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Hermitian modular forms and algebraic modular forms on $SO(6)$

Tomoyoshi Ibukiyama, Brandon Williams

TL;DR

The authors propose a Langlands‑type bridge between Hermitian modular forms of degree two and algebraic modular forms for SO(6) (via Spin(6)) using senary lattices attached to imaginary quadratic fields. They develop explicit Hecke theory, lift constructions, and theta maps on both sides, and they formulate a precise conjecture: nonlift Hermitian eigenforms correspond to non-Yoshida algebraic eigenforms with matching L‑functions and spinor characters. Evidence is gathered from mass/volume calculations and exact dimension data for small discriminants, plus explicit Euler factors and Borcherds/CM constructions illustrating the lifts and kernels. The results suggest a robust structural parallel between the two theories, with potential for a full correspondence once the Eichler basis problems for the relevant Weil representations are settled and all lift types are understood.

Abstract

We state conjectures that relate Hermitian modular forms of degree two and algebraic modular forms for the compact group $SO(6)$. We provide evidence for these conjectures in the form of dimension formulas and explicit computations of eigenforms.

Hermitian modular forms and algebraic modular forms on $SO(6)$

TL;DR

The authors propose a Langlands‑type bridge between Hermitian modular forms of degree two and algebraic modular forms for SO(6) (via Spin(6)) using senary lattices attached to imaginary quadratic fields. They develop explicit Hecke theory, lift constructions, and theta maps on both sides, and they formulate a precise conjecture: nonlift Hermitian eigenforms correspond to non-Yoshida algebraic eigenforms with matching L‑functions and spinor characters. Evidence is gathered from mass/volume calculations and exact dimension data for small discriminants, plus explicit Euler factors and Borcherds/CM constructions illustrating the lifts and kernels. The results suggest a robust structural parallel between the two theories, with potential for a full correspondence once the Eichler basis problems for the relevant Weil representations are settled and all lift types are understood.

Abstract

We state conjectures that relate Hermitian modular forms of degree two and algebraic modular forms for the compact group . We provide evidence for these conjectures in the form of dimension formulas and explicit computations of eigenforms.

Paper Structure

This paper contains 23 sections, 7 theorems, 207 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Hermitian modular forms of degree two
  3. Hermitian modular forms
  4. Hecke operators
  5. Formulas for Hecke operators
  6. Inert primes
  7. Split primes
  8. Liftings of elliptic modular forms
  9. Atkin--Lehner theory
  10. Dimensions
  11. Algebraic modular forms
  12. Algebraic modular forms on SO
  13. Hecke operators on algebraic modular forms
  14. L-functions for orthogonal modular forms
  15. The theta map
  16. ...and 8 more sections

Key Result

Theorem 3

The dimensions of Hermitian modular forms of degree two for discriminants $\Delta = -3, -4, -7, -8, -11$ have the following generating series. (i) For $\Delta = -3$, (ii) For $\Delta = -4$, (iii) When $\Delta = -7$, with the polynomial (iv) When $\Delta= -8$, with the polynomial (v) When $\Delta = -11$, with the polynomial

Figures (6)

  • Figure 1: Hermitian eigenforms for discriminant $-3$. Note that the first eigenform with spinor character ${\mathrm{spin}}_3 \otimes \mathrm{det}$ occurs in weight 45 and therefore does not appear in the table.
  • Figure 2: Hermitian eigenforms for discriminant $-4$
  • Figure 3: Hermitian eigenforms for discriminant $-7$
  • Figure 4: Hermitian eigenforms for discriminant $-8$
  • Figure 5: Hermitian eigenforms for discriminant $-11$
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1
  • Example 2
  • Theorem 3
  • proof
  • Remark 4
  • Conjecture 5
  • Remark 6
  • Example 7
  • Example 8
  • Conjecture 9: Main conjecture
  • ...and 11 more