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

1. Problem Statement

The paper addresses the classical problem of determining the number of integer solutions to quadratic forms under specific congruence conditions. Specifically, it focuses on the quadratic form $Q(\vec{x}) = x_1^2 + x_2^2 + x_3^2 + x_4^4$ where the variables $x_i$ are constrained by congruences modulo $M$ (where $M \in \{2, 3, 4, 6\}$).

The core objective is to establish a precise, linear relationship between:

1. The number of representations of an integer $n$ by these congruent quadratic forms (denoted as $r_{s,M}(n)$).
2. The number of rational points on specific elliptic curves over finite fields $\mathbb{F}_{p^k}$.

While the relationship between standard theta series and modular forms is well-known, this paper investigates how congruent theta series (theta series with congruence conditions) decompose into Eisenstein series and cusp forms, and how the cusp form components relate to the arithmetic of elliptic curves with Complex Multiplication (CM).

2. Methodology

The author employs a multi-step methodology rooted in the theory of modular forms:

• Congruent Theta Series: The paper defines the theta series associated with the congruent quadratic form $Q_{s,M}$ as:
  $$\theta_{s,M}(\tau) = \sum_{\substack{\vec{x} \in \mathbb{Z}^4 \\ x_i \equiv s \pmod M}} q^{x_1^2+x_2^2+x_3^2+x_4^2}$$
  These series are shown to be modular forms of weight 2 on specific congruence subgroups $\Gamma_0(N) \cap \Gamma_1(M)$.

• Operator Construction (S and V): To analyze these forms, the author utilizes two specific operators introduced by Bringmann and Kane:

  • Sieving Operator ($S_{M,m}$): Extracts Fourier coefficients $c(n)$ where $n \equiv m \pmod M$.
  • Scaling Operator ($V_d$): Maps $f(\tau) \to f(d\tau)$.
    These operators are applied to the non-holomorphic Eisenstein series $\widehat{E}_2(\tau)$ to construct a basis for the Eisenstein space $E_2(\Gamma_0(p^k))$ for levels $2^k$ ($k \le 7$), $3^\ell$ ($\ell \le 3$), and odd primes $p$.

• Decomposition: The congruent theta series $\theta_{s,M}$ is decomposed into an Eisenstein series part ($E_{s,M}$) and a cusp form part ($f_{s,M}$):
  $$\theta_{s,M} = E_{s,M} + f_{s,M}$$
  The Eisenstein part is explicitly calculated using the constructed basis and values at the cusps. The cusp form part is identified with specific Hecke eigenforms.

• Elliptic Curve Connection: The cusp forms identified (specifically related to $\eta(6\tau)^4$) are linked to the elliptic curve $E: y^2 = x^3 + 1$ defined over $\mathbb{Q}$. This curve has Complex Multiplication by $\mathbb{Q}(\sqrt{-3})$. The Fourier coefficients of the cusp form correspond to the trace of Frobenius on $E$.

3. Key Contributions

A. Construction of Eisenstein Bases

The paper provides a constructive method to generate bases for $E_2(\Gamma_0(p^k))$ using $S$ and $V$ operators applied to $\widehat{E}_2$.

• Theorem 1.4: Establishes that for $p=2$ and $k \le 3$, and $p=3$ and $k \le 1$, these operators generate a full basis. For higher powers or other primes, the basis is incomplete, requiring additional elements.
• The author provides explicit lists of these basis elements in Appendices A and B, including their Fourier expansions and values at cusps.

B. Decomposition of Congruent Theta Series

The paper explicitly decomposes $\theta_{s,M}$ for $M \in \{2, 3, 4, 6\}$:

• For many cases (e.g., $\theta_{0,2}, \theta_{1,2}, \theta_{0,4}$), the cusp form part vanishes, and the solution count is purely determined by divisor sums (Eisenstein series).
• For the case $\theta_{1,3}$ (representations of $n \equiv 1 \pmod 3$), the series contains a non-trivial cusp form part proportional to $\eta(6\tau)^4$.

C. Point Counting Relations

The most significant contribution is the derivation of exact formulas linking the number of solutions $r_{1,3}(p)$ to the number of points $N_p(k)$ on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_{p^k}$.

Theorem 1.7: For primes $p \equiv 1 \pmod 6$:
$$r_{1,3}(p) = \frac{1}{3} N_p(1)$$
where $N_p(1)$ is the number of $\mathbb{F}_p$-rational points on $E$ (including the point at infinity).

Generalization (Theorem 1.9): The paper extends this to composite numbers $n = \prod p_i^{\alpha_i} \equiv 1 \pmod 6$, providing a recursive decomposition of $r_{1,3}(n)$ involving the Fourier coefficients of the cusp form and divisor sums.

4. Key Results

1. Explicit Formulas for Solution Counts:
   The paper derives closed-form expressions for $r_{s,M}(n)$ for various $s, M$. For example, for $n \equiv 1 \pmod 6$:
   $$r_{1,3}(n) = \frac{1}{3} \left( \sigma_1(n) - a(n) \right)$$
   where $\sigma_1(n)$ is the sum of divisors and $a(n)$ is the $n$-th Fourier coefficient of the cusp form $\eta(6\tau)^4$.

2. Three-Term Recurrence:
   A three-term relation is established connecting $r_{1,3}(p^k)$, $N_p(k)$, and previous terms:
   $$3r_{1,3}(p^k) = N_p(k) + 3p \cdot r_{1,3}(p^{k-2})$$
   This allows the calculation of solution counts for prime powers using point counts on the elliptic curve.

3. Divisibility Properties:
   It is proven that for $p \equiv 1 \pmod 6$, $N_p(1)$ is divisible by 12. This follows from the fact that $r_{1,3}(p)$ is an integer and the relation $3r_{1,3}(p) = N_p(1)$.

4. Generalization to Other Forms:
   Table 1 and Table 2 in the paper list similar relations for other quadratic forms (e.g., $x_1^2 + x_2^2 + 2x_3^2 + 2x_4^2$) and their associated elliptic curves (e.g., $y^2 = x^3 + 4x$), demonstrating that this phenomenon is not isolated to the specific form $x_1^2 + \dots + x_4^2$.

5. Significance

• Bridging Number Theory and Arithmetic Geometry: The paper provides a concrete computational bridge between the combinatorial problem of counting integer solutions to quadratic forms with congruence conditions and the geometric problem of counting points on elliptic curves.
• Algorithmic Utility: By expressing solution counts in terms of elliptic curve point counts (which can be computed efficiently using Schoof's algorithm or similar methods), the paper offers a new, potentially more efficient way to calculate $r_{s,M}(n)$ for large $n$, bypassing direct enumeration.
• Modular Forms Theory: The work advances the understanding of congruent theta series by explicitly constructing bases for Eisenstein spaces at higher levels using $S$ and $V$ operators, a technique that can be applied to other modular form problems.
• CM Connections: It highlights the role of Complex Multiplication (specifically by $\mathbb{Q}(\sqrt{-3})$) in determining the arithmetic properties of these specific quadratic forms, reinforcing the deep connection between CM elliptic curves and theta series.

In summary, Mondal's work successfully translates the problem of counting constrained quadratic representations into the language of modular forms and elliptic curves, yielding explicit formulas and recursive relations that unify these distinct areas of number theory.