Torsion groups of elliptic curves that appear infinitely often over septic, octic and nonic fields

Abstract

We determine the sets $Φ^{\infty}(n)$ of abelian groups that appear as torsion groups of infinitely many elliptic curves, up to $\overline \Q$-isomorphism, over number fields of degree $n=7,8$ and $9$. The proof follows the strategy of Derickx and Sutherland for degrees 5 and 6: we reduce the problem to low-degree points on the modular curves $X_1(m,n)$, construct the required maps using modular units, and eliminate the remaining candidates using gonality computations over finite fields and maps to lower-genus modular curves. The curve \(X_1(37)\) in degree \(9\) requires an additional argument: we show that \(W^0_9(X_1(37))\) contains no translate of the positive-rank elliptic factor of \(J_1(37)\) arising from the quotient \(X_1(37)\to X_0^+(37)\simeq 37.a1\).