Author: Eiko

Time: 2025-03-04 13:04:45 - 2025-03-04 13:04:45 (UTC)

Let $C/\mathbb{Q}$ be a curve of genus $g$ over $\mathbb{Q}$,

$$y^2=x^{2g+1}+\cdots +b_0$$

Adn let $J=J(C)$ the Jacobian, we have $J\subset {\mathbb{P}}^{3^g-1}$ via $|3\cdot \theta |$.

• $g=2$, $J\subset {\mathbb{P}}^8$, birationally it is basically a symetric square of $C$, $J\sim {\mathrm{Sym}}^{2}C$., $PJ() $ is represented by $\{P_1,P_2\},P_i\in C(\mathbb{Q})$, actually $P_1+P_2-2\mathrm{\infty }$.

$$\theta =\{P,-\mathrm{\infty },P\}$$

$V_9=\mathcal{O}(9)$ is the space of linear forms on my projective space ${\mathbb{P}}^8$, consider

$${\wedge }^3{V}_9\to \wedge V_9\otimes V_9$$

for example $e_1\wedge e_2\wedge e_3\mapsto x_1\otimes e_2\wedge e_3-x_2\otimes e_1\wedge e_2+x_3\otimes e_2\wedge e_3$.

$$\left(\begin{array}{ccc}0& x_1& -x_3\\ -x_1& 0& x_2\\ x_3& -x_2& 0\end{array}\right)$$

Theorem. There is a ${\alpha }_J\in {\wedge }^{3}V$, such that

$$J\subset {\mathbb{P}}^8:\{v\in {\mathbb{P}}^8:\mathrm{rank}\mathrm{\Phi }({\alpha }_J)(v)\le 4\}$$

3-Descent

is thought as the following sequence

$$J(\mathbb{Q})/3J(\mathbb{Q})\to {\mathrm{Sel}}^{(3)}(J/\mathbb{Q})\to \mathrm{Sha}(J/\mathbb{Q})[3]$$

$${\mathrm{Sel}}^{(3)}(J/\mathbb{Q})=\{T\subset {\mathbb{P}}^8:T\text{ twist of }J\subset {\mathbb{P}}^8,|3\cdot \theta |\}.$$