Siegel Eisenstein series of level two and its applications

Abstract

In this paper, we construct a holomorphic Siegel modular form of weight 2 and level 2, and compute its Fourier coefficients explicitly. Moreover, we prove that this modular form equals the generating function of the representative number [Formula: see text] associated with the maximal order in the quaternion algebra [Formula: see text]. As a corollary, we can give a new proof of the famous formula for the sums of three squares. As applications, we give an explicit formula for the numbers of solutions of two systems of Diophantine equations related with Sun’s “1-3-5 conjecture”. Furthermore, we show that “a perfect square” in the integral condition version of Sun’s conjecture can be replaced by “a power of 4”.