FoCM 2014 conference
Workshop A3 - Computational Number Theory
December 12, 15:50 ~ 16:30 - Room A21
Computing twists of Shioda modular surfaces of level 4 related to visibility of Sha
Nils Bruin
Simon Fraser University, Canada - nbruin@sfu.ca
One of the most mysterious objects associated to an elliptic curve $E$ is its Tate-Shafarevich group $Sha(E)$. Its elements can be represented by classes in the Galois-cohomology group $H^1(Q,E[n])$, for various $n$.
If two distinct elliptic curves $E$ and $E'$ have isomorphic $n$-torsion, then a single class $\xi$ in $H^1(Q,E[n])$ can represent a trivial element in $Sha(E)$ and a non-trivial one in $Sha(E')$. In the terminology of Mazur, the element of $Sha(E')$ is made {\it visible} by $E$. Mazur showed that for $n=3$, all elements of $Sha$ can be made visible. In general, the question translates into whether a rational point lies on a certain twist of the Shioda modular surface, obtained by taking the universal elliptic curve over the modular curve $X(n)$ of full level $n$.
The case $n=4$ is particularly interesting. The curve $X(4)$ is rational, but the relevant surface over it is not. It is a K3 surface. Further complications in determining the correct surface arise from the fact that 4 is even. We will discuss how to compute a model of the relevant surface given $\xi$ and give some examples of the various obstructions to rational points that can arise on these surfaces.
Joint work with Tom Fisher (Cambridge, United Kingdom).