##
**Vinberg’s representations and arithmetic invariant theory.**
*(English)*
Zbl 1321.11045

Bhargava and his collaborators have found striking results about the average size of Selmer groups of Jacobians of algebraic curves over \(\mathbb{Q}\) as these curves vary in some natural families. Their method relies on a relation between rational orbits in coregular representations and Galois cohomology of Jacobian of the algebraic curves. In the work under review, the author generalise this relation to certain special Vinberg’s representation. The main result is the following:

Let \(k\) be a field of characteristic \(0\), \(G\) a split simple adjoint group over \(k\) of type \(A,\,D,\,E\). Let \(\theta\) be a stable involution of \(G\). Let \(G_0=G^\theta,\, \mathfrak{g}_1=\{a\in \mathfrak{g}\,|\,d\theta(a)=-a\}\). Let \(B=\mathfrak{g}_{1}//G_{0}\), where \(G_{0}\) acts on \(\mathfrak{g}_{1}\) by conjugation. Fix a sub regular nilpotent element \(e\in \mathfrak{g}_{1}\), let \(X\) be the intersection of \(\mathfrak{g}_{1}\) with the Slodowy slice passing through \(e\). Consider the projection \(X\subset \mathfrak{g}_{1}\to B=\mathfrak{g}_{1}//G_{0}\), which can be shown to be a family of affine reduced connected curves. This family can be compactified to a family of projective curves \(Y\to B\). Over \(B\), there exists a finite commutative group scheme \(Z\to B\), whose fiber \(Z_{b}\) over \(b\in \mathfrak{g_{1}}^{\mathrm{reg}}//G_{0}\subset B\) is isomorphic to the stabiliser in \(G_{0}\) of any regular element in \(\mathfrak{g}_{1,b}\). The first key result is that one has a natural isomorphism \(J_{Y_{b}}[2]\cong Z_{b}\) for any \(b\in B^{\mathrm{rs}}\), where \(J_{Y_{b}}\) is the Jacobian of the curve \(Y_{b}\). From this one deduce a bijection \[ \mathfrak{g}_{1,b}(k)/G_{0}(k)\cong \ker\{H^{1}(k, \,J_{Y_{b}}[2])\to H^{1}(k, G_{0})\}, \] by the usual machinery of Galois cohomology. The author goes on to show that the map \[ X_{b}(k)\subset \mathfrak{g}_{1,b}(k) \to \mathfrak{g}_{1,b}(k)/G_{0}(k) \] can be interpreted in terms of 2-descent on the Jacobian \[ J_{Y_{b}}(k) \to H^{1}(k,\,J_{Y_{b}}[2]). \]

Let \(k\) be a field of characteristic \(0\), \(G\) a split simple adjoint group over \(k\) of type \(A,\,D,\,E\). Let \(\theta\) be a stable involution of \(G\). Let \(G_0=G^\theta,\, \mathfrak{g}_1=\{a\in \mathfrak{g}\,|\,d\theta(a)=-a\}\). Let \(B=\mathfrak{g}_{1}//G_{0}\), where \(G_{0}\) acts on \(\mathfrak{g}_{1}\) by conjugation. Fix a sub regular nilpotent element \(e\in \mathfrak{g}_{1}\), let \(X\) be the intersection of \(\mathfrak{g}_{1}\) with the Slodowy slice passing through \(e\). Consider the projection \(X\subset \mathfrak{g}_{1}\to B=\mathfrak{g}_{1}//G_{0}\), which can be shown to be a family of affine reduced connected curves. This family can be compactified to a family of projective curves \(Y\to B\). Over \(B\), there exists a finite commutative group scheme \(Z\to B\), whose fiber \(Z_{b}\) over \(b\in \mathfrak{g_{1}}^{\mathrm{reg}}//G_{0}\subset B\) is isomorphic to the stabiliser in \(G_{0}\) of any regular element in \(\mathfrak{g}_{1,b}\). The first key result is that one has a natural isomorphism \(J_{Y_{b}}[2]\cong Z_{b}\) for any \(b\in B^{\mathrm{rs}}\), where \(J_{Y_{b}}\) is the Jacobian of the curve \(Y_{b}\). From this one deduce a bijection \[ \mathfrak{g}_{1,b}(k)/G_{0}(k)\cong \ker\{H^{1}(k, \,J_{Y_{b}}[2])\to H^{1}(k, G_{0})\}, \] by the usual machinery of Galois cohomology. The author goes on to show that the map \[ X_{b}(k)\subset \mathfrak{g}_{1,b}(k) \to \mathfrak{g}_{1,b}(k)/G_{0}(k) \] can be interpreted in terms of 2-descent on the Jacobian \[ J_{Y_{b}}(k) \to H^{1}(k,\,J_{Y_{b}}[2]). \]

Reviewer: Zongbin Chen (Lausanne)

### MSC:

11E72 | Galois cohomology of linear algebraic groups |

20G30 | Linear algebraic groups over global fields and their integers |

14L24 | Geometric invariant theory |