Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions
Koustav Mondal
TL;DR
The paper develops a bridge between congruent representations by the quaternary form $Q(oldsymbol{x})=x_1^2+x_2^2+x_3^2+x_4^2$ and elliptic-curve point counts over finite fields. It builds explicit weight-2 Eisenstein bases for levels $2^k$, $3^{ u}$, and odd primes using Sieving and V-operators, then analyzes the corresponding congruent theta series $ heta_{k,M}$ for $M=2,4,3,6$, obtaining exact formulas for the representation counts $r_{s,M}(n)$ in terms of divisor sums and cusp contributions. A central result connects $r_{1,3}(p^k)$ to $N_p(k)$ for the CM elliptic curve $E: y^2=x^3+1$, with a three-term relation and generalizations to higher exponents and composite $n$, via the CM cusp form $ ext{η}(6 au)^4$. The work thus provides a concrete modular-form framework to translate quadratic-form representations with congruence conditions into elliptic-curve point-count data, enabling arithmetic conclusions and Grössencharacter interpretations for higher exponents.
Abstract
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 $k \le 7$), $3^{\ell}$ (with $\ell \le 3$), 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 $p$.