On higher regulators of Siegel varieties

Cauchi, Antonio; Lemma, Francesco; Jacinto, Joaquín Rodrigues

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 767-804

Inspired by Beilinson conjectures, we provide a construction of classes in the middle degree plus one motivic cohomology of the Siegel Shimura variety of almost any dimension. We compute their image by Beilinson’s higher regulator in terms of Rankin–Selberg type automorphic integrals. Our construction generalizes the one for $\mathrm{GSp}\left(4\right)$ in [21] and for $\mathrm{GSp}\left(6\right)$ in [4]. For Siegel varieties associated to small genus symplectic groups, we also show how these integrals unfold.

Inspirés par les conjectures de Beilinson, nous construisons des classes en degré moitié plus un dans la cohomologie motivique des variétés modulaires de Siegel de presque toute dimension. Nous calculons leur image par le régulateur supérieur de Beilinson en termes d’intégrales adéliques de type Rankin–Selberg. Notre construction généralise celle pour $\mathrm{GSp}\left(4\right)$ de [21] et celle pour $\mathrm{GSp}\left(6\right)$ de [4]. Dans le cas de variétés de Siegel associées à des groupes symplectiques de petit genre, nous démontrons également que ces intégrales se déploient.

Classification : 11G18, 11G16, 11G40, 11F46, 11F67, 19F27
Keywords: Siegel Shimura varieties, Beilinson conjectures, Rankin–Selberg integrals

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