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Ranks of Elliptic Curves Twisted by Quadratic Forms

This paper proves that for any elliptic curve over $\mathbb{Q}$ , there exist infinitely many quadratic twists by sums of two squares that have rank 1, a result established through the analysis of moments of derivatives of modular $L$ -functions and which yields new insights into specific elliptic fibrations.

Original authors: Mohammad H. Hamdar, Cihan Sabuncu

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohammad H. Hamdar, Cihan Sabuncu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical, wiggly shape called an Elliptic Curve. In the world of math, these shapes are like secret codes. One of the biggest mysteries in math is figuring out how many "hidden keys" (called the rank) are needed to unlock all the secrets inside the shape.

Usually, these shapes are stubborn. Sometimes they have zero keys, sometimes one, sometimes many. Mathematicians have a big theory called the Birch and Swinnerton-Dyer conjecture that says if you look at the shape's "sound wave" (its L-function), you can tell exactly how many keys it has. If the sound wave hits a specific note perfectly, the shape has a certain number of keys.

The Great Twist

Now, imagine you can take this shape and give it a "twist." Think of it like putting a different colored lens on a pair of glasses. You can twist the shape using a special number, let's call it $d$ . When you twist the shape with $d$ , it becomes a new shape, $E_d$ .

The big question the authors, Mohammad H. Hamdar and Cihan Sabuncu, asked was: Can we find a twist $d$ that makes the shape have exactly one key?

But here's the catch: they didn't just want any twist. They wanted twists that are sums of two squares.
Think of a number like 5. You can write 5 as $1^2 + 2^2$ (1 plus 4). That's a sum of two squares. Numbers like 3 or 7 cannot be written this way. The authors wanted to find infinitely many numbers $d$ that are sums of two squares, which, when used to twist the curve, result in a shape with exactly one key (rank 1).

The Problem with the "Old Map"

Before this paper, a mathematician named Munshi tried to solve a similar puzzle. He looked at twists that are sums of two squares, but his tools were a bit like a flashlight that only shines on the very top of a mountain. He could see the "completed" version of the sound wave, but he missed the lower parts—the derivatives (the slopes and changes in the wave).

The authors say: "We need to look at the lower parts of the wave, the parts Munshi left in the dark."

The New Tool: The "Voronoi" Net

To find these special twists, the authors built a new mathematical net. They used something called the Voronoi summation formula.

The Analogy: Imagine you are trying to count how many fish are in a huge, dark ocean. You can't see them all.
- Munshi's method was like throwing a net that only catches the biggest fish at the very surface.
- The authors' method is like using a special sonar that bounces off the fish and creates a "ghost image" on the other side of the ocean. This ghost image is a two-dimensional Gauss sum. It's a bit like seeing a reflection in a funhouse mirror that is twice as long as the real thing, but it reveals the fish that were hiding deep down.

By using this "ghost image" technique, they could analyze the lower derivatives of the sound wave (the $L'$ function) instead of just the main wave.

The Big Discovery

After doing some very heavy lifting with complex math (involving things like Bessel functions and Fourier transforms), they proved a solid fact:

There are infinitely many numbers $d$ that are sums of two squares such that the twisted curve $E_d$ has a rank of 1.

This means there is an endless supply of these special "two-square" twists that unlock exactly one key.

What They Didn't Do (and What They Ruled Out)

It is important to know what this paper is not saying:

They did not prove this for every elliptic curve. They proved it specifically for curves where the "root number" (a specific sign in the math equation) is 1, AND the conductor (a measure of the curve's complexity) is not a perfect square. If the conductor is a perfect square, their formula says the answer is zero, so they ruled out that specific case for this method.
They did not just guess or simulate this. They provided a rigorous proof. They didn't run a computer simulation to say "it looks like it works"; they used logic to show it must work.
They did not solve the whole Birch and Swinnerton-Dyer conjecture. They just took a big step forward for a specific family of twists.

The Takeaway

Think of the elliptic curve as a locked treasure chest. The "twist" is a key. The authors showed that if you look for keys made of "two-square numbers," you will never run out of keys that open the chest to reveal exactly one secret. They did this by inventing a new way to listen to the chest's "sound wave," catching the whispers that previous methods missed.

They didn't just suggest it might be true; they proved it. And while they didn't solve the entire mystery of elliptic curves, they definitely found a whole new room in the mansion of math that was previously dark and empty.

