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Gross's conjecture: the dihedral case

Petar Bakić, Aleksander Horawa, Siyan Daniel Li-Huerta, Naomi Sweeting

Abstract

Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic modular forms on $\mathsf{G}_2$ associated via functoriality with certain modular forms on $\mathrm{PGL}_2$, Gross conjectured in 2000 that their Fourier coefficients encode $L$-values of cubic twists of the modular form (echoing Waldspurger's work on Fourier coefficients of half-integral weight modular forms). We prove Gross's conjecture when the modular forms are dihedral, giving the first examples for which it is known.

Gross's conjecture: the dihedral case

Abstract

Quaternionic modular forms on carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic modular forms on associated via functoriality with certain modular forms on , Gross conjectured in 2000 that their Fourier coefficients encode -values of cubic twists of the modular form (echoing Waldspurger's work on Fourier coefficients of half-integral weight modular forms). We prove Gross's conjecture when the modular forms are dihedral, giving the first examples for which it is known.

Paper Structure

This paper contains 34 sections, 30 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Quaternionic modular forms on $\mathsf{G}_2$
  3. Gross's conjecture
  4. Related work
  5. Idea of proof
  6. Unitary group theta lifts
  7. Unitary groups and algebra embeddings
  8. The seesaw and local representations of $G'$
  9. Spherical vectors for $G'$
  10. Local torus periods
  11. Local vectors for $G'$
  12. Archimedean case
  13. Global torus periods
  14. Formulas for Fourier coefficients
  15. Quaternionic discrete series on $\mathsf{G}_2$
  16. ...and 19 more sections

Key Result

Theorem 1.2

Assume that $L(\frac{1}{2},f)\neq0$. For all $\mathcal{E}\in\mathbb X(\mathbb Q)$, the Fourier coefficient $a_{\mathcal{E}}(\mathcal{F}^\epsilon)$ vanishes unless $\mathcal{E}\in\mathbb X(\mathbb Z)$. Moreover, if $\mathcal{E}\in\mathbb X(\mathbb Z)$ corresponds to the ring of integers of a totally

Theorems & Definitions (75)

  • Conjecture 1.1: Gross ChaoLi
  • Theorem 1.2: Theorem \ref{['thm:refined']}
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • ...and 65 more