On the distribution of the Picard ranks of the reductions of a \(K3\) surface. (English) Zbl 1471.14080

Costa, Edgar; Elsenhans, Andreas-Stephan; Jahnel, Jörg

Res. Number Theory 6, No. 3, Paper No. 27, 25 p. (2020).

Let \(S/K\) be a \(K3\) surface over a number field \(K\) and \(\overline{K}\) a fixed algebraic closure of \(K\). Let \(\mathcal{O}_K\) denote the ring of integers of \(K\), \(\mathfrak{p}\in \mathcal{O}_K\) a prime, \(\mathbb{F}_{\mathfrak{p}}\) the residue field of \(\mathfrak{p}\), and \(\overline{\mathbb{F}}_\mathfrak{p}\) a fixed algebraic closure of \(\mathbb{F}_{\mathfrak{p}}\). Assume that \(\mathfrak{p}\) is a prime of good reduction for \(S\) and let \(S_{\mathbb{F}_\mathfrak{p}}/\mathbb{F}_\mathfrak{p}\) denote the reduction of \(S\) modulo \(\mathfrak{p}\). It is well known that the geometric Picard number of \(S_{\mathbb{F}_\mathfrak{p}}\), denoted by \(\rho (S_{\overline{\mathbb{F}}_\mathfrak{p}}) := \mathrm{rk}\, \mathrm{Pic} ( S_{\overline{\mathbb{F}}_\mathfrak{p}})\), is always greater than the geometric Picard number of \(S_{\overline{K}}\). In other words, we always have the following inequality: \[ \rho (S_{\overline{\mathbb{F}}_\mathfrak{p}}) \geq \rho (S_{\overline{K}}). \]
In the paper under review, the authors investigate the primes of good reduction for \(S\) for which the inequality above is strict, called jump primes. We know of two cases for which every prime of good reduction is a jump prime:

1

when \(\rho (S_{\overline{K}})\) is odd, by the (now proven) Tate conjecture, and

2

when \(S\) has real multiplication by an endomorphism field \(E\) and the integer \((22-\rho (S_{\overline{K}}))/[E:\mathbb{Q}]\) is odd, by work of [F. Charles, Algebra Number Theory 8, No. 1, 1–17 (2014; Zbl 1316.14069)].

The same work by F. Charles also shows that these are the only cases in which all primes of good reduction are jump primes.
The authors of the paper under review introduce the quantities \(\Delta_{H^2}(S)\) and \(\Delta_{\mathrm{Pic}}(S)\), together with the quadratic character \[ \tau_S\colon \mathfrak{p}\mapsto \left( \frac{\Delta_{H^2}(S)\Delta_{\mathrm{Pic}}(S)}{\mathfrak{p}}\right), \] called the transcendental character of the \(K3\) surface \(S\). The main result of the paper shows that if \(S\) has even geometric Picard number and \(\tau_S (\mathfrak{p})=-1\), then \[ \rho (S_{\overline{\mathbb{F}}_\mathfrak{p}}) \geq \rho (S_{\overline{K}})+2\; . \] Using this, the authors prove that the experimental observation by [E. Costa and Y. Tschinkel, Exp. Math. 23, No. 4, 475–481 (2014; Zbl 1311.14039)] on the density of jump primes is correct. The authors also present an algorithm to compute \(\tau_S\) for a given \(K3\) surface embedded in a projective space, alongside with explicit examples. Finally, as an application of the main result, they show that if \(\rho (S_{\overline{K}})\) is even, \(S_{\overline{K}}\) has neither real nor complex multiplication, and the product \(\Delta_{H^2}(S)\Delta_{\mathrm{Pic}}(S)\) is not a square in \(K\), then \(S_{\overline{K}}\) contains infinitely many rational curves.

Reviewer: Dino Festi (Milan)

MSC:

14J28	\(K3\) surfaces and Enriques surfaces
14F20	Étale and other Grothendieck topologies and (co)homologies
11G35	Varieties over global fields
14G25	Global ground fields in algebraic geometry

Keywords:

characteristic polynomial of Frobenius; functional equation; \(K3\) surface; Picard rank

Citations:

Zbl 1316.14069; Zbl 1311.14039

Software:

NTL; Magma; SageMath; FLINT

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