Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose geometric fibers are (smooth) K3 surfaces, endowed with polarization of some fixed degree.) Are there only finitely many isomorphism classes, as would be the case for curves of fixed genus? If one doesn't know (or expect) finiteness, does one have an upper bound for the number of such K3 surfaces X/Q of bounded height?
[Math] K3 surfaces with good reduction away from finitely many places
ag.algebraic-geometryalgebraic-surfacesarithmetic-geometryk3-surfacesnt.number-theory
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I don't think the answer to the first question is known.
Will has already pointed out the trivial answer to the second question. However this is not the right question. I mean this is kind of trivial. The interesting question is if you fix the genus and require that the curve over $K$ has good reduction everywhere (outside a fixed set of primes). If you ask it in that way, then the answer for curves is negative (by Faltings) and so the easy fix to do it in higher dimensions does not work.
Here are some comments and references to Theorems 1,2,3:
Theorem 1 is known in more general context.
It does not need "strong", non-isotrivial is enough.
Relevant references are:
Kovács, Sándor J.(1-UT) Smooth families over rational and elliptic curves. J. Algebraic Geom. 5 (1996), no. 2, 369–385.
Kovács, Sándor J.(1-MIT) On the minimal number of singular fibres in a family of surfaces of general type. J. Reine Angew. Math. 487 (1997), 171–177.
Kovács, Sándor J.(1-CHI) Algebraic hyperbolicity of fine moduli spaces. J. Algebraic Geom. 9 (2000), no. 1, 165–174.
Viehweg, Eckart(D-ESSN); Zuo, Kang(PRC-CHHK) On the isotriviality of families of projective manifolds over curves. J. Algebraic Geom. 10 (2001), no. 4, 781–799.
Kovács, Sándor J.(1-WA) Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties. Compositio Math. 131 (2002), no. 3, 291–317.
There are also generalizations for families over higher dimensional bases. See for instance:
Viehweg, Eckart(D-ESSN); Zuo, Kang(PRC-CHHK) Base spaces of non-isotrivial families of smooth minimal models. Complex geometry (Göttingen, 2000), 279–328, Springer, Berlin, 2002.
Kebekus, Stefan(D-KOLN); Kovács, Sándor J.(1-WA) Families of canonically polarized varieties over surfaces. (English summary) Invent. Math. 172 (2008), no. 3, 657–682.
Kebekus, Stefan(D-FRBG); Kovács, Sándor J.(1-WA) The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. Duke Math. J. 155 (2010), no. 1, 1–33.
Patakfalvi, Zsolt(1-PRIN) Viehweg's hyperbolicity conjecture is true over compact bases. (English summary) Adv. Math. 229 (2012), no. 3, 1640–1642.
Theorem 2:
This is a triviality unless you fix some invariants. On the other hand for relative dimension $1$ and fixed genus this is not true. This is the geometric version of Shavarevich's conjecture and was first proved by Parshin:
Paršin, A. N. Algebraic curves over function fields. I. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 32 1968 1191–1219,
and then in a more general case by Arakelov:
Arakelov, S. Ju. Families of algebraic curves with fixed degeneracies. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293.
In higher dimensions, the statement is true indeed by taking the product of an arbitrary family of curves and an arbitrary curve (each of the appropriate genus). The second curve can be moved in moduli which gives even a continuous family of families.
In fact, this was what led to the notion of strong isotriviality.
Some relevant references are:
Kovács, Sándor J.(1-WA) Strong non-isotriviality and rigidity. Recent progress in arithmetic and algebraic geometry, 47–55, Contemp. Math., 386, Amer. Math. Soc., Providence, RI, 2005.
Kovács, Sándor J.(1-WA) Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture. Algebraic geometry—Seattle 2005. Part 2, 685–709, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
Kovács, Sándor J.(1-WA); Lieblich, Max(1-WA) Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich's conjecture. (English summary) Ann. of Math. (2) 172 (2010), no. 3, 1719–1748.
Zsolt Patakfalvi Arakelov-Parshin rigidity of towers of curve fibrations, connections to the infinitesimal Torelli problem http://arxiv.org/abs/1010.3069
Theorem 3:
as I explained above, even this is not true without the "strong" assumption.
