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Explicit Hecke eigenform product identities for Hilbert modular forms

Zeping Hao, Chao Qin, Yang Zhou

Abstract

Let $F$ be a totally real number field, and $g,f,h$ be Hilbert modular forms over $F$ that are Hecke eigenforms satisfying $g=f\cdot h$. We characterize such product identities among all real quadratic fields of narrow class number one, proving they occur only for $F=\mathbb Q(\sqrt{5})$, with precisely two such identities. We also shed some light on the general totally real case by showing that no such identity exists when both $f$ and $h$ are Eisenstein series of distinct weights.

Explicit Hecke eigenform product identities for Hilbert modular forms

Abstract

Let be a totally real number field, and be Hilbert modular forms over that are Hecke eigenforms satisfying . We characterize such product identities among all real quadratic fields of narrow class number one, proving they occur only for , with precisely two such identities. We also shed some light on the general totally real case by showing that no such identity exists when both and are Eisenstein series of distinct weights.

Paper Structure

This paper contains 7 sections, 14 theorems, 64 equations, 3 tables.

Table of Contents

  1. Introduction and Statement of the Main Theorem
  2. Preliminaries
  3. The case of two Eisenstein series
  4. The unequal-weight case
  5. The equal-weight case
  6. The case of Eisenstein series and cusp forms
  7. On totally real number fields beyond the quadratic case

Key Result

Theorem 1

Over all real quadratic fields $F$ of narrow class number one and full-level Hecke eigenforms of parallel weights, eigenform product identities exist only for $F=\mathbb Q(\sqrt 5)$ under the grand Riemann hypothesis, with explicit identities as established in Joshi-Zhang2019Hilbert. Here $E_k$ is the normalized Hecke eigenform of weight $k$ in the Eisenstein subspace, and $h_6, h_8$ are the uniq

Theorems & Definitions (29)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Proposition 3.8
  • proof
  • ...and 19 more