Alladi Ramakrishnan Hall

3-Selmer groups, ideal class groups and the rational cube sum problem

Pratiksha Shingavekar

IIT Madras

Given an elliptic curve E over a number field F and an isogeny phi of E defined over F, the study of the phi-Selmer group has a copious history. Let E/Q be an elliptic curve with a rational 3-isogeny phi. In this talk, we give an upper and a lower bound on the rank of the phi-Selmer group of E over Q(\sqrt(−3)) in terms of the 3-part of the ideal class group of certain quadratic extension of Q(\sqrt(−3)). As an application of our bounds, we are able to produce an infinite family of elliptic curves having arbitrary large 3-Selmer rank over Q(\sqrt(−3)). We also exhibit infinitely many imaginary quadratic and biquadratic fields having class numbers divisible by 3. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. This Diophantine problem asks: which integers can be written as a sum of cubes two rational numbers? This is a joint work with Dipramit Majumdar and Somnath Jha.