Number theory seminar in memory of F. Momose

Genus two lifts of $\mathbb{Q}$-curves whose jacobians have $\sqrt{−2}$ multiplication over $\mathbb{Q}$.


Kiichiro Hashimoto


An elliptic curve $E$ de ned over a number field $K$ is called a $\mathbb{Q}$-curve if for each  $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, there exists an isogeny  $\phi_{\sigma}: E \rightarrow ^{\sigma}$E. A $\mathbb{Q}$-curve $E$ is of degree $d$ if $$d = min \{deg(\phi_\sigma ) : \phi_\sigma : E \rightarrow ^{\sigma}E, id \neq \sigma \in Aut(K/ \mathbb{Q})\}.$$ In 1995, F. Momose gave us the following problems on $\mathbb{Q}$-curves over a quadratic fi eld K:

$\bullet$ (i) When is $Res_{K/\mathbb{Q}}(E)$ of $GL_2$-type ?
$\bullet$ (ii) Find as many $E/K$ as possible satisfying (i). 
$\bullet$ (iii) For each $E/K$ of (ii), fi nd a genus two curve $C$ over $\mathbb{Q}$ such that $Jac(C)$ is $\mathbb{Q}$-isogenous to $Res_{K/\mathbb{Q}}(E)$.

We shall discuss these problems for $\mathbb{Q}$-curves of degree $d = 2$. If $K$ is a quadratic fi eld and the non-trivial $\phi := \phi_\sigma$ is de ned over $K$, then $E$ is called minimal; called $\varepsilon$-minimal if $^{\sigma}\phi \circ \phi = \varepsilon d \cdot 1$ ($\varepsilon = \pm 1$). Then the answer to (i) is now well known : $Res_{K/\mathbb{Q}}(E)$ is of $GL_2$-type iff it is minimal. On the other hand, it seems that problems (ii) and (iii) are not fully studied, especially for those of $(-1)$-minimal $\mathbb{Q}$-curves. Another way of looking at the problem (iii) for (-1)-minimal Q-curves, through the modularity conjecture proved by Khare, is stated as follows.

$\bullet$ (iii)' For each normalized Hecke eigenform $f\in S_2\left(N,\left(\frac{N}{\cdot}\right) \right)$ with $K_f=\mathbb{Q}(a_n,n\in \mathbb{N})=\mathbb{Q}(\sqrt{-2})$, find a genus two curve $C$ over $ \mathbb{Q}$ such that $Jac(C)$ is $\mathbb{Q}$-isogenous to $A_f$ , Shimura's abelian surface.

We shall also discuss the problem of constructing an algebraic correspondence on $C$ de ned over $ \mathbb{Q}$ which induces the endomorphism $\sqrt{-2}$ on $Jac(C)$.

Example: ($N=24$) The hyperelliptic curve $$C: y^2= (x^2-6x+6)(x^4-6x^3+18x^2-36x+36)$$ is corresponding to the normalized Hecke eigenform $f$ in $S_2\left(24,\left(\frac{24}{\cdot}\right) \right)$ whose fi rst few Fourier coecients are $\begin{array}{cccccccc} a_2 & a_3 & a_5 & a_7 & a_11 & a_{13} & a_{17} & a_{19} \\ -a & a-1 & 0 & 0 & -2a & 0 & 4a & 2 \end{array}$
with $a =\sqrt{-2}$. The curve $C$ covers the elliptic curve $E_f$ , $$y^2+\sqrt{6}xy+1+\sqrt{6}y=x^3+1-\sqrt{6}x^2-3\sqrt{6}+1x-1-2\sqrt{6}$$ attached to $f$, which is obtained by Cremona, and $Jac(C)$ is isogenous to $E_f \times ^{\sigma}E_f$. The algebraic correspondence $T$ of the curve $C$ given by $$T= \{(x,y),(u,w)\in C\times C : A(x,u)=0, B(w,y,x,u)=0 \},$$ $$ \begin{array}{c} A(x, u) =3(12 - 6x + x^2) - 3u(6 - 4x + x^2) + u^2(3 - 3x + x^2) , \\ B(w, y, x, u) =wx^2(-144 + 108u - 18u^2 + 36x - 24ux + u^3x) \\ -(6 - 6u + u^2)(12 - 6u + u^2)^2y \end{array}$$ gives a map $T$ on $Div(C)$, $$ (u, v) \mapsto (x_1, y_1) + (x_2, y_2) ,\quad ((u, v), (x_i, y_i)) \in T $$ which induces an endomorphism $T^2 = (-2)id$ on $Pic^0(C)$.


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