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Atkin-Lehner Decompositions for Quaternionic modular forms

Siddharth Ramakrishnan Cherukara

TL;DR

This work develops Atkin–Lehner–style decompositions for spaces of quaternionic modular forms on definite quaternion algebras by translating the problem into a representation-theoretic framework and applying Jacquet–Langlands transfer. It provides explicit local decompositions at ramified primes, including odd conductors and ramified quadratic extensions, and then globalizes these results to yield precise isomorphisms between quaternionic cusp spaces and corresponding Hilbert/cuspidal $GL_2$ spaces. The authors obtain detailed decompositions of new and old subspaces, with exact formulas for how even and odd level components assemble from supercuspidal and dihedrally induced representations. These results generalize prior work by accommodating ramified local data and odd-level phenomena, enabling refined basis problems, computational approaches, and isolation of modular forms with fixed local Galois types. The work thus strengthens the bridge between quaternionic and classical Hilbert modular forms and provides practical tools for studying automorphic representations on both sides via Brandt matrices and the unramified Hecke algebra.

Abstract

In this paper, we obtain Atkin--Lehner decompositions for spaces of modular forms on definite quaternion algebras. Similar to Casselman's approach our methods are representation theoretic. Using Jacquet--Langlands correspondence we also obtain isomorphisms between spaces of quaternionic modular forms and corresponding spaces of Hilbert modular forms.

Atkin-Lehner Decompositions for Quaternionic modular forms

TL;DR

This work develops Atkin–Lehner–style decompositions for spaces of quaternionic modular forms on definite quaternion algebras by translating the problem into a representation-theoretic framework and applying Jacquet–Langlands transfer. It provides explicit local decompositions at ramified primes, including odd conductors and ramified quadratic extensions, and then globalizes these results to yield precise isomorphisms between quaternionic cusp spaces and corresponding Hilbert/cuspidal spaces. The authors obtain detailed decompositions of new and old subspaces, with exact formulas for how even and odd level components assemble from supercuspidal and dihedrally induced representations. These results generalize prior work by accommodating ramified local data and odd-level phenomena, enabling refined basis problems, computational approaches, and isolation of modular forms with fixed local Galois types. The work thus strengthens the bridge between quaternionic and classical Hilbert modular forms and provides practical tools for studying automorphic representations on both sides via Brandt matrices and the unramified Hecke algebra.

Abstract

In this paper, we obtain Atkin--Lehner decompositions for spaces of modular forms on definite quaternion algebras. Similar to Casselman's approach our methods are representation theoretic. Using Jacquet--Langlands correspondence we also obtain isomorphisms between spaces of quaternionic modular forms and corresponding spaces of Hilbert modular forms.

Paper Structure

This paper contains 10 sections, 16 theorems, 37 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Applications
  4. Preliminaries
  5. Local Preliminaries
  6. One dimensional representations
  7. Higher dimensional representations
  8. Global Preliminaries
  9. Local Results
  10. Global Results

Key Result

Theorem 1.1

Let $\pi_{\mathfrak{p}}$ be an irreducible smooth representation of $B^{\times}_{\mathfrak{p}}$ of odd conductor and trivial central character, dihedrally induced from $K_{\mathfrak{p}}$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 18 more