# Talk 2: Wanlin Li, Ceresa cycle and hyperellipticity

:::{.remark}
A hyperbolic curve is determined by its $\pi_1^\et$.
:::

:::{.remark}
Recall $y^2=f(x)$ defines a hyperelliptic curve $C$, which admits an involution $(x,y)\to(x,-y)$ and produces a degree 2 map
\[
C &\to \PP^1 \\
(x,y) &\mapsto x
.\]

Let $\kbar$ be a separable closure of $k$.
There is a fibration induced by taking a geometric point of $\spec k$ and pulling back:

\begin{tikzcd}
	{C_{\kbar}} && C \\
	\\
	{\spec \kbar} && {\spec k}
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJDX3tcXGtiYXJ9Il0sWzIsMCwiQyJdLFsyLDIsIlxcc3BlYyBrIl0sWzAsMiwiXFxzcGVjIFxca2JhciJdLFszLDJdLFsxLDJdLFswLDFdLFswLDNdLFswLDIsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==)

As in topology, this induces a LES in homotopy, which here splits into SESs.
In particular, 
\[
1\to \pi_1(C_{\kbar})\to \pi_1(C_k) \to \Gal(\kbar/k)
.\]
A point induces a section and thus a map $\Gal(\kbar/k) \to \Aut \pi_1(C_{\kbar})$.
Take the lower central series of $\pi \da \pi_1(C_{\kbar})$, this induces
\[
1\to L^2\pi/L^3\pi \to \pi/L^3\pi \to \pi/L^1 \pi = \pi^\ab \to 1
\]
where the first term is abelian.

> See Davis-Pries-Wickelgren for applications to Fermat curves.

:::

:::{.question}
This extension corresponds to an element in $\mu(C) \in H^1(G_{\kbar}; \Hom(\pi^\ab, L^2\pi/L^3\pi ) )$, and when $C$ is hyperelliptic $\mu(C) = 0$.
Does the converse hold?
:::

:::{.theorem title="Bisogno-L-Litt-Srinivasan"}
There exist non-hyperelliptic curves  $C$ over $k$ such that $\mu(C)$ is torsion -- in particular, the *Fricke-Macbeath* curve, which is genus 7 Hurwitz.
Moreover if $C_1\to C_2$ then $\mu(C_1)$ torsion implies $\mu(C_2)$ torsion.

> See <https://arxiv.org/abs/2004.06146>?

:::

:::{.theorem title="Harris-Pulte (Hain-Matsumoto)"}
$\mu(C)$ is the $\ell\dash$adic cycles class associated to the **Ceresa cycle**.

> See [Hain-Matsumoto](https://arxiv.org/abs/math/0306037) and [Pulte](https://projecteuclid.org/journals/duke-mathematical-journal/volume-57/issue-3/The-fundamental-group-of-a-Riemann-surface--Mixed-Hodge/10.1215/S0012-7094-88-05732-8.short)

:::

:::{.remark}
For $C\slice k$ and $p\in C(K)$, the Abel-Jacobi map yields
\[
\AJ: C &\injects \Jac(C) \\
q &\mapsto [q-p]
.\]
So define the **Ceresa cycle** as 
\[
\tilde c \da \AJ(C) - \bar{ \AJ(C) } \da [q-p] - [p-q]
.\]
Note that $\tilde c$ is homologically trivial in Chow, but algebraically *nontrivial* for a very general $C\slice \CC$ with $g\geq 3$.
:::

:::{.theorem title="Beauville"}
There is an explicit non-hyperelliptic curve $C$ with $\tilde c$ torsion:
\[
x^4 + xz^3 + y^3z = 0 \subseteq \PP^2, \qquad p = \tv{0,0,1}
.\]

> See <https://arxiv.org/abs/2105.07160>?

:::

:::{.remark}
Consider curves over the local field $K = \CC\fps{t}$.
Note $\Gal(\bar{\CC\fps t}/\CC\fps{t}) = \hat \ZZ$, so this resembles a circle, and one can degenerate a family over the punctured disc.
Apply nonabelian Picard-Lefschetz due to Asada-Matsumoto-Oda: if $C$ has semistable reduction then the monodromy of $C/\CC\fps{t}$ is given by a multi-twist, i.e. a product of Dehn twists about simple closed curves.
One can explicitly compute the Ceresa class in this situation.
The degeneration data can be encoded as a tropical curve (essentially the dual graph of the special fiber).

> See [Asada-Matsumoto-Oda](https://www.sciencedirect.com/science/article/pii/002240499400112V)

:::

:::{.theorem title="?"}
$\mu(C)$ is always torsion for $C$ defined over $\CC\fps{t}$.
:::

:::{.remark}
There is a notion of "hyperelliptic" for tropical curves: quotienting by the involution yields a tree.
:::