Köhler's Conjecture on Hecke Theta Series of Weight One

TL;DR

The work addresses when a Hecke theta series of weight one determines the underlying quadratic field and proves a refined Köhler conjecture by translating the problem into a 2-dimensional induced Galois representation framework. It develops a complete representation-theoretic picture for Ind_H^G(χ), including irreducibility criteria, kernels, and a full classification of possible image groups as quotients by the center, linking these to explicit matrix realizations. By relating Dirichlet characters on quadratic fields to weight-one theta series via Artin reciprocity, the authors connect theta-series identities to isomorphism classes of induced Galois representations and establish equivalences among conjectural statements in the primitive case. The results illuminate how equal theta series across distinct quadratic fields correspond to the same irreducible Galois representation, providing a concrete mechanism to understand the quadratic-field ambiguity in Hecke theta series and contributing to the broader understanding of weight-one modular forms and their Galois realizations.

Abstract

In this article, we prove a conjecture of Günter Köhler on the ambiguity of the quadratic field in the definition of Hecke theta series by deriving it from a similar statement on two-dimensional Galois representations induced from characters of quadratic fields.

Theorems & Definitions (14)

• Proposition 2.1
• proof
• Proposition 2.2
• Remark 2.3
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• Theorem 3.4
• Theorem 3.5: Artin Reciprocity Law
• Theorem 3.6