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**Equivariant birational geometry and modular symbols.**
*(English)*
Zbl 1518.14016

The paper under review constructs two infinite sequences of abelian groups attached to birational equivalenace classes of smooth, irreducible algebraic varieties \(X\) equipped with a birational and generically free action by a finite abelian group \(G\).

The construction proceeds by identifying the \(G\)-fixed locus as a disjoint union of irreducible subvarieties \(F_\alpha\). To each \(F_\alpha\) the authors associate an element of the abelian group \(\mathcal{B}_n(G)\) and an element of the abelian group \(\mathcal{M}_n(G)\), where \(n\) is the dimension of \(X\). The two groups \(\mathcal{B}\) and \(\mathcal{M}\) are defined by the authors using natural relations satisfied by the characters of \(G\) arising from the action of \(G\) on the tangent space to \(X\) at points of \(F_\alpha\). The authors prove that \(\mathcal{B}_n(G)\) is a \(G\)-equivariant birational invariant.

The authors conjecture that there are deep connections between this construction and automorphic forms. They provide evidence for these connections, including the construction of analogues of Hecke operators on \(\mathcal{B}_n(G)\) and \(\mathcal{M}_n(G)\) and the construction of a natural representation \(\mathcal{F}_n\) corresponding to a cohomological automorphic form, amongst many others.

The authors also point the way to future work by making various specific conjectures on the behaviour and structure of the groups \(\mathcal{B}_n(G)\) and \(\mathcal{M}_n(G)\).

The construction proceeds by identifying the \(G\)-fixed locus as a disjoint union of irreducible subvarieties \(F_\alpha\). To each \(F_\alpha\) the authors associate an element of the abelian group \(\mathcal{B}_n(G)\) and an element of the abelian group \(\mathcal{M}_n(G)\), where \(n\) is the dimension of \(X\). The two groups \(\mathcal{B}\) and \(\mathcal{M}\) are defined by the authors using natural relations satisfied by the characters of \(G\) arising from the action of \(G\) on the tangent space to \(X\) at points of \(F_\alpha\). The authors prove that \(\mathcal{B}_n(G)\) is a \(G\)-equivariant birational invariant.

The authors conjecture that there are deep connections between this construction and automorphic forms. They provide evidence for these connections, including the construction of analogues of Hecke operators on \(\mathcal{B}_n(G)\) and \(\mathcal{M}_n(G)\) and the construction of a natural representation \(\mathcal{F}_n\) corresponding to a cohomological automorphic form, amongst many others.

The authors also point the way to future work by making various specific conjectures on the behaviour and structure of the groups \(\mathcal{B}_n(G)\) and \(\mathcal{M}_n(G)\).

Reviewer: David McKinnon (Waterloo)

### MSC:

14E07 | Birational automorphisms, Cremona group and generalizations |

11F75 | Cohomology of arithmetic groups |

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\textit{M. Kontsevich} et al., J. Eur. Math. Soc. (JEMS) 25, No. 1, 153--202 (2023; Zbl 1518.14016)

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