Köhler's Conjecture on Hecke Theta Series of Weight One

Technical Summary: Köhler's Conjecture on Hecke Theta Series of Weight One

Problem Statement
The paper addresses a fundamental question regarding the uniqueness of the quadratic field associated with a Hecke theta series of weight one. By definition, such a series is constructed from a Hecke character of a specific quadratic field $K$. The central problem is to determine under what conditions the quadratic field $K$ is uniquely determined by the theta series itself. This inquiry is framed by a conjecture of Günter Köhler ([Köh11, Conjecture 5.5], [Köh18]), which posits specific conditions under which a theta series can be equivalently represented by characters of distinct quadratic fields.

Methodology
The authors, Mahima Kumar and Gabor Wiese, adopt a representation-theoretic approach to resolve Köhler's conjecture. Their methodology proceeds in three distinct stages:

1. Group-Theoretic Reduction: The authors first abstract the problem to the level of 2-dimensional Galois representations. They analyze representations $\rho$ induced from a character $\chi$ of an index-2 subgroup $H$ of a finite group $G$. Specifically, they investigate the conditions under which such a representation can also be induced from a different index-2 subgroup $H' \subset G$.
2. Explicit Matrix Realization: Unlike previous treatments that rely on abstract existence proofs, the authors provide explicit matrix realizations of these induced representations. They classify the possible images of these representations, identifying them as specific 2-groups (including dihedral groups $D_4$, quaternion groups $Q_8$, and modular maximal-cyclic groups). This allows for a concrete determination of when a representation admits multiple inducing subgroups.
3. Translation to Number Theory: Using the Artin Reciprocity Law and the theory of theta series, the authors translate the group-theoretic results back into the language of Dirichlet characters and Hecke theta series. They establish a correspondence between the ambiguity of the inducing subgroup in the Galois representation and the ambiguity of the underlying quadratic field in the theta series definition.

Key Contributions and Results

• Proof of a Modified Conjecture: The paper proves a slightly modified version of Köhler's conjecture (Theorem 4.1). The main result establishes the equivalence of the following statements for a Hecke theta series $\Theta_1(K, \xi, z)$:

  • (A) There exists a second quadratic field $K' \neq K$ and a character $\xi'$ such that $\Theta_1(K, \xi, z) = \Theta_1(K', \xi', z)$.
  • (B) The character $\xi$ satisfies specific algebraic conditions: it is not induced from a character of $\mathbb{Q}$ via the norm, and for any prime ideal $\mathfrak{p}$ coprime to the modulus, $\xi(\mathfrak{p}) = \pm \xi(\sigma(\mathfrak{p}))$, where $\sigma$ is the non-trivial automorphism of $K$.
  • (C) There exist two distinct quadratic fields $K', K''$ (different from $K$) yielding the same theta series.

• Characterization of Ambiguity: The authors demonstrate that if a theta series arises from a primitive character satisfying condition (B), it is associated with exactly three distinct quadratic fields. This corresponds to the group-theoretic finding that the image of the associated Galois representation is a group with three distinct index-2 subgroups (e.g., $D_4$, $Q_8$, or specific modular groups).

• Refinement of Rohrlich's Classification: In Proposition 2.2, the authors provide an explicit classification of the images of 2-dimensional representations induced by characters of two different index-2 subgroups. They refine the list of "almost abelian 2-groups" found in [Roh19a], offering concrete matrix generators and presentations for the groups involved (distinguishing cases based on the order of generators and the structure of the kernel).

• Counterexamples for Non-Primitive Characters: The paper clarifies that the equivalence of the conjecture's statements holds strictly for primitive characters. The authors construct explicit counterexamples showing that for non-primitive characters, the implication from the algebraic conditions (B) to the existence of alternative fields (A) may fail. This is due to the behavior of Fourier coefficients at primes dividing the modulus but not the conductor.

Significance and Claims
The authors claim that their work provides a complete proof of Köhler's conjecture (in its modified form) by deriving it from a well-known but previously unapplied group-theoretic fact regarding triply imprimitive representations.

• Independence and Explicitness: The paper emphasizes that its presentation is independent of prior works on "triply monomial" representations (e.g., [Roh19a], [ST18]) and is distinguished by its use of explicit matrix realizations. This concreteness allows for a direct verification of the conditions under which the quadratic field is ambiguous.
• Clarification of Conditions: The work clarifies the necessity of the "primitive" assumption in Köhler's original formulation. By removing an item from the original conjecture that the authors show does not always hold (regarding the order of characters), they provide a more precise and robust statement of the ambiguity phenomenon.
• Connection to Modular Forms: The results confirm that the ambiguity of the quadratic field is intrinsically linked to the structure of the associated Galois representation. Specifically, the theta series fails to uniquely determine the field if and only if the associated Galois representation is induced from characters of three distinct quadratic fields, a condition characterized by the specific group-theoretic properties of the image of the representation.

The paper concludes that the ambiguity of the quadratic field in the definition of Hecke theta series is a direct consequence of the representation-theoretic property where a 2-dimensional irreducible representation can be induced from multiple index-2 subgroups.