Where does this elliptic curve come from?

Question

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.

Answer 1

You can say something about their particular $E_\lambda$. Their choice of $E_\lambda$ is not something totally out of the blue, although it may seem that way in the paper. It's something that arises fairly naturally from the character POV. Essentially, the results using character sums suggest that the K3 surface $X_\lambda$ is related to the product $E_\lambda \times E_\lambda$.

Here's one rough way to think about it. The family $X_\lambda$ has pretty big Picard number (19 generically, 20 at singular points). A result of Shioda-Inose says that if the Picard number of an algebraic K3 surface is 20, it can be viewed as a double-cover of a Kummer surface. You can generalize this notion to other (say, large Picard rank) cases with the notion of "Shioda-Inose structure." Furthermore, in the rank 20 case, the two isogenous elliptic curves associated to the surface by such a structure have complex multiplication and so have natural Hecke characters associated to them, whence the link.

The answer for the particular example you consider is all spelled out in great detail in Ling Long's "On a Shioda-Inose structure of a family of K3 surfaces." She even points out the connection to hypergeometric solutions to associated Picard-Fuchs equations. Her related paper "On Shioda-Inose structures of one-parameter families of K3 surfaces" gives a little more of the theoretical background that I alluded to.