On K3 surfaces defined over Q — Correction to the paper [1] in vol.43 (1994)

Tetsuji Shioda

1. In a private correspondence, Matthias Schuett has pointed out a mistake in my paper [1]. Namely it is in the proof of Theorem 1 which states that, for any K3 surface defined over the rational number field Q, the Q-Picard number cannot attain the maximum value h^{1,1} = 20. Moreover he has given an explicit example of a K3 surface with Q-Picard number 20, by using a K3 surface with maximal singular fibre studied in my recent paper [2]. According to Klaus Hulek, he and Verrill have found other example of such, too.

Thus “Theorem 1” of [1] should be cancelled. Also “Corollary 3” of that paper should be cancelled which asserted that, for any elliptic curve E/Q(t) arising from an elliptic K3 surface, the Mordell-Weil rank of E(Q(t)) is less than 18. As a consequence, “Question 2” (which motivated that paper) is still open: Does there exist an elliptic K3 surface defined over Q such that E/Q(t) has rank 18 (E being the generic fibre)?

2. What was wrong with the proof of Theorem 1 in [1]?

We argued as follows. Assume there is a K3 surface, say X, over Q with Q-Picard number 20. The reduction X(p) of X modulo p is a K3 surface over the prime field F_{p} such that ρ(X(p)/F_{p}) ≥ 20 for almost all p. Over the algebraic closure k_{p} of F_{p}, the Picard number ρ(X(p)) = ρ(X(p)/k_{p}) is known to be either 20 or 22.

The gap in the proof was precisely at the point where we overlooked the possibility for ρ(X(p)/F_{p}) = 21, as pointed out by Schuett and Hulek. As a matter of fact, the proof there implies that ρ(X(p)/F_{p}) = 21 occurs for an infinitely many prime p, instead of leading to a contradiction as claimed in the paper [1].

3. It should be remarked that the corresponding proof is correct, if we replace a K3 by an abelian surface. Namely we have Proposition 1 For any abelian surface A over Q, the Q-Picard number cannot attain the maximum value h^{1,1}(A) = 4, i.e. ρ(A/Q) < 4.

Proof Arguing as above, we claim that the possibility ρ(A(p)/F_{p}) = 5 cannot occur. Indeed, it would imply in terms of the standard notation (cf. [1])

P_{2}(A(p)/F_{p}, T) = (1 − pT)^{5}(1 + pT).  (1)

But this is impossible, because H^{2} ≅ Λ^{2}(H^{1}) together with Frobenius action holds for the etale cohomology of A(p)/k_{p}. q.e.d.

Remark In the case of a K3 surface X as in 2, on the other hand, we have

P_{2}(X(p)/F_{p}, T) = (1 − pT)^{21}(1 + pT).  (2)

for infinitely many p. In particular, this gives an example of the sign change in the functional equation for P_{2}(T) = P_{2}(X(p)/F_{p}, T):

P_{2}(\frac{1}{p^{2}T}) = −\frac{1}{(pT)^{b_{2}}} P_{2}(T).  (3)

At any rate, it seems that the mistake is caused by very delicate difference between K3 surfaces and abelian surfaces even in the “singular” case (ρ = h^{1,1}) where their structure is very closely related (cf. [3]).

4. Let us mention the related work of Shafarevich [4] who has formulated a very interesting finiteness conjecture:

Conjecture 2 For all K3 surfaces over a given number field, their Néron-Severi lattice NS(X) belong to a finite set of isomorphism classes.

He has proven this for singular K3 (and abelian) surfaces in a stronger sense:

Theorem 3 There is only a finite number of isomorphism classes of singular K3 surfaces over the complex number field which is defined over a number field of given degree.

In particular, it follows that the number of isomorphism classes of singular K3 surfaces defined over Q is finite, and so is the number of those with Q-Picard number 20. It will be an interesting question to determine these numbers.

Finally I would like to thank M. Schuett and K. Hulek for pointing out my error in their nice communications, and T. Katsura for useful discussion.

References

[1] Shioda, T.: On the rank of elliptic curves over Q(t) arising from K3 surfaces, Comment. Math. Univ. St. Pauli 43, 117–120 (1994).

[2] — : The elliptic K3 surfaces with a maximal singular fibre, C. R. Acad. Sci. Paris, Ser. I 336, 461–466 (2003).

[3] Inose, H., Shioda, T.: On singular K3 surfaces, in: Complex Analysis and Algebraic Geometry, Iwanami Shoten and Cambridge Univ. Press, 119–136 (1977).

[4] Shafarevich, I.R.: On the arithmetic of singular K3 surfaces, in: Algebra and Analysis, Proc. Chebotarev Conf. in 1994 /Kazan, Walter de Gruyter & Co., 103–108 (1996).

Department of Mathematics Rikkyo University 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501 Japan (shioda@rkmath.rikkyo.ac.jp)