `\mathbb{F}_q[t]`

Wei-Chen Yao

Department of Mathematics

University of Taipei

yao@Utaipei.edu.tw

    Let `\mathbb{F}_q` be a finite field of odd characteristic. For `A,B,C\in\mathbb{F}_q[t]`, a binary quadratic form `f(x,y)=Ax^2+Bxy+Cy^2` is called definite if the discriminant `\mathcal{D}=B^2-4AC` has either odd degree or has even degree and non-square leading coefficient in `\mathbb{F}_q`. For `m\in\mathbb{F}_q[t]` and `f` is a definite binary quadratic form, we define `N(f,m)` by the number of representations of `m` in `f`. Let `\{f_1,\cdots, f_h,\cdots,f_H\}` be a representative set of properly equivalent classes of definite binary quadratic forms for given discriminant `\mathcal{D}` and `f_1,\cdots,f_h` are primitive. We define

`R(\mathcal{D},m)=\frac{1}{2}\sum_{i=1}^{H}N(f_i,m)\quad\text{and}\quad r(\mathcal{D}, m)=\frac{1}{2}\sum_{i=1}^{h}N(f_i, m).`

In this talk, we will discuss properties of these functions and present formulas for these functions.