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**Beyond the Manin obstruction. – Appendix A by S. Siksek: 4-descent. – Appendix B: The Grothendieck spectral sequence and the truncation functor.**
*(English)*
Zbl 0951.14013

Let \(k\) be a field of characteristic \(0\) and let \({\mathbb A}_k\) be the adèle ring of \(k\). A smooth variety \(X\) over \(k\) such that \(X({\mathbb A}_k)\neq \emptyset\) whereas \(X(k)=\emptyset\) is a counterexample to the Hasse principle. The classical way of constructing counterexamples to the Hasse principle is via the Manin obstruction. Let \(\text{Br}(X)=H^2(X,{\mathbb G}_m)\) be the Grothendieck-Brauer group of \(X\). Define \(X({\mathbb A}_k)^{\text{Br}}\) to be the subset of points of \(X({\mathbb A}_k)\) orthogonal to all elements of \(\text{Br}(X)\). The image of \(X(k)\) under the diagonal embedding \(X(k)\hookrightarrow X({\mathbb A}_k)\) is contained in \(X({\mathbb A}_k)^{\text{Br}}\). A counterexample to the Hasse principle is accounted for by the Manin obstruction if already \(X({\mathbb A}_k)^{\text{Br}}=\emptyset\).

In this paper, a counterexample to the Hasse principle not accounted for by the Manin obstruction is constructed. Such a counterexample is given by a smooth proper surface defined over \({\mathbb Q}\) of Kodaira dimension \(\kappa=0\). More specifically, the construction of such a family of smooth proper surfaces over \(k\) is based on two pieces of data:

(1) an elliptic curve \(E\) over \(k\), a \(2\)-isogeny \(\psi: C\to E\) which lifts to a \(4\)-isogeny \(\psi'': C''\to E\) and such that \(C\) has a zero-cycle of degree \(2\) over \(k\),

(2) an unramified double covering \(\phi:D\to D''\) of curves of genus one.

Let \(Y=C\times D\), and let \(X\) be the quotient of \(Y\) by the fixed point free involution \((\sigma, \rho)\), \(f:Y\to X\) where \(\sigma:C\to C\) is the hyperelliptic involution, and \(\rho:D\to D\) is the fixed point free involution interchanging the sheets of the covering \(\phi:D\to D''\). Then \(X\) is a smooth proper geometrically integral surface with \(\kappa = 0\), \(p_g=0\), \(g=1\), \(K_X^2=0\) and \(b_1=b_2=2\). Such a family of surfaces \(X\) may be defined by the affine equations \((x^2+1)y^2=(x^2+2)z^2=3(t^4-54t^2-117t-243)\) taking \(E: y^2=x^3-1221\), \(C: y^2 =3(t^4-54t^2-117t-243)\) and \(D\) defined by \(y^2=x^2+1\), \(z^2=x^2+2\). (In the appendix, S. Siksek exhibits \(4\)-descent of the curve \(E\), giving rise to an element of exact order \(4\).)

Theorem: For the family of surfaces \(X\) constructed above, \(X({\mathbb{Q}})=\emptyset\), but \(X({\mathbb{A}}_{\mathbb{Q}})^{\text{Br}}\neq\emptyset\).

A refinement of the Manin obstruction is introduced combining the theory of descent and the Manin obstruction. Then it is shown that this refined Manin obstruction is the only obstruction to the Hasse principle for the family of surfaces. This refinement is defined extending the descent theory of J.-L. Colliot-Thélène and J.-J. Sansuc [Duke Math. J. 54, 375-492 (1987; Zbl 0659.14028)] to arbitrary torsors under groups of multiplicative type.

In this paper, a counterexample to the Hasse principle not accounted for by the Manin obstruction is constructed. Such a counterexample is given by a smooth proper surface defined over \({\mathbb Q}\) of Kodaira dimension \(\kappa=0\). More specifically, the construction of such a family of smooth proper surfaces over \(k\) is based on two pieces of data:

(1) an elliptic curve \(E\) over \(k\), a \(2\)-isogeny \(\psi: C\to E\) which lifts to a \(4\)-isogeny \(\psi'': C''\to E\) and such that \(C\) has a zero-cycle of degree \(2\) over \(k\),

(2) an unramified double covering \(\phi:D\to D''\) of curves of genus one.

Let \(Y=C\times D\), and let \(X\) be the quotient of \(Y\) by the fixed point free involution \((\sigma, \rho)\), \(f:Y\to X\) where \(\sigma:C\to C\) is the hyperelliptic involution, and \(\rho:D\to D\) is the fixed point free involution interchanging the sheets of the covering \(\phi:D\to D''\). Then \(X\) is a smooth proper geometrically integral surface with \(\kappa = 0\), \(p_g=0\), \(g=1\), \(K_X^2=0\) and \(b_1=b_2=2\). Such a family of surfaces \(X\) may be defined by the affine equations \((x^2+1)y^2=(x^2+2)z^2=3(t^4-54t^2-117t-243)\) taking \(E: y^2=x^3-1221\), \(C: y^2 =3(t^4-54t^2-117t-243)\) and \(D\) defined by \(y^2=x^2+1\), \(z^2=x^2+2\). (In the appendix, S. Siksek exhibits \(4\)-descent of the curve \(E\), giving rise to an element of exact order \(4\).)

Theorem: For the family of surfaces \(X\) constructed above, \(X({\mathbb{Q}})=\emptyset\), but \(X({\mathbb{A}}_{\mathbb{Q}})^{\text{Br}}\neq\emptyset\).

A refinement of the Manin obstruction is introduced combining the theory of descent and the Manin obstruction. Then it is shown that this refined Manin obstruction is the only obstruction to the Hasse principle for the family of surfaces. This refinement is defined extending the descent theory of J.-L. Colliot-Thélène and J.-J. Sansuc [Duke Math. J. 54, 375-492 (1987; Zbl 0659.14028)] to arbitrary torsors under groups of multiplicative type.

Reviewer: N.Yui (Kingston/Ontario)