Group, Lie and Number Theory Seminar

Given nonzero integers d_1, d_2 and an elliptic curve E/Q, the Galois module E^{d_1d_2}[4] can be written as a subquotient of the Galois module (E \oplus E^{d_1} \oplus E^{d_2})[4], where E^d denotes E twisted by the quadratic character associated to Q(\sqrt{d}). This observation extends the fact that E[2] is isomorphic to E^{d_1}[2] and can be extended to 8-torsion, 16-torsion, etc. After applying this result to partially control the 2^k-Selmer groups in some k-dimensional hypercubes of twists, we will show how the combinatorial trick from Talk II can be adapted to give the distribution of 2^k-Selmer ranks in grids of twists. Speaker(s): Alexander Smith (Harvard University)