On higher regulators of Siegel varieties

Antonio Cauchi; Francesco Lemma; Joaquín Rodrigues Jacinto

doi:10.5802/jtnb.1295

Antonio Cauchi ¹ ; Francesco Lemma ² ; Joaquín Rodrigues Jacinto ³

¹ University College Dublin, School of Mathematics and Statistics Science Centre - South Belfield Dublin 4, Dublin Ireland
² Université Paris Cité, CNRS, IMJ–PRG bâtiment Sophie Germain, case 7012 75205 Paris Cedex 13 France
³ Aix–Marseille Université, CNRS, Centrale Marseille, I2M. Campus de Luminy, Avenue de Luminy, Case 930 13288 Marseille Cedex 9 France

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

Abstract (VO)
Résumé

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 $GSp (4)$ in [21] and for $GSp (6)$ 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 $GSp (4)$ de [21] et celle pour $GSp (6)$ 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.

Reçu le : 2022-08-17
Révisé le : 2024-08-28
Accepté le : 2024-09-20
Publié le : 2025-03-31

DOI : 10.5802/jtnb.1295

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

Affiliations des auteurs :

Antonio Cauchi ¹ ; Francesco Lemma ² ; Joaquín Rodrigues Jacinto ³

¹ University College Dublin, School of Mathematics and Statistics Science Centre - South Belfield Dublin 4, Dublin Ireland
² Université Paris Cité, CNRS, IMJ–PRG bâtiment Sophie Germain, case 7012 75205 Paris Cedex 13 France
³ Aix–Marseille Université, CNRS, Centrale Marseille, I2M. Campus de Luminy, Avenue de Luminy, Case 930 13288 Marseille Cedex 9 France

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On higher regulators of {Siegel} varieties},
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Antonio Cauchi; Francesco Lemma; Joaquín Rodrigues Jacinto. On higher regulators of Siegel varieties. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 767-804. doi : 10.5802/jtnb.1295. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1295/

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