# Enriques surfaces over $\mathbb Z$

Does there exist a smooth proper morphism $$E \to \operatorname{Spec} \mathbb Z$$ whose fibers are Enriques surfaces?

By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-Kodaira classification of complex surfaces, the only possible surface of Kodaira dimension $$0$$ or $$1$$ that can appear this way is an Enriques surface.

In particular, no K3 surfaces are smooth over $$\mathbb Z$$. Hence the K3 double cover of $$E$$ is not smooth. The cover is etale away from $$\mathbb F_2$$, hence must be singular over $$\mathbb F_2$$. This means $$E$$ most be a classical Enriques surface (a $$\mu_2$$-torsor rather than a $$\mathbb Z/2$$-torsor or $$\alpha_2$$-torsor).

It is possible to get some other information about $$E$$:

The Picard group of this surface has rank $$10$$. The Galois action on the Picard group must be unramified at each prime, hence trivial, so the full lattice of cycles is defined over $$\mathbb Q$$. Thus by the Lefschetz trace formula, $$E$$ has exactly $$25$$ $$\mathbb F_2$$-points.

Some other questions that might be helpful to solve this one are:

How many K3 surfaces are there with good reduction away from $$2$$ (and Picard rank at least $$10$$, and a fixed-point-free involution, etc.)?

Given an Enriques surface over $$\mathbb Q_2$$, what are obstructions to good reduction over $$\mathbb Z_2$$, other than ramification of the cohomology?

Which classical Enriques surfaces over $$\mathbb F_2$$ are liftable to $$\mathbb Z_2$$? Can something be said about the singularities and Galois representations of their $$K3$$ double covers?

Can we compute the discriminants of explicit families of Enriques surfaces and try to solve the Diophantine equation $$\Delta=1$$?

One example of an Enriques surface over $$\mathbb Z[1/2]$$ whose cohomology is unramified at $$2$$ can be constructed as the quotient of a Kummer surface with good reduction away from $$2$$. Let $$E_1$$ and $$E_2$$ be two elliptic curves that are either $$y^2=x^3-x$$ or $$y^2=x^3-4x$$. Let $$e_1$$ and $$e_2$$ be $$2$$-torsion points on $$E_1$$ and $$E_2$$ respectively. Then we can construct (Example 3.1) a fixed-point free involution on the Kummer surface of $$E_1 \times E_2$$, giving an Enriques surface.

Because $$E_1$$ and $$E_2$$ have good reduction away from $$2$$, this surface has good reduction also. $$H^2$$ of the Kummer surface comes from $$H^2(E_1 \times E_2)$$ plus the exceptional classes of the 16 blown-up $$2$$-torsion points. Because these points are defined over $$\mathbb Q$$, the cohomology classes are unramified. $$H^2(E_1)$$ and $$H^2(E_2)$$ are unramified as well, so the only ramified part of the cohomology of the Kummer surface is $$H^1(E_1) \times H^1(E_2)$$. Because the involution acts as reflection on $$E_1$$ and translation on $$E_2$$, it acts as $$-1$$ on $$H^1(E_1) \times H^1(E_2)$$, so that does not descend to the Enriques surface, hence its cohomology is unramified. This surface also has $$\mathbb Q$$-points, thus $$\mathbb Q_2$$ points.

Thus I cannot see any obstruction to good reduction at $$2$$. However, the construction certainly does not produce a smooth model of the surface over $$2$$.

Does this surface have good reduction at $$2$$?

• Ekedahl and Shepherd-Barron (arxiv.org/abs/math/0405510) have given examples of classical Enriques surfaces with non-trivial vector fields and these need not have unramified lifts. You might also find the paper by Liedtke "Arithmetic moduli and lifting of Enriques surfaces" useful.
– naf
Mar 28, 2015 at 6:16
• Is the Galois action on the (geometric) Picard group of the surface in your example trivial? From what you write this is clear rationally but it seems possible to me that it could be non-trivial integrally (though I am not sure whether knowing this is useful). Also, a couple of typos: in the second last paragraph Kunneth should presumably be Kummer and $H^2(E)$ should be $H^1(E_2)$.
– naf
Mar 29, 2015 at 7:01
• @ulrich I'm not sure. I think the exceptional classes are invariant even integrally, but the classes coming from $H^1(E)$ and $H^2(E)$ might not be. One may be able to figure this out by studying the two elliptic vibrations over $\mathbb P^1$. If one did get a nontrivial action, because it only shows up in $2$-adic cohomology, you might need to use $p$-adic Hodge theory to see if it's compatible with good reduction. Mar 29, 2015 at 14:06
• If the action is non-trivial, then I think one gets an obstruction to good reduction if the fibre over 2 is classical (since the Picard scheme is smooth in this case). However, I don't see why the special fibre cannot be an $\alpha_2$ surface.
– naf
Apr 1, 2015 at 6:27
• @ulrich The Picard scheme is a group scheme over $\mathbb Z$. But the group scheme $\alpha_2$ does not lift to $\mathbb Z$ or even $\mathbb Z_2$. I'll try to figure out what the Galois action mod $2$ is in my case. Apr 1, 2015 at 17:41

## 1 Answer

A preprint by Stefan Schröer came out today with the answer to this question: arXiv:2004.07025.

No such Enriques surface exists. In fact, there is no classical Enriques surface over $$\mathbb F_2$$ with 25 $$\mathbb F_2$$-points (and with the extension of $$\mathbb Z^{10}$$ by $$\mathbb Z/2$$ split in the Picard group, which you can also deduce).