Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions

In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular, non-holomorphic Eisenstein series of weight $2$, we construct a basis for the Eisenstein space for levels $2^k$ (with $k \le 7$), $3^{\ell}$ (with $\ell \le 3$), and $p$, where $p>3$ is an odd prime. Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp-form part of the theta series corresponding to $Q$, we establish a relation between the number of integer solutions to the equation $Q(\mathbf{x}) = p$ and the number of $\mathbb{F}_p$-rational points on the associated elliptic curve under certain congruence conditions on $p$.