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On converse theorems for Hilbert modular forms assuming unramified twists

Pengcheng Zhang

TL;DR

The paper develops converse theorems for Hilbert modular forms over totally real fields by linking L-series to modularity via unramified twists. It first shows that holomorphic continuation and functional equations for all unramified twists force the associated $h$-tuple of functions to exhibit Fricke symmetry, and a Vaserstein-type argument then yields invariance under a large congruence subgroup, proving Hilbert modularity at level $K_1(\frak r\frak n)$. It then leverages a partial Euler product and infinity-type Hecke operators to promote the form to the predicted level $K_0(\frak n)$, under suitable Euler-factor hypotheses. Collectively, the results extend classical converse theorems from GL$_2$ to Hilbert modular forms by eliminating ramified-twist requirements and clarifying how unramified data determines modularity. The approach connects functional equations, Euler products, and congruence-subgroup invariance to yield explicit level predictions and a path toward broader automorphic converse results.

Abstract

We prove two results on converse theorems for Hilbert modular forms over totally real fields of degree $r>1$. The first result recovers a Hilbert modular form (of some level) from an $L$-series satisfying functional equations twisted by all the unramified Hecke characters. The second result assumes both the 'unramified' functional equations and an Euler product, and recovers a Hilbert modular form of the expected level predicted by the shape of the functional equations. Our result generalizes the current converse theorems for $\mathrm{GL}_2$ in the case of Hilbert modular forms in that we completely remove the assumptions on ramified twists.

On converse theorems for Hilbert modular forms assuming unramified twists

TL;DR

The paper develops converse theorems for Hilbert modular forms over totally real fields by linking L-series to modularity via unramified twists. It first shows that holomorphic continuation and functional equations for all unramified twists force the associated -tuple of functions to exhibit Fricke symmetry, and a Vaserstein-type argument then yields invariance under a large congruence subgroup, proving Hilbert modularity at level . It then leverages a partial Euler product and infinity-type Hecke operators to promote the form to the predicted level , under suitable Euler-factor hypotheses. Collectively, the results extend classical converse theorems from GL to Hilbert modular forms by eliminating ramified-twist requirements and clarifying how unramified data determines modularity. The approach connects functional equations, Euler products, and congruence-subgroup invariance to yield explicit level predictions and a path toward broader automorphic converse results.

Abstract

We prove two results on converse theorems for Hilbert modular forms over totally real fields of degree . The first result recovers a Hilbert modular form (of some level) from an -series satisfying functional equations twisted by all the unramified Hecke characters. The second result assumes both the 'unramified' functional equations and an Euler product, and recovers a Hilbert modular form of the expected level predicted by the shape of the functional equations. Our result generalizes the current converse theorems for in the case of Hilbert modular forms in that we completely remove the assumptions on ramified twists.
Paper Structure (15 sections, 31 theorems, 112 equations)

This paper contains 15 sections, 31 theorems, 112 equations.

Table of Contents

  1. Introduction and the main result
  2. Background and introduction
  3. Setup and the main result
  4. Acknowledgements
  5. Preliminaries and notations
  6. Notations
  7. Hecke characters
  8. Hilbert modular forms
  9. A candidate for the Hilbert modular form
  10. Functional equations
  11. Generating sets of congruence subgroups
  12. Invariance of $f_\lambda$ under $\Gamma_1^+(\mathfrak{t}_\lambda,\mathfrak{r}\mathfrak{n})$
  13. The Euler product
  14. Invariance of $f_\lambda$ under $\Gamma_0^+(\mathfrak{t}_\lambda,\mathfrak{n})$
  15. Lemmas on prime ideals

Key Result

Theorem 1.1

Let $F$ be a totally real field of degree $r>1$. Let $\mathfrak{n}$ be a nonzero integral ideal of $\mathcal{O}_F$, $\mathbf{k}\in(2\mathbb{Z}^+)^r$, and $\epsilon=\pm1$. Let $\{\mathfrak{t}_1,\ldots,\mathfrak{t}_h\}$ be any set of representatives of $\mathrm{Cl}^+(F)$ and let $\mathfrak{r}=\mathfra where $|\mathbf{k}|=\sum_{j=1}^rk_j$. Then, $L(s)$ is the $L$-function of a Hilbert modular form of

Theorems & Definitions (59)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • ...and 49 more