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Mellin transform formulas for Drinfeld modules

Oğuz Gezmiş, Nathan Green

Abstract

We introduce formulas for the logarithms of Drinfeld modules using a framework recently developed by the second author. We write the logarithm function as the evaluation under a motivic map of a product of rigid analytic trivializations of $t$-motives. We then specialize our formulas to express special values of Goss $L$-functions as Drinfeld periods multiplied by rigid analytic trivializations evaluated under this motivic map. We view these formulas as characteristic-$p$ analogues of integral representations of Hasse-Weil type zeta functions. We also apply this machinery for Drinfeld modules tensored with the tensor powers of the Carlitz module, which serves as the Tate twist of a Drinfeld module.

Mellin transform formulas for Drinfeld modules

Abstract

We introduce formulas for the logarithms of Drinfeld modules using a framework recently developed by the second author. We write the logarithm function as the evaluation under a motivic map of a product of rigid analytic trivializations of -motives. We then specialize our formulas to express special values of Goss -functions as Drinfeld periods multiplied by rigid analytic trivializations evaluated under this motivic map. We view these formulas as characteristic- analogues of integral representations of Hasse-Weil type zeta functions. We also apply this machinery for Drinfeld modules tensored with the tensor powers of the Carlitz module, which serves as the Tate twist of a Drinfeld module.
Paper Structure (35 sections, 34 theorems, 294 equations)

This paper contains 35 sections, 34 theorems, 294 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The Mellin transform of Drinfeld modules and $L$-functions
  4. Comparison with the classical theta functions
  5. Tate twists of Drinfeld modules
  6. Generality of the Arguments
  7. Outline of the paper
  8. Preliminaries and background
  9. Anderson $t$-modules
  10. Anderson generating functions
  11. Anderson $t$-motives
  12. Anderson $t$-motive of Drinfeld modules
  13. Anderson $t$-motive of the tensor powers of the Carlitz module
  14. Anderson $t$-motive of the tensor product of Drinfeld modules with the tensor powers of the Carlitz module
  15. Dual $t$-motives
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\phi$ be a Drinfeld module given by so that ${\lvert k_i \rvert}\leq 1$ for each $1\leq i \leq r-1$ and $k_r\in \mathbb{F}_q^{\times}$. Let $\overline \pi := (\lambda_1,\dots,\lambda_r)$ be the vector of fundamental periods of $\phi$. Then Moreover, for any $\bold{z}\in \mathbb{C}_{\infty}$ in the domain of convergence of $\log_{\phi}$, we have

Theorems & Definitions (72)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Definition 2.1
  • Example 2.2
  • ...and 62 more