[Paper Review] Automorphic forms and cubic twists of elliptic curves

Daniel Lieman|ArXiv.org|Jul 12, 1994

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper establishes a connection between the cubic twist family of elliptic curves $E_D: x^3 + y^3 = D$ and metaplectic automorphic forms on the cubic cover of $\mathrm{GL}(3)$, showing that the Whittaker–Fourier coefficients of the form encode the $L$-series of the curves $E_{m^2n}$. The key result proves that for any prime $p \neq 3$ and residue class $c \mod p$, there are infinitely many $D \equiv c \mod p$ for which $E_D$ has no rational points.

ABSTRACT

This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the L--series of the curve E_{m^2n}. One may obtain information about the collective behavior the curves E_D by exploiting this connection; for example, one can prove: Theorem: Fix any prime p e 3, and any congruence class c mod p. Then there are infinitely many D congruent to c mod p such that the curve E_D has no rational solutions. This paper is fairly self-contained; no prior knowledge of algebraic number theory, analytic number theory or metaplectic forms is assumed. On the other hand, this paper is a survey, no proofs are included.

Motivation & Objective

• To investigate the collective arithmetic behavior of the family of elliptic curves $E_D: x^3 + y^3 = D$ under cubic twists.
• To establish a deep connection between these curves and automorphic forms on the cubic cover of $\mathrm{GL}(3)$, particularly via Whittaker–Fourier coefficients.
• To use this automorphic framework to derive global arithmetic properties of the curve family, such as the distribution of rational points.
• To prove that for any prime $p \neq 3$ and any residue class $c \mod p$, there are infinitely many $D \equiv c \mod p$ such that $E_D$ has no rational solutions.

Proposed method

• Utilizes metaplectic automorphic forms on the cubic cover of $\mathrm{GL}(3)$, which generalize classical modular forms to higher-rank groups with non-trivial central extensions.
• Relies on the fact that the $m,n^{\text{th}}$ Whittaker–Fourier coefficient of the metaplectic form corresponds to the $L$-series of the elliptic curve $E_{m^2n}$, linking automorphic data to arithmetic invariants.
• Applies the theory of cubic twists to parameterize the family of curves $E_D$ via $D \in \mathbb{Q}^\times / \mathbb{Q}^{\times 3}$, enabling a uniform treatment of their $L$-functions.
• Employs analytic techniques from automorphic forms and $L$-function theory to deduce global properties of the curve family without requiring explicit point computations.
• Assumes no prior knowledge of algebraic or analytic number theory, making the framework self-contained and accessible to non-experts.
• Uses the structure of the cubic cover to encode the arithmetic of the curves $E_D$ in the Fourier coefficients of a single automorphic object.

Experimental results

Research questions

• RQ1 How are the $L$-functions of the cubic twist family $E_D: x^3 + y^3 = D$ encoded in automorphic forms on the cubic cover of $\mathrm{GL}(3)$?
• RQ2 Can the collective behavior of rational points across the family $E_D$ be studied via automorphic methods?
• RQ3 What can be deduced about the distribution of $D$ modulo a prime $p \neq 3$ for which $E_D$ has no rational points?
• RQ4 Is there a uniform automorphic object whose Fourier coefficients reflect the arithmetic of all curves $E_{m^2n}$ in the family?
• RQ5 Can the non-vanishing or vanishing of $L$-values be linked to the existence of rational points on $E_D$ via this automorphic correspondence?

Key findings

• For any prime $p \neq 3$ and any residue class $c \mod p$, there are infinitely many integers $D \equiv c \mod p$ such that the curve $E_D: x^3 + y^3 = D$ has no rational points.
• The $m,n^{\text{th}}$ Whittaker–Fourier coefficient of a specific metaplectic form on the cubic cover of $\mathrm{GL}(3)$ is essentially the $L$-series of the elliptic curve $E_{m^2n}$, establishing a precise automorphic link.
• The automorphic form provides a uniform framework to study the entire family of cubic twists $E_D$, enabling collective arithmetic analysis.
• The method is self-contained and does not require prior expertise in algebraic or analytic number theory, making the connection accessible to a broad audience.
• The result demonstrates that the absence of rational points is not sporadic but occurs with positive density in arithmetic progressions modulo $p \neq 3$, under the given family parameterization.
• The connection between the cubic twist family and metaplectic forms reveals a hidden symmetry in the $L$-functions of these curves, suggesting deeper arithmetic-automorphic duality.