Near coincidences and nilpotent division fields

2024/09/02 by Harris Daniels, Jeremy Rouse, Daniels, Harris +1 · 1 voice
Mathematics · #math.NT

paper · pdf · doi:10.48550/arxiv.2409.00881

Abstract

Let E/ℚ be an elliptic curve. We say that E has a near coincidence of level (n,m) if m | n and ℚ(E[n]) = ℚ(E[m],ζ_n). We classify near coincidences of prime power level and use this result to give a classification of values of n for which \rm Gal(ℚ(E[n])/ℚ) is a nilpotent group. Along the way we prove a Gauss-Wantzel analog for the elliptic curve E\colon y² = x³-x, showing that ℚ(E[n])/ℚ is constructible if and only if φ(n) is a power of 2. Assuming that there are no non-CM rational points on the modular curves X_ns⁺(p) for primes p > 11, we show that \rm Gal(ℚ(E[n])/ℚ) nilpotent implies that n is a power of 2 or n ∈ \ 3, 5, 6, 7, 15, 21 \.

Discussions

• Completely forget about central series!!! Look at the theorem here... The definition of nilpotent should be that it's a direct product of Sylow groups! That's so clearly important! The image is a sc [bsky, 13 points, 2 comments]

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