A group theoretic perspective on entanglements of division fields

In this paper, we initiate a systematic study of entanglements division fields from group theoretic perspective. For positive integer n n and subgroup alttext="upper G subset-of-or-equal-to upper L 2 left-parenthesis double-struck Z slash n right-parenthesis"> G ⊆ L 2 ( Z / stretchy="false">) encoding="application/x-tex">G\subseteq GL_2( \mathbb {Z}/n\mathbb {Z}) with surjective determinant, provide definition for G"> encoding="application/x-tex">G to represent an alttext="left-parenthesis comma b a , b encoding="application/x-tex">(a,b) -entanglement give additional criteria explained or unexplained -entanglement. Using these new definitions, determine the tuples p q right-parenthesis T p q T encoding="application/x-tex">((p,q),T) , alttext="p greater-than element-of Z"> &gt; ∈<!-- ∈ encoding="application/x-tex">p&gt;q\in {Z} distinct primes T"> encoding="application/x-tex">T finite group, such that there are infinitely many non- mathvariant="double-struck">Q accent="false">¯<!-- ¯ </mml:mover> encoding="application/x-tex">\overline {\mathbb {Q}} -isomorphic elliptic curves over Q"> encoding="application/x-tex">\mathbb {Q} encoding="application/x-tex">(p,q) type . Furthermore, each possible combination entanglement level completely classify defined by constructing corresponding modular curve alttext="j"> j encoding="application/x-tex">j -map.