max-ab-extn-contained-in-div-flds

This repository contains code and data associated to the paper "Maximal abelian extensions contained in division fields of elliptic curves over $\mathbb{Q}$ with complex multiplication", by Asimina S. Hamakiotes, arXiv:2408.16164 (2024).

The ComputeMaxAbExtn.m file contains code that does the following:

• Given an elliptic curve $E/\mathbb{Q}$ with CM, ComputeTwist(E) returns the twisting factor $\alpha$ of $E$ such that $E^\alpha$ is one of the curves in Table 1 of the paper, and returns the curve that it is a twist of.
• Given an elliptic curve $E/\mathbb{Q}$ with CM and a prime $p$, MaxAbExtn(E,p) returns the maximal abelian extension contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$ for $n \geq 2$.
• VerifyAllExamples() verifies all of the examples in Section 2 of the paper.

The VerifyComputations.m file contains code that verifies the computations done in Section 5 of the paper, as well as the code used for the proof of Lemma 5.2. The file has the following:

• Theorem 1.1: VerifyThm11SplitIndex1Cases() and VerifyThm11SplitIndex3Case() verify the computations done for images that are contained in the normalizer of the split Cartan and have index 1 and 3, respectively; VerifyThm11NonSplitIndex1Cases() and VerifyThm11NonSplitIndex3Case() verify the computations done for images that are contained in the normalizer of the non-split Cartan and have index 1 and 3, respectively.
• Theorem 1.2: VerifyThm12Index1() and VerifyThm12Index2() verify the computations done for images that have index 1 and index 2 in the normalizer of Cartan, respectively.
• Theorem 1.3: VerifyThm13Index1() and VerifyThm13Index2() verify the computations done for images that have index 1 and 2 in the normalizer of Cartan, respectively; VerifyThm13Index3and6() verifies the computations done for images that have index 3 and 6 in the normalizer of Cartan.
• Proposition 5.3: This is part of Theorem 1.4.(a); VerifyProp53Split() and VerifyProp53NonSplit() verify the computations done for images that are contained in the normalizer of the split and non-split Cartan, respectively.
• Proposition 5.4: This is part of Theorem 1.4.(a); VerifyProp54Index1() and VerifyProp54Index3() verify the images that have index 1 and 3 in the normalizer of Cartan, respectively.
• Theorem 1.4.(b): VerifyThm14bi() verifies the computations done for $\Delta_Kf^2 = -12, -28$; VerifyThm14bii() verifies the computations done for $\Delta_Kf^2 = -4$; VerifyThm14biiiIndex1() and VerifyThm14biiiIndex2() verify the computations done for $\Delta_Kf^2 = -8, -16$.
• VerifyAllComputations() verifies all the computations mentioned above that are used in Section 5 of the paper.
• Lemma 5.2: VerifyLemma52() contains the proof of Lemma 5.2.