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

Antonio Cauchi, Francesco Lemma, Joaquín Rodrigues Jacinto

Abstract

We study instances of Beilinson-Tate conjectures for automorphic representations of $\mathrm{PGSp}_6$ whose Spin $L$-function has a pole at $s=1$. We construct algebraic cycles of codimension three in the Siegel-Shimura variety of dimension six and we relate its regulator to the residue at $s=1$ of the $L$-function of certain cuspidal forms of $\mathrm{PGSp}_6$. Using the exceptional theta correspondence between the split group of type $G_2$ and $\mathrm{PGSp}_6$ and assuming the non-vanishing of a certain archimedean integral, this allows us to confirm a conjecture of Gross and Savin on rank $7$ motives of type $G_2$.

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

Abstract

We study instances of Beilinson-Tate conjectures for automorphic representations of whose Spin -function has a pole at . We construct algebraic cycles of codimension three in the Siegel-Shimura variety of dimension six and we relate its regulator to the residue at of the -function of certain cuspidal forms of . Using the exceptional theta correspondence between the split group of type and and assuming the non-vanishing of a certain archimedean integral, this allows us to confirm a conjecture of Gross and Savin on rank motives of type .
Paper Structure (60 sections, 55 theorems, 309 equations, 1 table)

This paper contains 60 sections, 55 theorems, 309 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Overview of the proofs
  5. Structure of the manuscript
  6. Acknowledgements
  7. Preliminaries
  8. Algebraic groups and algebraic representations
  9. Compact Lie groups and representations
  10. Lie algebras
  11. Weyl groups
  12. Discrete series
  13. Shimura varieties
  14. Cohomology of Siegel sixfolds
  15. Hodge theory
  16. ...and 45 more sections

Key Result

Theorem 1.2

Let $\pi = \pi_\infty \otimes \pi_f$ be a cuspidal automorphic representation of $\mathrm{PGSp}_6(\mathbf{A})$ such that $\pi_\infty$ is a discrete series of Hodge type $(3,3)$ in the discrete series $L$-packet of $V^\lambda$. Then where $C$ is an explicit non-zero constant independent of $\pi$, $S$ is a sufficiently large set of places containing the ramified and archimedean places, $\Psi^{[\lam

Theorems & Definitions (128)

  • Conjecture 1.1
  • Theorem 1.2: Theorem \ref{['theoremcyclebetti1']}
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Theorem \ref{['TheoGS']}
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: Theorem \ref{['cuspidality']}, Proposition \ref{['prop:comparisonbetweenFC']}
  • Definition 2.1
  • Lemma 2.2
  • ...and 118 more