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A family of asymptotically bad wild towers of function fields

Maria Chara, Ricardo Toledano

TL;DR

The paper establishes a general ramification-based sufficient condition for the genus of wild non-skew towers of function fields over a perfect field to be infinite, extending prior work on wild ramification. It presents a framework (theorem wildramif) that uses a prescribed ramification pattern to force divergence of the cumulative different exponents, ensuring $\gamma(\mathcal{T})=\infty$. A concrete new example of a skew recursive wild tower with infinite genus is constructed via the relation $(y-a)^m+b(y-a)=(x-a)^m/g(x)$ with $m=q+1$, together with a ramification analysis based on Eisenstein and Abhyankar’s lemmas. These results broaden the catalog of infinite-genus towers and provide actionable criteria for identifying towers unsuitable for certain asymptotically good-code constructions.

Abstract

In a previous work general conditions were given to prove the infiniteness of the genus of certain towers of function fields over a perfect field. It was shown that many examples where particular cases of those general results. In this paper the genus of a family of wild towers of function fields will be considered together with a result with less restrictive sufficient conditions for a wild tower to have infinite genus.

A family of asymptotically bad wild towers of function fields

TL;DR

The paper establishes a general ramification-based sufficient condition for the genus of wild non-skew towers of function fields over a perfect field to be infinite, extending prior work on wild ramification. It presents a framework (theorem wildramif) that uses a prescribed ramification pattern to force divergence of the cumulative different exponents, ensuring . A concrete new example of a skew recursive wild tower with infinite genus is constructed via the relation with , together with a ramification analysis based on Eisenstein and Abhyankar’s lemmas. These results broaden the catalog of infinite-genus towers and provide actionable criteria for identifying towers unsuitable for certain asymptotically good-code constructions.

Abstract

In a previous work general conditions were given to prove the infiniteness of the genus of certain towers of function fields over a perfect field. It was shown that many examples where particular cases of those general results. In this paper the genus of a family of wild towers of function fields will be considered together with a result with less restrictive sufficient conditions for a wild tower to have infinite genus.
Paper Structure (3 sections, 1 theorem, 12 equations, 1 figure)

This paper contains 3 sections, 1 theorem, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Climbing a wild tower
  3. A new family of skew recursive wild towers with infinite genus

Key Result

Theorem 2.1

Let $\mathcal{T}=\{T_i\}_{i=0}^{\infty}$ be a non skew recursive tower of function fields over a perfect field $K$ of characteristic $p>0$ defined by a suitable bivariate polynomial $F\in K[x,y]$. Let $m=\deg_yF=\deg_xF$ and let us consider the basic function field $T=K(x,y)$ associated to $\mathcal Then $\gamma(\mathcal{T})=\infty$.

Figures (1)

  • Figure 1: Ramification of $P_i$ in $F_i$ in Theorem \ref{['wildramif']}

Theorems & Definitions (2)

  • Theorem 2.1
  • proof