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Quadratic Euler-Kronecker constants in positive characteristic

Amir Akbary, Félix Baril Boudreau

TL;DR

The paper analyzes the distribution of Euler-Kronecker constants $\gamma_D$ for geometric quadratic function-field extensions via the Dirichlet derivative $\frac{L'(1,\chi_D)}{L(1,\chi_D)}$, establishing a limiting distribution with a smooth positive density through a function-field probabilistic model. It derives integral moments of $-\frac{L'(1,\chi_D)}{L(1,\chi_D)}$, builds a Granville–Soundararajan–style random model, and compares Laplace transforms to obtain a discrepancy result that implies a positive proportion of discriminants with small absolute values. The work further proves unconditional omega results for extreme values of $L'(1,\chi_D)/L(1,\chi_D)$ and discusses implications for the stable Taguchi heights and logarithmic Weil heights of rank-2 Drinfeld modules with CM. Collectively, these results yield refined distribution statements for heights of CM Drinfeld modules and deepen the understanding of Euler-Kronecker constants in positive characteristic settings.

Abstract

In 2006, Ihara defined and systematically studied a generalization of the Euler-Mascheroni constant for all global fields, named the Euler-Kronecker constants. This paper examines their distribution across geometric quadratic extensions of a rational global function field, via the values of logarithmic derivatives of Dirichlet $L$-functions at 1. Using a probabilistic model, we show that the values converge to a limiting distribution with a smooth, positive density function, as the genii of quadratic fields approach infinity. We then prove a discrepancy theorem for the convergence of the frequency of these values, and obtain information about the proportion of the small values. Finally, we prove omega results on the extreme values. Our theorems imply new distribution results on the stable Taguchi heights and logarithmic Weil heights of rank 2 Drinfeld modules with CM.

Quadratic Euler-Kronecker constants in positive characteristic

TL;DR

The paper analyzes the distribution of Euler-Kronecker constants for geometric quadratic function-field extensions via the Dirichlet derivative , establishing a limiting distribution with a smooth positive density through a function-field probabilistic model. It derives integral moments of , builds a Granville–Soundararajan–style random model, and compares Laplace transforms to obtain a discrepancy result that implies a positive proportion of discriminants with small absolute values. The work further proves unconditional omega results for extreme values of and discusses implications for the stable Taguchi heights and logarithmic Weil heights of rank-2 Drinfeld modules with CM. Collectively, these results yield refined distribution statements for heights of CM Drinfeld modules and deepen the understanding of Euler-Kronecker constants in positive characteristic settings.

Abstract

In 2006, Ihara defined and systematically studied a generalization of the Euler-Mascheroni constant for all global fields, named the Euler-Kronecker constants. This paper examines their distribution across geometric quadratic extensions of a rational global function field, via the values of logarithmic derivatives of Dirichlet -functions at 1. Using a probabilistic model, we show that the values converge to a limiting distribution with a smooth, positive density function, as the genii of quadratic fields approach infinity. We then prove a discrepancy theorem for the convergence of the frequency of these values, and obtain information about the proportion of the small values. Finally, we prove omega results on the extreme values. Our theorems imply new distribution results on the stable Taguchi heights and logarithmic Weil heights of rank 2 Drinfeld modules with CM.

Paper Structure

This paper contains 10 sections, 21 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Notation and Prerequisites
  3. Affine Setting
  4. Projective Setting
  5. Computing Moments
  6. Probabilistic Model
  7. Comparing Laplace Transforms
  8. Exponential Decay
  9. Discrepancy and Proof of Theorem \ref{['Theorem_Application_Small_Absolute_Value']}
  10. Proof of Theorem \ref{['Theorem_Omega_Result']}

Key Result

Theorem 1.1

For $n \geq 1$, we have More precisely, there are $\gg_q q^n \log^2(\log(q^n))/\log(q^n)$ polynomials $D \in \mathcal{H}_n$ for which

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['Lemma_Estimate_On_Sums_of_Quadratic_Characters']} (i)
  • Remark 2.4
  • proof : Proof of Lemma \ref{['Lemma_Estimate_On_Sums_of_Quadratic_Characters']} (ii) (when $q \equiv 3 \pmod{4}$ and both $n$ and $d_f$ are odd)
  • ...and 32 more