Endomorphism of Jacobian of hyperelliptic curve

Question

Let $C$ be a hyperelliptic curve of genus $g=2＞1$, given by an equation, $u^2=1-4t+2t^2-6t^3+t^4+2t^5+t^6$. Let $J(C)$ be a Jacobian of $C$.

Why $End_ \Bbb{Q}(J(C)) \cong \Bbb{Z}[(-1+ \sqrt{5})/2]$ holds ?

$J(C)$ cannot be expressed as equation explicitly, so I don't have way to check this. Once equation is explicitly given, I will be able to find element of $End _\Bbb{Q}(J(C))- \Bbb{Z}$, but I cannot do this in this situation, so at a loss.

Answer 1

Let $g=2$ and let $J$ be the Jacobian of $C$. The complex points of an abelian variety form a complex torus, so $\newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} J(\C) \cong \C^g/\Lambda$ where $\Lambda \leq \C^g$ is the period lattice of $C$. An endomorphism $\varphi: J \to J$ induces an endomorphism $\varphi^*: \C^g/\Lambda \to \C^g/\Lambda$, which is really just a linear map $\varphi^*: \C^g \to \C^g$ such that $\varphi^*(\Lambda) \subseteq \Lambda$. Thus, to find all endomorphisms of the abelian variety $J$, we first find all endomorphisms of the torus $J(\C)$ and then check that these come from endomorphisms of the abelian variety $J$.

Choose a basis $\omega_1, \ldots, \omega_g$ for $H^0(C, \Omega_C)$ and a basis $\gamma_1, \ldots, \gamma_{2g}$ for $H_1(J(\C), \Z)$. Let $\Pi$ be the $g \times 2g$ period matrix with respect to these bases, so $$ \displaystyle \Pi = \left( \int_{\gamma_i} \omega_j \right)_{\substack{i = 1, \ldots, 2g\\ j = 1, \ldots, g}} $$ and $\Lambda = \Pi \Z^{2g}$. Then the linear map $\varphi^*$ with $\varphi^*(\Lambda) \subseteq \Lambda$ is given by multiplication by a $g \times g$ matrix $M$ such that $$ M \Pi = \Pi R $$ for some $2g \times 2g$ matrix $R$. (Since $\varphi^*(\Lambda) \subseteq \Lambda$, then $M \Pi$ must be a linear combination of the columns of $\Pi$; the coefficients in these linear combinations are given by $R$.) Such matrices $M$ can be found using numerical linear algebra. (The entries of the period matrix are complex numbers, hence can only be represented up to some numerical precision.)

These ideas have been implemented in Magma; here is some code for your example.

R<x> := PolynomialRing(QQ);
f := 1-4*x+2*x^2-6*x^3+x^4+2*x^5+x^6;
C := HyperellipticCurve(f);
J := AnalyticJacobian(C);
E := EndomorphismRing(J);
R!MinimalPolynomial(E.1);

The matrix E.1 generating the endomorphism ring is $$ \begin{pmatrix} 1 & 0 & 0 & 1\\ 0 & 0 & -1 & 0\\ 0 & -1 & 1 & 0\\ 1 & 0 & 0 & 0 \end{pmatrix} $$ and the above code returns $x^2 - x - 1$ as its minimal polynomial, which is the minimal polynomial for $\frac{1+\sqrt{5}}{2}$. (You can try the code yourself at the online Magma calculator.) The Magma implementation of the analytic Jacobian for hyperelliptic curves is described in

van Wamelen, Paul B. "Computing with the analytic Jacobian of a genus 2 curve." Discovering Mathematics with Magma: Reducing the Abstract to the Concrete.

The above calculations shows that there is an endomorphism of complex tori that acts as $\frac{1+\sqrt{5}}{2}$, but we still haven't rigorously shown that the endomorphism of tori comes from an endomorphism of abelian varieties. In the linked article, van Wamelen explains how this can be verified via interpolation. There are two alternative approaches to this verification step, one by Lombardo in Computing the geometric endomorphism ring of a genus 2 Jacobian and another by Costa et al. in Rigorous computation of the endomorphism ring of a Jacobian. Both are implemented in Magma; Lombardo has code on his webpage and Costa et al.'s code is on Github. (As a note, the methods and implementations of Costa et al. also work for curves of higher genus as well as non-hyperelliptic curves; see the examples in $\S8.2$.)

In this case, we can also verify the calculation using the LMFDB. As suggested by Aphelli in the comments, your curve $C$ is isomorphic to the hyperelliptic curve with LMFDB label 48841.a.48841.1 which has simplified equation $$ C': y^2 = x^6 + 4x^5 + 2x^4 + 6x^3 + x^2 - 2x + 1 \, . $$ Computing with Magma, we find the isomorphism \begin{align*} C &\overset{\sim}{\to} C'\\ (X : Y : Z) &\mapsto (-Z : (-1/2)X^3 + (1/2) X^2 Z + (1/2) Y + (1/2) Z^3 : X) \, . \end{align*} The LMFDB page states that $\operatorname{End}(\operatorname{Jac}(C')) \cong \mathbb{Z}\left[\frac{1 + \sqrt{5}}{2}\right]$, in agreement with our computations.