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 $k\le 7$), $3^{\mathrm{\ell }}$ (with $\mathrm{\ell }\le 3$), 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 $p$.