# Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces $X_\lambda$ that are the smooth complete model of the double cover of $\mathbb{P}^2$ branched over the six lines $$U_\lambda : XYZ(X+\lambda Y)(X+Z)(Y+Z) = 0,$$ where $\lambda\in \mathbb{Q}\setminus\{0,-1\}$. Given a few pages of manipulating character sums, they find a nice relationship when counting points on $X_\lambda$ and on the elliptic curve $$E_\lambda : y^2 = (x-1)\left(x^2 - \frac{1}{\lambda + 1}\right),$$ and can show that over any finite field $\mathbb{F}_p$ where $E_\lambda$ has good reduction, the zeta function of $X_\lambda$ is $$Z(X_\lambda/\mathbb{F}_p,T) = \frac{1}{(1-T)(1-p^2T)(1-pT)^{19}(1-\gamma pT)(1-\gamma \pi_{\lambda,p}^2T)(1-\gamma \bar{pi}_{\lambda,p}^2T)},$$ where $\pi_{\lambda,p}$ and $\bar{\pi}_{\lambda,p}$ are the eigenvalues of the Frobenius at $p$ on $E_\lambda$, and $\gamma$ is the Legendre symbol $\left(\frac{\lambda+1}{p}\right)$.

Now certainly one could say the elliptic curve $E_\lambda$ comes from the character sum manipulation and leave it at that, but there's something deeper going on.

If we go a little farther back, to the work of Jan Stienstra and Frits Beukers, On the Picard-Fuchs Equation and the Formal Brauer Group of Certain Elliptic K3-Surfaces, (on page 291) they associate to the K3 surface $X_1$, the elliptic curve $$E : y^2 = x^3 - 4x^2 + 2x,$$ and going through their references, one finds this curve comes from the cohomology of $X_1$. A result of Shioda shows that $E$ is very intrinsically attached to $X_1$ (over $\mathbb{C}$), as it is given by $$H^2(X_1,\mathcal{O})/j^*H^2(X_1,\mathbb{Z}),$$ where $j$ is the natural map in the exponential exact sequence $$0 \rightarrow \mathbb{Z} \stackrel{j}{\rightarrow} \mathcal{O} \rightarrow \mathcal{O}^* \rightarrow 0$$ and $j^*$ is the induced map on cohomology $$H^1(X_1,\mathcal{O}^*) \rightarrow H^2(X_1,\mathbb{Z}) \stackrel{j^*}{\rightarrow} H^2(X_1,\mathcal{O}).$$

Now the work of Shioda (and Inose) only guarantees a model of the elliptic curve over some sufficiently large field of characteristic $p$, and in many small cases, the curves $E$ and $E_1$ are not isomorphic over $\mathbb{F}_p$ (as per a checked cases in Sage). So, is there any way to explain the use of the $E_\lambda$ without resorting to character sums? Is the choice of $E_\lambda$, despite giving a great result, perhaps the ''wrong'' choice of elliptic curve, given there are examples where $E$ and $E_1$ are not isomorphic over the base field? Either way, the idea of choosing a model over $\mathbb{Q}$ or $\mathbb{F}_p$ from a given model over $\mathbb{C}$ is nontrivial, so something is, or may be going on here, and any thoughts would be appreciated.

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