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The \(L\)-series of a cubic fourfold. (English) Zbl 1183.11036

This paper computes the \(L\)-series of a cubic fourfold. A cubic fourfold \(X\) is called special if it contains a surface \(T\) which is not homologous to a multiple of the class \(h^2\), where \(h\) is the class of the hyperplane section. The discriminant of \(X\) is the discriminant of the saturated rank-\(2\) sublattice of \(H^4(X,{\mathbb Z})\) spanned by \(h^2\) and \(T\). B. Hassett [Compos. Math. 120, No. 1, 1–23 (2000; Zbl 0956.14031)] showed that if the discriminant \(d>6\) and if \(d \equiv 0,2\pmod 6\), then the special fourfolds of discriminant \(d\) are parametrized by a non-empty \(19\)-dimensional quasi-projective variety \({\mathcal C}_d\). Moreover, if \(d=2(n^2+n+1)\) with \(n\in{\mathbb N}\), then there exists a non-empty open subset of \({\mathcal C}_d\) such that the Fano variety \(F(X)\) of a cubic fourfold belonging to this open subset is isomorphic to the desingularized symmetry product \(S^{[2]}\) of some \(K3\) surface \(S\).
Fix such a \(K3\) surface \(S\), and consider the composition \[ \psi: S^{[2]}\hookrightarrow \text{Gr}(1,5)\hookrightarrow {\mathbb P}^{14}, \] where the first inclusion is given by \(F(X)\simeq S^{[2]}\), and the second by the Plücker embedding. To \(\psi\), one can associate a line bundle \({\mathcal L}\), which is isomorphic to \({\mathcal O}(n\Delta+f)\) where \(\Delta\) is the exceptional divisor on \(S^{[2]}\), \(n\) is a half-integer and \({\mathcal O}(f)\) is the line bundle associated to the isomorphism \(F(X)\simeq S^{[2]}\).
The main result of the paper under review is the following result:
Theorem. Let \(K\) be a number field. Let \(S/K\) be a \(K3\) surface. Suppose \(S^{[2]}(K)\neq \emptyset\). Assume that there is a cubic fourfold \(X\) over \({\mathbb C}\) such that \(F(X)\simeq S_{\mathbb C}^{[2]}\) and that the associated line bundle on \(S\) descends to \(K\). Then \(X\) has a model over \(K\), \(F(X)\simeq S^{[2]}\) and \[ H^4_{\text{ét}}(X,{\mathbb Q}_{l})= H^2_{\text{ét}}(S,{\mathbb Q}_{l})(-1)\oplus{\mathbb Q}_{ l}[\Delta](-1). \]
The \(L\)-series of the Fermat cubic fourfold is computed using this geometric structure, showing that there is a weight three form of level \(27\).

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G25 Global ground fields in algebraic geometry
14J35 \(4\)-folds

Citations:

Zbl 0956.14031
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References:

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