What question did this study set out to answer?

The research investigates the relationship between the number of integer solutions to a quadratic form and the point-counting of elliptic curves over finite fields.

February 19, 2026Open Access

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

Key Points

The research investigates the relationship between the number of integer solutions to a quadratic form and the point-counting of elliptic curves over finite fields.
Analyzed theta series related to the quadratic form.
Applied operators on modular, non-holomorphic Eisenstein series of weight 2.
Constructed a basis for Eisenstein space at specific levels, including powers of 2 and 3.
Examined the connection between the trace of Frobenius on elliptic curves and Fourier coefficients of cusp forms.
Established a relationship between integer solutions of the quadratic form and the number of rational points on elliptic curves under congruence conditions.
Identified specific cases for prime moduli and congruences to enhance understanding of the point-counting problem.

Abstract

Abstract In this paper, we analyze the theta series associated to the quadratic form Q (x): =x₁²+x₂²+x₃²+x₄² Q (x →): = x 1 2 + x 2 2 + x 3 2 + x 4 2 with congruence conditions on xᵢ 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 2ᵏ, k 7 2 k, k ≤ 7, 3^, 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 (x) = p Q (x →) = p and the number of Fₚ F p -rational points on the associated elliptic curve under certain congruence conditions on p.

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