Weak Siegel-Weil formula for $\mathbb{M}_{2}(\mathbb{Q})$ and arithmetic on quaternions

We prove a weak version of the Siegel-Weil formula on $\operatorname {SL}_2$ for the dual pair $(\operatorname {SL}_2, O_{2, 2})$, where $O_{2, 2}$ is the split orthogonal group. By this formula and the Siegel-Weil formula, we give explicit formulas for Hecke correspondence's degree and average representation numbers over genus associated to Eichler orders. At last, we give explicit formulas for representations of a number as sums of three squares and four squares by local Whittaker functions, and it turns out that these functions are exactly the local factors of Hardy's singular series.