[Paper Review] Classifying Brumer's quintic polynomials by weak Mordell-Weil groups

Masanari Kida, Yūichi Rikuna+1 more|ArXiv.org|Feb 1, 2008

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper develops a classification framework for Brumer’s dihedral quintic polynomials using Kummer theory over rational elliptic curves with a rational 5-isogeny. By associating each polynomial to an elliptic curve whose Mordell-Weil group encodes equivalent splitting fields, the authors show that infinitely many parameter pairs yield isomorphic splitting fields, generalizing prior results on parameter transformations and unramified extensions.

ABSTRACT

We develop a general classification theory for Brumer's dihedral quintic polynomials by means of Kummer theory arising from certain elliptic curves. We also give a similar theory for cubic polynomials.

Motivation & Objective

• To develop a general classification theory for Brumer’s dihedral quintic polynomials using Kummer theory arising from elliptic curves with rational 5-isogenies.
• To extend the classification method to generic cubic polynomials via analogous Kummer-theoretic constructions.
• To explain and generalize previous results on parameter transformations and unramified extensions in dihedral quintic fields.
• To establish a systematic link between rational points on elliptic curves and isomorphism classes of splitting fields of Brumer’s polynomials.

Proposed method

• Associate each Brumer’s quintic polynomial to an elliptic curve $ E_{a,b} $ defined over $ \mathbb{Q} $, with a rational 5-isogeny and a rational point $ P_0 $ of infinite order.
• Use the Mordell-Weil group $ E_{a,b}(\mathbb{Q}) $ to classify parameter pairs $ (\alpha, \beta) $ such that $ \mathrm{Bru}(\alpha, \beta; X) $ has the same splitting field as $ \mathrm{Bru}(a, b; X) $.
• Apply Kummer theory to the isogeny $ \phi: E_{a,b} \to E_{a,b}^* $, using the dual isogeny $ \phi^* $ to relate rational points to unramified extensions.
• Construct a correspondence between subgroups of $ E_{a,b}(\mathbb{Q}) / \phi^*(E_{a,b}^*(\mathbb{Q})) $ and isomorphism classes of splitting fields.
• Use the structure of the Mordell-Weil group modulo the image of the dual isogeny to enumerate unramified extensions and parameter transformations.
• Extend the method to cubic polynomials by constructing analogous elliptic curves with rational 3-isogenies and analyzing their Mordell-Weil groups.

Experimental results

Research questions

• RQ1 How can the set of parameter pairs $ (a,b) $ yielding isomorphic splitting fields for Brumer’s quintic polynomial be systematically classified?
• RQ2 What role does the Mordell-Weil group of an associated elliptic curve play in encoding equivalence of splitting fields?
• RQ3 Can Kummer theory over elliptic curves with rational isogenies be used to generate infinitely many equivalent parameter pairs for a given splitting field?
• RQ4 To what extent does this method generalize to other Galois groups, such as $ \mathcal{D}_3 $ for cubic polynomials?

Key findings

• For any given Brumer’s quintic polynomial $ \mathrm{Bru}(a,b;X) $, there are infinitely many distinct parameter pairs $ (\alpha, \beta) $ such that $ \mathrm{Bru}(\alpha, \beta; X) $ has the same splitting field, as guaranteed by the infinite order of the rational point $ P_0 $ on $ E_{a,b} $.
• The splitting field of $ \mathrm{Bru}(1, x(P); X) $ is isomorphic to that of $ \mathrm{Bru}(1,0;X) $ for all rational points $ P \in \mathcal{E}(\mathbb{Q}) $ with $ x(P) \neq 0 $, and there are infinitely many such points on the elliptic curve $ \mathcal{E}: 47y^2 = 4x^3 + 28x^2 + 24x + 47 $.
• The Mordell-Weil group $ E_{-3321607}(\mathbb{Q}) $ is isomorphic to $ \mathbb{Z}^{\oplus 6} $, and the quotient $ E_{-3321607}(\mathbb{Q}) / \phi^*(E_{-3321607}^*(\mathbb{Q})) $ is isomorphic to $ (\mathbb{Z}/3\mathbb{Z})^{\oplus 3} $, which has 13 subgroups of order 3, each corresponding to an unramified cubic extension over $ \mathbb{Q}(\sqrt{-3321607}) $.
• The 13 unramified cubic extensions correspond to the 13 isomorphism classes of monic irreducible cubic polynomials with discriminant $ -3321607 $, all arising from the Kummer construction over the elliptic curve.
• The method generalizes to cubic polynomials: for a given discriminant $ D $, the number of isomorphism classes of cubic splitting fields equals the number of subgroups of order 3 in the quotient of the Mordell-Weil group modulo the image of the dual isogeny.
• The construction yields a dihedral septic polynomial over $ \mathbb{Q}(a,b) $ via a 7-isogeny on a curve with a 7-torsion point, though its genericity for $ \mathcal{D}_7 $-extensions remains unproven.