Imaginary quadratic fields F with X0(15)(F) finite

Abstract

Caraiani and Newton have proven that if F is an imaginary quadratic number field such that $X_0\left(15\right)$ has rank 0 over F, then every elliptic curve over F is modular. This paper is concerned with the quadratic fields $F=Q\left(\sqrt{-p}\right)$ 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 $Q$ on the quadratic twist $X_0{\left(15\right)}_{-p}$. 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

. In fact we prove a more general connection between higher descents for elliptic curves which seems interesting in its own right.