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Rational Points on Diagonal Cubic Surfaces

Kazuki Sato

TL;DR

This work proves that certain diagonal cubic surfaces over ${f Q}$ possess a rational point, assuming the finiteness of the Tate–Shafarevich group for elliptic curves over ${f Q}$. The authors develop a descent framework on the CM curve $E_A: x^3+y^3=Az^3$ over $k={f Q}( alpha)$, compute the $igl( rac{-3}{ullet}igr)$-Selmer groups $S(A)$ and their subgroups $C(A)$, and analyze local solubility via cubic residue symbols to deduce global solvability for specific residue classes of primes $p_i$ modulo $9$. They then leverage Swinnerton-Dyer-type adelic criteria to connect local data with global rational points, including a refined treatment at the prime $3$. The results illuminate how Selmer-group computations and SD-type criteria can yield explicit rational points on diagonal cubic surfaces under a Tate–Shafarevich finiteness hypothesis, and identify cases where finiteness assumptions can be weakened to focus on particular twists. The methods have potential broader impact on understanding the Hasse principle and Brauer–Manin obstruction for diagonal surfaces in number theory.

Abstract

We show under the assumption that the Tate-Shafarevich group of any elliptic curve over the rational numbers is finite that the cubic surface $x_1^3 + p_1p_2x_2^3 + p_2p_3x_3^3 + p_3p_1x_4^3 = 0$ has a rational point, where $p_1, p_2$ and $p_3$ are rational primes congruent to $2$ or $5$ modulo $9$.

Rational Points on Diagonal Cubic Surfaces

TL;DR

This work proves that certain diagonal cubic surfaces over possess a rational point, assuming the finiteness of the Tate–Shafarevich group for elliptic curves over . The authors develop a descent framework on the CM curve over , compute the -Selmer groups and their subgroups , and analyze local solubility via cubic residue symbols to deduce global solvability for specific residue classes of primes modulo . They then leverage Swinnerton-Dyer-type adelic criteria to connect local data with global rational points, including a refined treatment at the prime . The results illuminate how Selmer-group computations and SD-type criteria can yield explicit rational points on diagonal cubic surfaces under a Tate–Shafarevich finiteness hypothesis, and identify cases where finiteness assumptions can be weakened to focus on particular twists. The methods have potential broader impact on understanding the Hasse principle and Brauer–Manin obstruction for diagonal surfaces in number theory.

Abstract

We show under the assumption that the Tate-Shafarevich group of any elliptic curve over the rational numbers is finite that the cubic surface has a rational point, where and are rational primes congruent to or modulo .

Paper Structure

This paper contains 4 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Descent on a Elliptic Curve
  3. Calculation of Selmer groups
  4. Corollary of Swinnerton-Dyer's theorem

Key Result

Theorem 1

Let $p_1, p_2, p_3$ be rational primes such that $p_1,p_2, p_3 \equiv 2, 5 \bmod 9$, and a cubic surface over ${\mathbb Q}$. Assume that the Tate-Shafarevich group of any elliptic curve over ${\mathbb Q}$ is finite. Then $V({\mathbb Q}) \neq \emptyset$.

Theorems & Definitions (25)

  • Theorem 1: Theorem \ref{['mainthm']}
  • Remark 2
  • Lemma 1.1: bf
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 15 more