[Paper Review] Counting Number Fields in Fibers

Yuri Bilu, Jean Gillibert|arXiv (Cornell University)|Jun 7, 2016

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper extends a result of Dvornicich and Zannier on the growth of degree in composita of number fields arising from fibers of rational functions on curves, generalizing from the rational base field ℚ to an arbitrary number field K. It establishes that for a curve X over K with a degree-n rational function t, the compositum of fields K(P_τ) for τ in the ring of integers of K with bounded height B has degree at least exp(cB^d / log B) over K, where d is the degree of K over ℚ and c depends on K, genus g, and n. This implies at least cB^d / log B distinct fields among the K(P_τ).

ABSTRACT

Let X be a projective curve over Q and t a non-constant Q-rational function on X of degree n>1. For every integer a pick a points P(a) on X such that t(P(a))=a. Dvornicich and Zannier (1994) proved that for large N the field Q(P(1), ..., P(N)) is of degree at least exp(cN/log N) over Q, where c>0 depends only on X and t. In this note we extend this result, replacing Q by an arbitrary number field.

Motivation & Objective

• To generalize a result by Dvornicich and Zannier on the degree growth of composita of number fields arising from fibers of rational functions on curves from the base field ℚ to an arbitrary number field K.
• To establish a lower bound on the degree of the compositum K(P₁, ..., P_N) over K, where P_τ are points in the fiber t⁻¹(τ) for τ in the ring of integers of K.
• To quantify the number of distinct fields among K(P_τ) for τ of bounded height, extending the logarithmic count from ℚ to arbitrary number fields.
• To provide a uniform bound in terms of the height of τ and the arithmetic invariants of the curve and function, including genus and degree.

Proposed method

• Use of height functions H(τ) on the ring of integers 𝒪_K to parameterize the fibers t⁻¹(τ) for τ ∈ 𝒪_K.
• Application of Hilbert's Irreducibility Theorem and its effective versions to ensure that for many τ, the field K(P_τ) has degree n over K.
• Employment of local analysis at finite and infinite places, including v-adic completions and strict henselization, to study ramification behavior.
• Utilization of Abhyankar's Lemma and tame cover theory to analyze ramification in local extensions K_v(P)/K_v(t(P)) at places v.
• Construction of a finite set S of primes (including 2 and primes dividing discriminants) where ramification may occur, and analysis of unramified extensions outside S.
• Use of the theory of thin sets and the Chebotarev density theorem to control the density of τ for which the fiber is irreducible or the extension is large.

Experimental results

Research questions

• RQ1 What is the minimal degree of the compositum K(P_τ) for τ ∈ 𝒪_K with H(τ) ≤ B, when X is a curve over a number field K and t is a non-constant rational function on X?
• RQ2 How many distinct fields among {K(P_τ) : τ ∈ 𝒪_K, H(τ) ≤ B} are there, as a function of B and the invariants of K, X, and t?
• RQ3 Can the lower bound on the degree of the compositum be extended from ℚ to an arbitrary number field K, and how does the degree [K:ℚ] affect the growth rate?
• RQ4 Under what conditions on τ and the local behavior of t at places v is the extension K_v(P)/K_v(t(P)) unramified or tamely ramified?
• RQ5 How does the ramification of the function t at branch points interact with the v-adic valuation of t(P) − α for α a branch point?

Key findings

• For any number field K of degree d over ℚ, and a curve X over K of genus g with a non-constant rational function t of degree n ≥ 2, there exist constants c > 0 and B₀ > 1 such that for all B ≥ B₀, the compositum of fields K(P_τ) for τ ∈ 𝒪_K with H(τ) ≤ B has degree at least exp(cB^d / log B) over K.
• The number of distinct fields among {K(P_τ) : τ ∈ 𝒪_K, H(τ) ≤ B} is at least cB^d / log B for some c > 0 depending on K, g, and n.
• The growth rate exp(cB^d / log B) is optimal in the sense that it matches the degree growth in the classical case where X is the curve y² = x and t = x, giving K(P_τ) = K(√τ), and the compositum is K(√p : p ≤ B) of degree 2^{π(B)} ≈ exp(cB / log B), which matches the bound up to the exponent d.
• The result holds uniformly regardless of the choice of points P_τ in the fiber t⁻¹(τ), showing the bound is robust under such choices.
• The proof relies on local analysis at places v, showing that if v(t(P) − α) > 0 for a branch point α and the valuation is not divisible by the ramification index e, then the extension K_v(P)/K_v(t(P)) is ramified, which contributes to the global degree growth.
• The appendix by Jean Gillibert provides a more canonical approach to the local ramification analysis, using strict henselization and formal schemes to clarify the structure of the pull-back of the cover along local rings.