Weak Siegel-Weil formula for 𝕄₂(ℚ) and arithmetic on quaternions

We prove a weak version of the Siegel-Weil formula on SL 2 \operatorname {SL}_2 for the dual pair ( SL 2 , O 2 , 2 ) (\operatorname {SL}_2, O_{2, 2}) , where O 2 , 2 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.