Regulators in the Arithmetic of Function Fields

Problem Statement

This paper addresses the development of a regulator theory for Anderson $A$-motives over function fields, aiming to establish analogues of Beilinson's conjectures known in the arithmetic of number fields. Specifically, the author seeks to define a regulator morphism for rigid analytically trivial Anderson $A$-motives and investigate the finiteness and rank properties of the associated motivic cohomology groups.

In the classical setting over number fields, Beilinson's conjectures relate the dimension of extension groups of mixed motives (specifically $\text{Ext}^1_{\mathbb{Z}}(1, M)$) to special values of $L$-functions via a regulator map to Hodge realization spaces. In the function field setting, previous work by the author ([Gaz22]) initiated the study of $A$-motivic cohomology but encountered obstacles: the natural module of extensions, $\text{Ext}^1_{A}(1, M)$, is generally not finitely generated. This failure stems from two sources: the non-exactness of taking Hodge filtrations and the infinite nature of the absolute Galois group $G_\infty$ at the place at infinity. The paper aims to resolve these issues by defining a refined module of extensions with "analytic reduction" and constructing a corresponding regulator to Hodge-Pink structures.

Methodology

The author employs a synthesis of rigid analytic geometry, Galois cohomology, and the theory of shtuka models. The methodology proceeds through several key stages:

1. Rigid Analytically Trivial A-Motives and Betti Realization: The study is restricted to the subcategory of rigid analytically trivial $A$-motives ($\text{AMot}^{\text{rat}}_K$). For such motives, the Betti realization functor $M \mapsto M_B$ is exact. The author establishes that $M_B$ carries a natural continuous action of the profinite Galois group $G_\infty = \text{Gal}(K_\infty^s | K_\infty)$.
2. Analytic Reduction and Regulated Extensions: To address the lack of finite generation, the author defines "analytic reduction at $\infty$" for extensions. An extension has analytic reduction if it splits as a representation of $G_\infty$. This leads to the definition of the module \text{Ext}^1_{A, \infty}^{\text{reg}}(1, M), the kernel of the map from regulated extensions to Galois cohomology $H^1(G_\infty, M_B)$.
3. Hodge-Pink Structures with Infinite Frobenii: The paper extends Pink's theory of Hodge-Pink structures by introducing "infinite Frobenii," which are continuous representations of $G_\infty$ compatible with the Hodge-Pink lattice. This allows for the construction of a Hodge-Pink realization functor $H^+$ from $\text{AMot}^{\text{rat}}_K$ to the category of Hodge-Pink structures with infinite Frobenii ($\text{HdgPk}^+$).
4. Shtuka Models: A central technical tool is the association of $A$-motives with shtuka models.
   • $C$-shtuka models: Associated to any $A$-motive over the surface $(\text{Spec } A) \times C$.
   • $C \times C$-shtuka models: Associated to $A$-motives with negative weights over $C \times C$.
     The author proves that the complex computing the motivic cohomology ($G_M$) is quasi-isomorphic to the coherent cohomology of these shtuka models. This connection allows the use of Grothendieck's theorem on formal functions and the finiteness of coherent cohomology on proper schemes to deduce finiteness results.
5. Regulator Construction: The exactness of the Hodge-Pink realization functor induces a morphism (the regulator) from the module of regulated, analytically reduced extensions of $A$-motives to the space of Hodge-additive, analytically reduced extensions of Hodge-Pink structures.

Key Contributions and Results

The paper establishes two main theorems regarding the structure of $A$-motivic cohomology and the properties of the regulator:

1. Finiteness Theorem (Theorem 6.1):

   • The $A$-modules \text{Ext}^1_{A, \infty}^{\text{reg}}(1, M) (integral, regulated extensions with analytic reduction) and the class module $\text{Cl}(M)$ (the cokernel of the analytic reduction map) are finitely generated.
   • If the weights of $M$ are all negative, the class module $\text{Cl}(M)$ is finite.
   • This result serves as the function field analogue of the expectation that $\text{Ext}^1_{\mathbb{Z}}(1, M)$ is finite-dimensional in the number field setting.

2. Rank Equality Theorem (Theorem 6.2):

   • Assuming the weights of $M$ are all negative, the rank of \text{Ext}^1_{A, \infty}^{\text{reg}}(1, M) as an $A$-module equals the dimension of the target space \text{Ext}^1_{\infty}^{\text{ha}}(1^+, H^+(M)) as a $K_\infty$-vector space.
   • This dimension is explicitly computable in terms of the positive Hodge-Pink weights of $M$.

3. Failure of Full Rank:

   • Contrary to the conjectural picture over number fields (where the regulator is expected to have dense/full rank), the author demonstrates that the image of the Hodge-Pink regulator $\text{Reg}(M)$ may fail to have full rank in its target.
   • This failure is observed even in simple cases, such as the $p$-th Carlitz twist. Consequently, a direct analogue of Beilinson's conjecture regarding the density of the image does not hold in this setting.

Significance and Claims

The paper claims to provide a rigorous framework for motivic cohomology in the function field setting by overcoming the infinite generation issues present in earlier formulations. By introducing the concept of analytic reduction and utilizing shtuka models, the author successfully proves finiteness results that mirror classical expectations.

However, the paper modestly notes a significant divergence from the number field analogy: the regulator does not necessarily have full rank. The author suggests that this discrepancy implies that the standard Hodge-Pink structures might not be the correct objects for formulating a full Beilinson-type conjecture in function fields. Instead, the author proposes that "function field Hodge structures" (without the lattice data) might need to be rehabilitated to define a regulator with full rank. The paper concludes that while the rank equality (Theorem 6.2) holds, the precise formulation of the analogue to Beilinson's conjecture (relating the regulator image to special $L$-values) remains an open problem requiring further investigation. The work is presented as a natural sequel to [Gaz22], bridging the gap between Anderson $A$-motives and the conjectural landscape of mixed motives.