On large Picard groups and the Hasse principle for curves and \(K3\) surfaces.

*(English)*Zbl 0877.14005Let \(X\) be a proper, geometrically integral variety over a perfect field \(k\). As usual, \(\overline{k}\) will be an algebraic closure of \(k\) and we write \({\mathcal G}=\text{Gal} (\overline{k}/k)\). Further, we define \(\overline{X}= X\times_{\text{Spec }k} \text{Spec }\overline{k}\) and denote by \(\text{Pic }X\) the Picard group of linear equivalence classes of Cartier divisors on \(X\). There is an obvious injective map \(\beta:\text{Pic }X\to (\text{Pic } \overline{X})^{\mathcal G}\). For many diophantine questions, it is important to known that \(\beta\) is surjective, i.e., that every divisor class which is stable under the Galois action actually comes from a \(k\)-divisor. It seems to us that this condition is of sufficient interest to deserve a special same. So we denote it by BP and call it the “BigPic” condition.

Definition. \(\text{BP}(X,k) \Leftrightarrow \text{Pic }X= (\text{Pic } \overline{X})^{\mathcal G}\).

In spite of its regular appearance in many papers, this condition does not seem to have been much investigated for its own merits. So, for instance, it is well known that \[ X(k)\neq \emptyset\Rightarrow \text{BP}(X,k), \] but what is known exactly when \(X\) has no \(k\)-rational point? – In the present paper we begin by collecting several properties of BP. Most of them are well known, but they are difficult to find all at one place in the literature. In spite of its simplicity, this study already raises several questions. In section 2 we investigate rational curves and smooth quadrics in \(\mathbb{P}_k^3\). We see that BP behaves in a rather unexpected fashion (example 2.9). Over a number field, this is intimately connected with the Hasse principle.

Then we use the condition BP as a leading thread through the arithmetic maze of projective curves and K3 surfaces. In fact, on trying to locate some explicit examples where BP does or does not hold (§3), we are naturally led to some particularly nice counterexamples to the Hasse principle for curves, some with a point of degree 3, others with no point in any odd-degree extension of \(\mathbb{Q}\). – Many examples involve some special intersections of quadrics. This feature allows us to produce various families of curves of arbitrary genus for which the Hasse principle fails to hold, including families with fixed coefficients by varying genus (§4). At the end we exhibit some K3 surfaces with points everywhere locally, but none over \(\mathbb{Q}\). In fact, we give an example in every class consisting of complete intersections (§5). Such examples appear to be new.

Definition. \(\text{BP}(X,k) \Leftrightarrow \text{Pic }X= (\text{Pic } \overline{X})^{\mathcal G}\).

In spite of its regular appearance in many papers, this condition does not seem to have been much investigated for its own merits. So, for instance, it is well known that \[ X(k)\neq \emptyset\Rightarrow \text{BP}(X,k), \] but what is known exactly when \(X\) has no \(k\)-rational point? – In the present paper we begin by collecting several properties of BP. Most of them are well known, but they are difficult to find all at one place in the literature. In spite of its simplicity, this study already raises several questions. In section 2 we investigate rational curves and smooth quadrics in \(\mathbb{P}_k^3\). We see that BP behaves in a rather unexpected fashion (example 2.9). Over a number field, this is intimately connected with the Hasse principle.

Then we use the condition BP as a leading thread through the arithmetic maze of projective curves and K3 surfaces. In fact, on trying to locate some explicit examples where BP does or does not hold (§3), we are naturally led to some particularly nice counterexamples to the Hasse principle for curves, some with a point of degree 3, others with no point in any odd-degree extension of \(\mathbb{Q}\). – Many examples involve some special intersections of quadrics. This feature allows us to produce various families of curves of arbitrary genus for which the Hasse principle fails to hold, including families with fixed coefficients by varying genus (§4). At the end we exhibit some K3 surfaces with points everywhere locally, but none over \(\mathbb{Q}\). In fact, we give an example in every class consisting of complete intersections (§5). Such examples appear to be new.