Frobenius Traces for Rank-2 Drinfeld Modules, Higher-Dimensional Galois Representations, and a Strong Multiplicity One Theorem in Positive Characteristic

Technical Summary: Frobenius Traces for Rank-2 Drinfeld Modules, Higher-Dimensional Galois Representations, and a Strong Multiplicity One Theorem in Positive Characteristic

Problem Statement
The paper addresses the extent to which Frobenius traces determine Galois representations up to isomorphism in the context of positive characteristic. In characteristic 0, the Brauer-Nesbitt theorem establishes that two semisimple representations are isomorphic if their Frobenius traces agree for all but finitely many places. However, in positive characteristic, this principle is not generally valid because the reconstruction of characteristic polynomials from traces via Newton identities requires divisibility conditions (specifically, that the characteristic of the field is greater than the dimension of the representation) which often fail. The paper investigates the specific conditions under which Frobenius trace equality still forces isomorphism for representations associated with Drinfeld modules and, more generally, for semisimple Galois representations over local fields of positive characteristic.

Methodology
The author employs a combination of algebraic number theory, the theory of Drinfeld modules, and algebraic group theory.

1. Drinfeld Module Specifics: For rank-2 Drinfeld modules, the author utilizes the Weil pairing to establish a relationship between the Frobenius determinant and the image of a representation associated with a twisted Carlitz module. This allows the reconstruction of the characteristic polynomial (which depends on both trace and determinant) without relying on Newton identities, thereby bypassing the divisibility constraints of positive characteristic.
2. Reduction to Finite Extensions: The proofs frequently involve restricting representations to a finite extension $L/K$ where the image of the Galois group becomes sufficiently "open" or irreducible. Lemma 3.1 demonstrates that for rank-2 non-CM Drinfeld modules, there exists a finite extension where the determinants of the representations become equal, reducing the problem to comparing traces on a subgroup where the representations are irreducible.
3. Algebraic Chebotarev Density: To generalize results to higher dimensions and arbitrary semisimple representations, the paper introduces an algebraic version of the Chebotarev density theorem. This involves analyzing the Zariski closure $H$ of the image of the Galois representation in $GL_r \times GL_r$. The author defines a "componentwise thinness" condition (CT) regarding the Haar measure of the intersection of the image with specific Zariski-closed subsets.
4. Burnside's Theorem and Schur's Lemma: In the general case (Proposition 3.4), the proof relies on Burnside's Theorem to show that the algebra generated by an absolutely irreducible representation is the full endomorphism ring. This is used to construct a non-zero equivariant map between two representations with matching traces, which, via Schur's Lemma, yields an isomorphism.

Key Contributions and Results

• Main Theorem 1 (Theorem 3.3): For two non-CM rank-2 Drinfeld modules over a finite extension of a function field, if their associated $\ell$-adic Galois representations have equal Frobenius traces for all but finitely many places, then the modules are isogenous, and the representations are isomorphic. This result holds without the assumption that the characteristic of the coefficient field is greater than the rank.
• Generalization to Higher Ranks (Corollary 3.7): The result is extended to non-CM Drinfeld modules of arbitrary rank $r$. If the Frobenius traces agree for all but finitely many places, the modules are isogenous. This relies on the existence of a prime $\tilde{\ell}$ of sufficiently large degree such that the associated representations are absolutely irreducible (a consequence of the Open Image Theorem for Drinfeld modules).
• Strong Multiplicity One for Positive Characteristic (Theorem 4.4): The paper formulates and proves a function field analogue of Rajan's strong multiplicity one theorem. It establishes that for two semisimple Galois representations $\rho_1, \rho_2$ over a local field of positive characteristic:
  • If the Zariski closure of the product representation satisfies a specific connectivity condition ($H^\circ \subset X$, where $X$ is the locus where characteristic polynomials agree), then the representations become isomorphic after restriction to a finite extension.
  • If $\rho_1$ is absolutely irreducible and its Zariski closure is connected, then $\rho_2$ is isomorphic to $\rho_1$ twisted by a finite character.
• Density Criteria (Corollaries 4.5 & 4.7): The paper provides criteria based on Dirichlet density. If the set of places where the characteristic polynomials agree has a density greater than $1 - 1/|\Phi|$ (where $\Phi$ is the component group of the Zariski closure), the representations are isomorphic (up to a finite character twist). This contrasts with Rajan's characteristic 0 results, noting that the strategy relying on unitary matrices and maximal compact subgroups does not directly apply in positive characteristic.

Significance
The paper claims to resolve the ambiguity of Frobenius trace determination in positive characteristic for a significant class of representations (those arising from non-CM Drinfeld modules). By avoiding the Newton identities, it provides a robust isogeny theorem for rank-2 and higher-rank Drinfeld modules. Furthermore, it establishes a rigorous framework for strong multiplicity one theorems in positive characteristic, identifying the precise algebraic conditions (irreducibility and Zariski connectivity) required to ensure that trace equality implies isomorphism. The work highlights the structural differences between characteristic 0 and positive characteristic settings, particularly regarding the necessity of irreducibility assumptions and the behavior of the Zariski closure of Galois images.