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On the Formal Carlitz Module

Bruno Anglès

TL;DR

The paper develops a $P$-adic analogue of Taelman’s class formula for the Carlitz module by introducing $P$-adic zeta values $\zeta_{P,O_L}(1)$ and a regulator $R_{P,L}$ in the real function-field setting, leveraging the $P$-adic $z$-deformation of Anderson–Drinfeld-type modules. It constructs the $P$-adic Carlitz module, its $z$-deformation, and a regulator mechanism that yields the main formula $\zeta_{P,O_L}(1)= [H(O_L)]_A \frac{[C(O_L/PO_L)]_A}{[O_L/PO_L]_A} R_{P,L}$ for finite real extensions $L/K$, with a parallel to the ∞-adic theory. The work also defines and analyzes the $P$-adic Leopoldt defect $\delta_P(U(O_L))$, connecting rank considerations of unit modules to the validity of the $P$-adic class formula in various cases and outlining open questions about its vanishing. Overall, the results bridge function-field arithmetic of the Carlitz module with $P$-adic and Iwasawa-theoretic perspectives, providing a framework for further study of $P$-adic regulators and Stark-type phenomena in function fields.

Abstract

For finite extensions of a rational function field over a finite field, we prove a "P-adic class formula" in the spirit Taelman's work.

On the Formal Carlitz Module

TL;DR

The paper develops a -adic analogue of Taelman’s class formula for the Carlitz module by introducing -adic zeta values and a regulator in the real function-field setting, leveraging the -adic -deformation of Anderson–Drinfeld-type modules. It constructs the -adic Carlitz module, its -deformation, and a regulator mechanism that yields the main formula for finite real extensions , with a parallel to the ∞-adic theory. The work also defines and analyzes the -adic Leopoldt defect , connecting rank considerations of unit modules to the validity of the -adic class formula in various cases and outlining open questions about its vanishing. Overall, the results bridge function-field arithmetic of the Carlitz module with -adic and Iwasawa-theoretic perspectives, providing a framework for further study of -adic regulators and Stark-type phenomena in function fields.

Abstract

For finite extensions of a rational function field over a finite field, we prove a "P-adic class formula" in the spirit Taelman's work.

Paper Structure

This paper contains 12 sections, 19 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Carlitz-Goss $\infty$-adic Zeta values
  4. Carlitz-Goss $P$-adic Zeta values
  5. Taelman's class formula
  6. The Carlitz module
  7. Units and class modules
  8. $\infty$-adic $z$-deformation
  9. The $P$-adic Carlitz module
  10. $P$-adic $z$-deformation
  11. A $P$-adic class formula
  12. Leopoldt's defect

Key Result

Lemma 3.1

Let $n\in \mathbb Z, n\leq 0.$ We denote by $\ell_q(-n)$ the sum of the digits of $-n$ in base $q.$ Then $Z_{O_L}(n;z)\in A[z],$ and : where $g_L$ is the genus of $L.$

Theorems & Definitions (43)

  • Lemma 3.1
  • proof
  • Definition 3.2
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Lemma 4.4
  • ...and 33 more