Imaginary quadratic fields $F$ with $X_0(15)(F)$ finite

Abstract

Caraiani and Newton have proven that if $F$ is an imaginary quadratic number field such that $X_0(15)$ has rank $0$ over $F$, then every elliptic curve over $F$ is modular. This paper is concerned with the quadratic fields $F=\mathbb{Q}(\sqrt{-p})$ for a prime number $p$. We give explicit conditions on $p$ under which the rank is $0$, and prove that these conditions are satisfied for $87,5\%$ of the primes for which the rank is expected to be even based on the parity conjecture. We also show these conditions are satisfied if and only if rank $0$ follows from a $4$-descent over $\mathbb{Q}$ on the quadratic twist $X_0(15)_{-p}$. To prove this, we perform two consecutive $2$-descents and prove this gives rank bounds equivalent to those obtained from a $4$-descent using visualisation techniques for $\mathrm{Sha}[2]$. In fact we prove a more general connection between higher descents for elliptic curves which seems interesting in its own right.