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 $$Q(\vec {x}) :=x_1^2+x_2^2+x_3^2+x_4^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>4</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> with congruence conditions on $$x_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> 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 $$2^k, k\le 7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> , $$3^{\ell }, \ell \le 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>3</mml:mn> <mml:mi>ℓ</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> 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 $$Q(\vec {x}) = p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> and the number of $$\mathbb {F}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -rational points on the associated elliptic curve under certain congruence conditions on p .