Three-Selmer groups for elliptic curves with 3-torsion

Abstract

We consider a specific family of elliptic curves with rational 3-torsion subgroup. We arithmetically define 3-Selmer groups through isogeny and 3-descent maps, then associate the image of the 3-descent maps to solutions of homogeneous cubic polynomials affiliated with the elliptic curve E and an isogenous curve E′. Thanks to the work of Cohen and Pazuki, we have solubility conditions for the homogeneous polynomials. Using these conditions, we give a graphical approach to computing the size of 3-Selmer groups. Finally, we translate the conditions on graphs into a question concerning ranks of matrices and give an upper bound for the rank of the elliptic curve E by calculating the size of the Selmer groups.