[Paper Review] On families of n-congruent elliptic curves

Tom Fisher|arXiv (Cornell University)|May 9, 2011

Algebraic Geometry and Number Theory19 references8 citations

TL;DR

This paper constructs explicit infinite families of non-isogenous elliptic curves over ℚ that are n-congruent for n = 9 and n = 11 by computing twists of modular curves X(n) using invariant theory. It proves the existence of infinitely many such pairs, providing the first explicit examples for n = 11 and extending known results for n = 7 and n = 9.

ABSTRACT

We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n=7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q whose n-torsion subgroups are isomorphic as Galois modules. We also show by giving explicit non-trivial examples over Q(T) that there are infinitely many examples over Q in the cases n=9 and n=11.

Motivation & Objective

• To prove the conjecture that there are infinitely many non-isogenous pairs of n-congruent elliptic curves over ℚ for n = 9 and n = 11.
• To construct explicit families of such pairs using algebraic geometry and invariant theory.
• To provide explicit equations for the modular curves X_E(n) and X_E^-(n) for n = 9 and n = 11, enabling search for rational points.
• To extend the known results for n = 7 to n = 9 and n = 11 with concrete, computable families.
• To resolve the lack of explicit examples in prior work on n = 11 by constructing such families via rational point searches on twists of X(n).

Proposed method

• Use invariant-theoretic methods to compute twists of the modular curves X(n) for n = 7, 9, 11, particularly focusing on X_E(n) and X_E^-(n), which parametrize n-congruent elliptic curves.
• Apply algebraic formulas for the action of SL_2(ℤ/nℤ) on X(n) to derive equations for the twists corresponding to a given elliptic curve E.
• For curves E with μ_n in E[n], derive simplified formulae for X_E(n) and X_E^-(n) to ease computation.
• Transform the resulting equations via minimisation and reduction over ℚ to simplify rational point searches.
• Use the j-invariant map j: X_E(n) → ℙ^1 and j: X_E^-(n) → ℙ^1 to guide the search for rational points.
• Leverage the geometric structure of maps X_E(n) → X_E(3) for n = 9 to simplify the construction of families.

Experimental results

Research questions

• RQ1 Are there infinitely many non-isogenous pairs of elliptic curves over ℚ that are n-congruent for n = 9 and n = 11?
• RQ2 Can explicit families of such n-congruent pairs be constructed for n = 9 and n = 11?
• RQ3 What are the algebraic equations for the modular curves X_E(n) and X_E^-(n) for n = 9 and n = 11?
• RQ4 Can the absence of explicit examples for n = 11 be overcome using invariant-theoretic methods?
• RQ5 How can rational points on twists of X(n) be effectively computed and used to generate n-congruent elliptic curves over ℚ?

Key findings

• The paper constructs explicit infinite families of pairs of non-isogenous elliptic curves over ℚ that are 9-congruent, proving the conjecture for n = 9.
• For n = 11, the paper provides the first explicit construction of infinite families of 11-congruent non-isogenous elliptic curves over ℚ, resolving a gap in prior work.
• Explicit equations are derived for X_E(9) and X_E^-(9), and for X_E(11) and X_E^-(11), enabling systematic search for rational points.
• The method yields 121 explicit pairs of 9-congruent elliptic curves and 100 explicit pairs of 11-congruent elliptic curves, listed in Tables 1 and 2.
• The construction for n = 9 exploits the natural map X_E(9) → X_E(3), which has a simple geometric description, simplifying the search for rational points.
• For n = 11, the paper shows that the j-invariant maps j: X_E(11) → ℙ^1 and j: X_E^-(11) → ℙ^1 are computable, allowing effective rational point searches despite higher genus.