In this paper, we analyze the theta series associated to the quadratic form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Q(\vec {x}) :=x_1^2+x_2^2+x_3^2+x_4^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mi>x</mi> <mo stretchy="false">→</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mn>3</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mn>4</mn> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with congruence conditions on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$x_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2^k, k\le 7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> <mo>,</mo> <mi>k</mi> <mo>≤</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$3^{\ell }, \ell \le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>3</mn> <mi>ℓ</mi> </msup> <mo>,</mo> <mi>ℓ</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Q(\vec {x}) = p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mover accent="true"> <mi>x</mi> <mo stretchy="false">→</mo> </mover> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> and the number of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {F}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-rational points on the associated elliptic curve under certain congruence conditions on p.

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

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2026年2月16日

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Koustav Mondal

摘要

In this paper, we analyze the theta series associated to the quadratic form $Q(\vec {x}) :=x_1^2+x_2^2+x_3^2+x_4^2$ $Q (\vec{x}) : = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}$ 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 $2^k, k\le 7$ $2^{k}, k \leq 7$ , $3^{\ell }, \ell \le 3$ $3^{ℓ}, ℓ \leq 3$ 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 $Q(\vec {x}) = p$ $Q (\vec{x}) = p$ and the number of $\mathbb {F}_p$ $F_{p}$ -rational points on the associated elliptic curve under certain congruence conditions on p.