For an elliptic curve $E$ defined over a number field $K$ and $L/K$ a Galois extension, Akbary and Murty introduced the notion of minimal subfields related to the rank of the curve. In this short talk we will discuss some new results in the analytic side of things, extending their work and applying it to generalize a theorem of Kolyvagin, investigating how the order of vanishing of the corresponding $L$-function at $s=1$ changes, as the rank increases by 1.

Bio: Samprit Ghosh is a Postdoctoral Fellow at the University of Calgary. He earned his doctorate from University of Toronto last year (2023), under the supervision of Prof Kumar Murty. His primary area of research is analytic number theory and his thesis was on Higher Euler-Kronecker constants.