On the Picard number of K3 surfaces over number fields

Abstract

14 pages, comments welcomeInternational audienceWe discuss some aspects of the behavior of specialization at a finite place of Néron-Severi groups of K3 surfaces over number fields. We give optimal lower bounds for the Picard number of such specializations, thus answering a question of Elsenhans and Jahnel. As a consequence of these results, we show that it is possible to explicitly compute the Picard number of any given K3 surface over a number field

Point counting on K3 surfaces and an application concerning real and complex multiplication

“…Moreover, one hopes to find a prime such that rk Pic X p is actually equal to rk Pic X K or rk Pic X K + 1. As was observed by F. Charles [5], the existence of such primes is related to whether X has real multiplication or not. More precisely, existence is provided unless X has real multiplication and…”

Examples of surfaces with real multiplication

“…There is a theoretical algorithm to prove real (or complex) multiplication for a given K3 surface under the assumption of the Hodge conjecture; cf. the indications given a.-s. elsenhans and j. jahnel in the proof of [4,Theorem 6]. Its main idea is to inspect the Hilbert scheme of X × X; it is far from realistic to do this in practice.…”

“…Several of them are K3. Quite recently, Charles [4] provided a theoretical analysis on the existence of primes fulfilling condition (1). The result is that such primes always exist, unless X has real multiplication by a number field E such that (22 − rk Pic X Q )/[E : Q] is odd.…”

“…Associated to every abelian surface X over Q, there is an algebraic group G ⊂ End(H 1 et (X Q , Q l )) ∼ = GSp 4 (Q l ) such that the image of the natural operation of Gal(Q/Q) on H 1 et (X Q , Q l ) is Zariski dense in G. The reader might compare the theorem of Tankeev and Zarhin (Theorem 4.1), which gives an analogous statement for K3 surfaces. Corresponding to G, there is a compact subgroup of USp 4 .…”

Picard numbers of quintic surfaces

“…Applied to sufficiently many primes, this approach would usually succeed in returning the Picard number of a K3 surface S over a number field. A notable obstacle comes from the endomorphism algebra End(T (S)), see 2.5 and for details [Cha14]. We point out that Charles' ideas in [Cha14] heavily depend on the fact that h 2,0 = 1 for a K3 surface; hence they do not carry over to quintics or most other surfaces of general type.…”