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

Abstract In this paper, we analyze the theta series associated to the quadratic form $$Q(\vec {x}) :=x_1^2+x_2^2+x_3^2+x_4^2$$ Q ( x → ) : = x 1 2 + x 2 2 + x 3 2 + x 4 2 with congruence conditions on $$x_i$$ 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 Eisenstein space for levels $$2^k, k\le 7$$ 2 k , k ≤ 7 , $$3^{\ell }, \ell \le 3$$ 3 ℓ , ℓ ≤ 3 and p , for odd prime p . Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp form part of theta series corresponding to Q , we establish relation between the number of integer solutions to the equation $$Q(\vec {x}) = p$$ Q ( x → ) = p and the number of $$\mathbb {F}_p$$ F p -rational points on the associated elliptic curve under certain congruence conditions on p .