arXiv subjects

Jiali Yan

Publications and source records attributed to Jiali Yan.

3 recordsLinked to original sources

arXiv2023

Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian

We describe a method for computing the Cassels-Tate pairing on the 2-Selmer group of the Jacobian of a genus 2 curve. This can be used to improve the upper bound coming from 2-descent for the rank of the group of rational points on the Jacobian. Our method remains practical regardless of the Galois action on the Weierstrass points of the genus 2 curve. It does however depend on being able to find a rational point on a certain twisted Kummer surface. The latter does not appear to be a severe restriction in practice. In particular, we have used our method to unconditionally determine the ranks of all genus 2 Jacobians in the L-functions and modular forms database (LMFDB).

math.NT↗

arXiv2021

Computing the Cassels-Tate Pairing in the Case of a Richelot Isogeny

In this paper, we study the Cassels-Tate pairing on Jacobians of genus two curves admitting a special type of isogenies called Richelot isogenies. Let $ϕ: J \rightarrow \widehat{J}$ be a Richelot isogeny between two Jacobians of genus two curves. We give an explicit formula as well as a practical algorithm to compute the Cassels-Tate pairing on $\text{Sel}^{\widehatϕ}(\widehat{J}) \times \text{Sel}^{\widehatϕ}(\widehat{J})$ where $\widehatϕ$ is the dual isogeny of $ϕ$. The formula and algorithm are under the simplifying assumption that all two torsion points on $J$ are defined over $K$. We also include a worked example demonstrating we can turn the descent by Richelot isogeny into a 2-descent via computing the Cassels-Tate pairing.

math.NT↗

arXiv2021

Computing the Cassels-Tate Pairing for Genus Two Jacobians with Rational Two Torsion Points

In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption that all points in $J[2]$ are $K$-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating we can indeed improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing.

math.NT↗