2024/03/05 by Tori Day, Rylan Gajek-Leonard, Day, Tori +1
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic and Geometric Analysis #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Primary 11G05 #Secondary 11F80
paper · pdf · doi:10.48550/arxiv.2403.03073
openalex publication_date 2024/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix distinct primes p and q and let E be an elliptic curve defined over a number field K. The (p,q)-entanglement type of E over K is the isomorphism class of the group Gal(K(E[p])∩ K(E[q])/K). The size of this group measures the extent to which the image of the mod pq Galois representation attached to E fails to be a direct product of the mod p and mod q images. In this article, we study how the (p,q)-entanglement group varies over different base fields. We prove that for each prime ℓ dividing the greatest common divisor of the size of the mod p and q images, there are infinitely many fields L/K such that the entanglement over L is cyclic of order ℓ. We also classify all possible (2,q)-entanglement types that can occur as the base field L varies.