Algebraic cycles and functorial lifts from G2 to PGSp6

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

doi:10.2140/ant.2025.19.551

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Algebraic cycles and functorial lifts from $G_2$ to $\mathrm{PGSp}_6$

Antonio Cauchi, Francesco Lemma and Joaquín Rodrigues Jacinto

Vol. 19 (2025), No. 3, 551–616

DOI: 10.2140/ant.2025.19.551

Abstract

We study instances of Beilinson–Tate conjectures for automorphic representations of ${PGSp}_{6}$ whose spin $L$ -function has a pole at $s = 1$ . We construct algebraic cycles of codimension 3 in the Siegel–Shimura variety of dimension 6 and we relate its regulator to the residue at $s = 1$ of the $L$ -function of certain cuspidal forms of ${PGSp}_{6}$ . Using the exceptional theta correspondence between the split group of type $G_{2}$ and ${PGSp}_{6}$ and assuming the nonvanishing of a certain archimedean integral, this allows us to confirm a conjecture of Gross and Savin on rank- $7$ motives of type $G_{2}$ .

Keywords

theta correspondence, algebraic cycles, Beilinson conjecture, Tate conjecture

Mathematical Subject Classification

Primary: 11F46, 11F67, 14G10, 14G35

Milestones

Received: 26 April 2023

Revised: 25 January 2024

Accepted: 29 April 2024

Published: 20 February 2025

Authors

Antonio Cauchi
Deptartment of Mathematics Tokyo Institute of Technology Tokyo Japan

Francesco Lemma
Université Paris Cité, CNRS, IMJ–PRG Paris France

Joaquín Rodrigues Jacinto
Aix–Marseille Université Marseille France

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Vol. 19, No. 3, 2025