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

Abstract

We determine the sets ${\Phi }^{\infty }(n)$ 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 translates the problem into one about low-degree points on modular curves $X_1(m,n)$. We construct the infinite families using modular units, and eliminate the remaining candidates using finite-field gonality computations, covering arguments, and a specialization argument for $W_d^0$. The most difficult case is $X_1(37)$ in degree $9$, where the Jacobian has positive rank. We handle this case by showing that $W^0_9(X_1(37)_{\F_2})$ contains no translate of the positive-rank elliptic factor induced by the morphism $X_1(37)\to X_0^{+}(37)$.

Improved exposition. 12 pages