Title:
Computing the Cassels-Tate Pairing on 2-Selmer Groups of Jacobians of Genus Two Curves

Abstract:
Let $J$ be the Jacobian variety of a genus two curve defined over a
number field $K$. We are interested in computing the Cassels-Tate
pairing $\langle \; ,  \; \rangle_{CT}$ on $\text{Sel}^2(J) \times
\text{Sel}^2(J)$ following the homogeneous space definition of the
pairing. For $\epsilon, \eta \in \text{Sel}^2(J)$, we give a computable
formula with a practical algorithm for $\langle \epsilon,  \eta
\rangle_{CT}$ both in the case where all points in $J[2]$ are defined
over $K$ and in the case where the twisted Kummer surface
$\mathcal{K}_{\eta}$ has a $K$-rational point. In both cases, we
calculate examples for which computing the Cassels-Tate pairing
improves the rank bound of $J$ obtained by carrying out standard
descent calculations. We also give techniques to reduce the degree of
the number field needed in the algorithm for computation.