Technical Summary: Ranks of Elliptic Curves Twisted by Quadratic Forms

Problem Statement
The paper addresses the problem of establishing the existence of infinitely many quadratic twists $E_d$ of a given elliptic curve $E$ over $\mathbb{Q}$ that possess algebraic rank 1. Specifically, the authors focus on twists where the twisting parameter $d$ is a sum of two squares ( $d = a^2 + b^2$ ). This family of twists is non-linear and corresponds to the study of the elliptic surface defined by $(1+t^2)y^2 = f(x)$ . While previous works (e.g., Coates–Li–Tian–Zhai) have established results for twists with specific prime factorizations, and Munshi [16] studied the family of sums of two squares, Munshi's methods yielded asymptotics for higher derivatives of the completed $L$ -function but failed to provide results on the ranks of $E_d$ for this specific family in the non-Complex Multiplication (non-CM) case. The central challenge is to prove that the first derivative of the $L$ -function, $L'(1/2, E \otimes \chi_d)$ , does not vanish for infinitely many such $d$ with the correct root number ( $w(E \otimes \chi_d) = -1$ ).

Methodology
The proof follows the general strategy of Soundararajan and Young [23], utilizing the approximate functional equation for the derivative of the twisted $L$ -function. The core technical innovation lies in the treatment of the weight function associated with the representation of integers as sums of two squares, denoted by $r(d)$ .

Approximate Functional Equation: The authors express $L'(1/2, f \otimes \chi_d)$ as a sum over Fourier coefficients $\lambda_f(n)$ weighted by a smooth function $W$ . The sum is split into a "short range" (where $n$ is small) and a "long range."
Voronoi Summation for Sums of Two Squares: For the short range, the authors apply a Voronoi summation formula specifically adapted for the function $r(d)$ (Proposition 8). Unlike standard Poisson summation, this formula introduces a dual sum involving two-dimensional Gauss sums $G(k_1, k_2)(n)$ . This step effectively doubles the length of the dual sum, making the analysis resemble a second-moment problem rather than a first-moment one.
Analysis of Gauss Sums: A significant portion of the technical work (Sections 3) is dedicated to analyzing the properties of these two-dimensional Gauss sums. The authors establish their multiplicativity (Proposition 10) and derive explicit local bounds (Lemma 13). They utilize the Voronoi formula to transform the sum over $d$ into a sum over frequencies $(k_1, k_2)$ .
Main Term and Error Terms:
- The main term arises from the zero frequency $(k_1, k_2) = (0, 0)$ . This yields an asymptotic formula involving the symmetric square $L$ -function $L(1, \text{sym}^2 f)$ and a constant $C$ dependent on the root number and level.
- The off-diagonal terms (non-zero frequencies) are handled by expressing the Bessel transform of the weight function via Mellin inversion. The resulting sums are bounded using Li's unconditional estimate [15] for the second moment of twisted $L$ -functions.
- The long range contribution is controlled via Cauchy–Schwarz and results from Zhou [26].

Key Results
The paper establishes the following asymptotic formula for the first moment of the derivative of the twisted $L$ -function over the family of square-free integers $d$ that are sums of two squares:

$\sum_{\substack{d \text{ SF} \\ (d,2N)=1 \\ w(f \otimes \chi_{8d})=-1}} r(d) L'(1/2, f \otimes \chi_{8d}) \psi\left(\frac{8d}{X}\right) = C X \log X + O(X (\log X)^{1/2} (\log \log X)^3)$

where $C$ is a constant explicitly defined in terms of the Fourier transform of the test function $\psi$ , the symmetric square $L$ -function, and local factors. Crucially, the authors show that $C \neq 0$ provided the root number $w(f) = -1$ (or $w(f)=1$ and $N$ is not a perfect square).

Corollary on Ranks
As a direct consequence of the non-vanishing of the main term, the authors prove Corollary 2: Let $E$ be an elliptic curve over $\mathbb{Q}$ of odd conductor $N$ . If the global root number $w(E) = 1$ (and $N$ is not a perfect square), there exist infinitely many fundamental discriminants $d = a^2 + b^2$ such that the twist $E_d$ has algebraic rank 1.

Significance and Claims
The paper claims to be one of the first results to successfully handle lower derivatives (specifically the first derivative) of $L$ -functions within the family of twists by sums of two squares, a gap left by Munshi's earlier work [16] which focused on higher derivatives of the completed $L$ -function.

The authors highlight two primary contributions:

Extension to Non-CM Cases: While Munshi [17] obtained rank results for this family in the CM case, this paper extends such results to the general non-CM case.
Methodological Distinction: The authors note that their approach differs substantially from concurrent work by Huang and Wei [6]. While Huang and Wei use Mellin inversion to study averages of Dirichlet $L$ -functions, this paper employs a Voronoi summation formula for sums of two squares, leading to the analysis of two-dimensional Gauss sums. The authors suggest that the Voronoi formula established in Proposition 8 and the properties of the associated Gauss sums may be of independent interest.

The results provide concrete information regarding the generic rank and Picard number of the elliptic surface $(1+t^2)y^2 = f(x)$ , linking analytic properties of $L$ -functions to the geometry of elliptic fibrations.