Representing endomorphisms via correspondences

$$\begin{aligned} \alpha_C : \ & C \xrightarrow{\ \ AJ\ \ } J \xrightarrow{\ \ \alpha\ \ } J \dashrightarrow \operatorname{Sym}^g(C) \\ & P \mapsto \{Q_1, \ldots, Q_g\} \Longleftrightarrow \alpha([P - P_0]) = \left[ \sum_{i=1}^g Q_i - P_0 \right] \end{aligned}$$

This traces out a divisor on $C \times C$, which determines $\alpha$.

This divisor is a certificate of containment for $\alpha \in \operatorname{End} \overline{J}$.

Theorem (C-Mascot-Sijsling-Voight)

We give an algorithm for nondegenerate $\alpha \in \mathrm{M}_g(\overline{k})$

$$\alpha \mapsto \begin{cases} \texttt{true} & \text{if } \alpha \in \operatorname{End} \overline{J}, \text{ and a certificate} \\ \texttt{false} & \text{if } \alpha \notin \operatorname{End} \overline{J} \end{cases}$$

By interpolation via $\alpha_C$ or by locally solving a differential equation on $C \times C$.

SOURCE: L:L241-279

Nondegeneracy is a hypothesis on the input, not a branch of the output. A degenerate alpha that does lie in End Jbar is outside the algorithm's scope; it is not returned false.

What it does. The template. Numerical candidate, then an exact object that witnesses it. Everything in 3.3 is this pattern with quadric surfaces in place of correspondences.

[figure: images/certificatew.png] The source frame sets this certificate icon, labelled alpha, next to the words "certificate of containment" and inside the theorem's true branch. The deck carries no images, so the icon is dropped and the word "certificate" carries it.

Spoken: Close the gap: realise a candidate as a divisor on $C \times C$, which is an object one can verify.

Spoken: An algorithm which, given a nondegenerate $\alpha \in M_g(\overline{\mathbf{Q}})$, returns true with a certificate if $\alpha \in \operatorname{End} \overline{J}$, and false if $\alpha \notin \operatorname{End} \overline{J}$.

TRANSCRIBED FROM: five-nomial-quartics/slides/mukai_leiden.tex:241-279