Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
121 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A group theoretic perspective on entanglements of division fields (2008.09886v3)

Published 22 Aug 2020 in math.NT

Abstract: In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer $n$ and a subgroup $G\subseteq \text{GL}_2(\mathbb{Z}/{n}\mathbb{Z})$ with surjective determinant, we provide a definition for $G$ to represent an $(a,b)$-entanglement and give additional criteria for $G$ to represent an explained or unexplained $(a,b)$-entanglement. Using these new definitions, we determine the tuples $((p,q),T)$, with $p<q\in\mathbb{Z}$ distinct primes and $T$ a finite group, such that there are infinitely many non-$\bar{\mathbb{Q}}$-isomorphic elliptic curves over $\mathbb{Q}$ with an unexplained $(p,q)$-entanglement of type $T$. Furthermore, for each possible combination of entanglement level $(p,q)$ and type $T$, we completely classify the elliptic curves defined over $\mathbb{Q}$ with that combination by constructing the corresponding modular curve and $j$-map.

Summary

We haven't generated a summary for this paper yet.