For strongly non-isomorphic families it is proven in
Kovács, Sándor J.(1-WA); Lieblich, Max(1-WA) Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich's conjecture. (English summary) Ann. of Math. (2) 172 (2010), no. 3, 1719–1748.
I would expect it to be true for a somewhat larger class of families, but the actual class still needs to be defined. The key modulo this paper is rigidity.
For more details see
Kovács, Sándor J.(1-WA) Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture. Algebraic geometry—Seattle 2005. Part 2, 685–709, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
or
Chapter III of Hacon, Christopher D.(1-UT); Kovács, Sándor J.(1-WA) Classification of higher dimensional algebraic varieties. Oberwolfach Seminars, 41. Birkhäuser Verlag, Basel, 2010. x+208 pp. ISBN: 978-3-0346-0289-1
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).
Best Answer
Some thoughts.
There are no such varieties when S = 1. This is a consequence of a theorem of Fontaine, MR1274493 (Schémas propres et lisses sur Z).
I think that one should only expect finitely many such varieties for any fixed S. Let me give an argument that uses every possible conjecture I know. There may be an unconditional proof, but that would probably require knowing something about K3-surfaces.
I first want to claim that the ramification at primes q|S is "bounded" independently of X. The corresponding fact for elliptic curves will be that the power of the conductor for each q|N is bounded by 2 (if p > 3) or (if p = 2 or 3) by some fixed number I can't remember.
The most obvious argument along these lines is to consider the representation on inertia I
_q acting on the p-adic etale cohomology groups H^2(X). These correspond to Galois representations with image in GL_22(Z_p). The argument I have in mind for elliptic curves works directly in this case, providing that one has "independence of p" statement for the Weil-Deligne representations at q (quick hint: the image of wild inertia divides the gcd of the orders of GL_22(F_p) over all primes p). This may require the existence of semi-stable models, which one certainly has for elliptic curves, but I don't know for K3-surfaces.The next step is to use a Langlands-type conjecture. The p-adic representation V on H^2(X) may be reducible, but at least we know that each irreducible chunk will correspond to an irreducible Galois representation of Q into GL
_n(Z_p) for some n (at most 22). Each of these, conjecturally, will correspond to a cuspidal automorphic form of fixed weight and level divisible only by q|S. Moreover, from the previous paragraph, the level will be bounded at q|S. Thus there will only be finitely many representations which can occur as H^2(X) for any K3-surface X/Z[1/S]. (Maybe I am assuming here that the Galois representation acting on H^2(X) is semi-simple --- let us do so, since this is a conjecture of Grothendieck and Serre.)Finally, I want to deduce from any equality H^2(X) = H^2(X') that X is (essentially) X'. From the Tate conjecture we deduce the existence of correspondences X~~>X' and X'~~>X over Q whose composition induces an isomorphism on H^2(X) --- and now hopefully some knowledge of the geometry of K3 surfaces is enough to show that these sets of "isogenous" K3 surfaces form a finite set.
EDIT:
As Buzzard points out, I obscured the fact in the last paragraph that some more arithmetic may be necessary. What I meant to say is that understanding isogeny classes of K3's over Q will first require understanding isogeny classes over C, and hopefully this second task will be the hard part.
As David points out, the Torelli theorem for K3 will surely be relevant here. I think there can be non-isomorphic isogenous K3s, however. If one takes an isogeny of abelian surfaces A->B then one can presumably promote this to an isogeny of the associated Kummer surfaces.
EDIT:
Here is another thought. Deligne proves the Weil conjecture for K3 surfaces:
http://www.its.caltech.edu/~clyons/DeligneWeilK3trans.pdf
The philosophy is that there should be an inclusion of motives H^2(X) --> H^1(A) tensor H^1(A) for some abelian variety A (possibly of some huge but uniformly bounded dimension, like 2^19). It may be possible (conjecturally or otherwise) to reduce your question to the analogous statement for A, for which it is known. (Prop 6.5 is relevant here). It may well be possible to show that the variety A is defined over Z[1/2S], for example. I could make this edit more coherent but I'm off to lunch, so treat this as a thought fragment.