[Paper Review] Galois group at Galois point for genus-one curve

Mitsunori Kanazawa, Hisao Yoshihara|arXiv (Cornell University)|May 9, 2011

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper classifies all possible Galois groups at outer Galois points for genus-one curves with singular points, showing that abelian Galois groups have order at most nine, while non-abelian groups can be unbounded. It provides explicit defining equations for curves when the Galois group is abelian, using field extensions induced by projections from Galois points and analyzing automorphism groups of elliptic curves via lattice structures and group actions.

ABSTRACT

We show all the possible structures of finite subgroups of the automorphism groups of elliptic curves. Here the automorphism means the biholomorphic transformation. Using the result, we determine every Galois group G at outer Galois point for genus-one curve with singular points. In particular, if G is abelian, then its order is at most nine. However if not so, the order is unbounded. Furthermore, we give every defining equation of the curve with the Galois point in the case where G is abelian.

Motivation & Objective

• To classify all finite subgroups of the automorphism group of elliptic curves.
• To determine the structure of Galois groups at outer Galois points for genus-one curves with singular points.
• To provide explicit defining equations for such curves when the Galois group is abelian.
• To investigate the boundedness of Galois group orders, particularly distinguishing between abelian and non-abelian cases.
• To extend known results on Galois points from smooth curves to singular genus-one curves.

Proposed method

• Analyzes the automorphism group of elliptic curves via the semi-direct product structure of translations and complex multiplications.
• Uses the exact sequence $1 \to T(E) \to A(E) \to A(E)_0 \to 1$ to decompose Galois groups into translation and multiplier components.
• Applies field extension techniques: for a Galois point $P$, the extension $k(C)/\bar{\pi}_P^*(k(\mathbb{P}^1))$ is analyzed to determine Galois group $G$.
• Constructs defining equations by finding invariants under group actions, e.g., using $s = u^2$ and $t = 1/y$ to generate function fields.
• Employs explicit rational parametrizations and divisor computations to verify field equality $\mathbb{C}(x,y) = \mathbb{C}(s,t)$.
• Uses examples with specific curves like $y^2 = x^3 + x$, $y^2 = x^3 + 1$, and $y^2 = x(x-1)(x-b)$ to derive equations for abelian Galois groups.

Experimental results

Research questions

• RQ1 What are all possible finite subgroups of the automorphism group of an elliptic curve over $\mathbb{C}$?
• RQ2 For a genus-one curve with singular points, what are the possible structures of the Galois group at an outer Galois point?
• RQ3 If the Galois group is abelian, what is the maximum possible order, and what are the defining equations of the curve?
• RQ4 Can non-abelian Galois groups at Galois points be unbounded in order?
• RQ5 How can one explicitly construct defining equations for genus-one curves with abelian Galois groups at Galois points?

Key findings

• The Galois group at an outer Galois point for a genus-one curve with singular points is either abelian of order at most nine or non-abelian with unbounded order.
• All possible finite subgroups of the automorphism group of an elliptic curve are classified, with $A(E)_0$ being cyclic of order 2, 4, or 6.
• For abelian Galois groups, the paper provides explicit defining equations in terms of invariants like $s = u^2$ and $t = 1/y$, such as $s(s-2)^2y^4 = (y^4 + s)^2$ for $G \cong \mathbb{Z}_2 \oplus \mathbb{Z}_4$.
• The group $G$ is a semi-direct product $G_T \rtimes G_0$, where $G_T$ is the translation part and $G_0$ is the multiplier part.
• For $G \cong \mathbb{Z}_3^2$, the defining equation is $s^2x^6 = (x^3 + 1)(x^3 - 8)^2$, derived from the invariant $s = -y(y^2 - 9)/(y^2 - 1)$.
• In the case $G \cong \mathbb{Z}_2^3$, the defining equation is $(4t^4 + 5st^2 - 1)^2 = st^2(s + 8t^2)^2$, with $t = 1/y$ and $s = u^2$, $u = (x^2 - 1)/y$.