[Paper Review] Chevalley-Weil Theorem and Subgroups of Class Groups

Yuri Bilu, Jean Gillibert|arXiv (Cornell University)|Jun 9, 2016

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper establishes an 'absolute' version of the Chevalley-Weil theorem for étale covers of curves over number fields, proving infinitely many specializations into unramified extensions. Using Class Field Theory and Jacobian torsion subgroups over roots of unity, it constructs number fields with arbitrarily large $n$-ranks in their class groups, significantly improving prior methods by eliminating the unit group penalty present in earlier results.

ABSTRACT

We prove, under some mild hypothesis, that an étale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an "absolute" version of the Chevalley-Weil theorem. Using this result, we are able to generalize the techniques of Mestre, Levin and the second author for constructing and counting number fields with large class group.

Motivation & Objective

• To establish an absolute version of the Chevalley-Weil theorem for étale covers of curves over number fields.
• To construct infinitely many number field extensions with large $n$-ranks in their class groups, independent of unit group rank.
• To generalize and improve upon Mestre, Levin, and Bilu's methods for constructing number fields with large class groups.
• To provide a quantitative lower bound on the number of such fields with bounded discriminant.
• To apply the results to cyclotomic fields and Fermat-type curves to produce explicit constructions with high $p$-rank class groups.

Proposed method

• Prove that an étale cover of curves over a number field has infinitely many specializations into everywhere unramified extensions, under mild hypotheses.
• Use Class Field Theory to relate the $n$-rank of the class group of a specialization field to the $μ_n$-torsion in the Jacobian of the curve.
• Define $\operatorname{rk}_{\mu_n}J(\mathcal{C})$ as the maximal rank of a $\mathrm{Gal}(\bar{K}/K)$-submodule isomorphic to $\mu_n^r$.
• Apply the theory of torsors and specialization of Galois covers to ensure that the class group of the specialization field inherits $\mu_n$-torsion from the Jacobian.
• Use the function $t$ defining the degree-$d$ morphism $\mathcal{C} \to \mathbb{P}^1$ to parametrize the specializations and control discriminant growth.
• Derive a quantitative lower bound of $cX^{\ell/(2m(d-1))}/\log X$ for the number of such fields with discriminant $\mathcal{D}(L/K) \leq X$, where $\ell = [K:\mathbb{Q}]$.

Experimental results

Research questions

• RQ1 Can one prove the existence of infinitely many number field extensions of fixed degree with arbitrarily large $n$-rank in the class group, independent of the unit group rank?
• RQ2 Can the negative contribution from the unit group rank in prior constructions be eliminated via a new cohomological or class field theoretic approach?
• RQ3 To what extent can the $n$-rank of the class group of a number field be controlled by the $μ_n$-torsion in the Jacobian of a curve over the base field?
• RQ4 What is the asymptotic density of number fields with large $n$-rank class groups among all extensions of fixed degree?
• RQ5 Can explicit families of curves be constructed such that their specializations yield number fields with $p$-rank class groups of size linear in the degree?

Key findings

• There exist infinitely many number fields $L/K$ of degree $d$ such that $\operatorname{rk}_n\operatorname{Cl}(L) \geq \operatorname{rk}_{\mu_n}J(\mathcal{C}) + \operatorname{rk}_n\operatorname{Cl}(K)$, with no unit group penalty.
• For a prime $p \geq 3$ and $d \leq p-1$, there exist infinitely many extensions $L/K$ with $[L:K] = d$ and $\operatorname{rk}_p\operatorname{Cl}(L) \geq 3 + \operatorname{rk}_p\operatorname{Cl}(K)$, when $K$ contains $\zeta_p$.
• When $K$ contains $\zeta_p$, the curve $y^p = x^{d-1}(1-x)$ yields $\operatorname{rk}_{\mu_p}J(\mathcal{C}) \geq d-1$, leading to $\operatorname{rk}_p\operatorname{Cl}(L) \geq d-1 + \operatorname{rk}_p\operatorname{Cl}(K)$ for infinitely many $L$.
• The number of such fields $L$ with $\mathcal{D}(L/K) \leq X$ is at least $cX^{\ell/(2m(d-1))}/\log X$, where $\ell = [K:\mathbb{Q}]$ and $c > 0$ depends on the curve and functions.
• The result recovers and improves upon Mestre’s construction: for $K = \mathbb{Q}$, the method yields $\operatorname{rk}_5\operatorname{Cl}(L) \geq 3$ for infinitely many quadratic fields $L$, with $cX^{\ell/22}$ such fields up to discriminant $X$.
• The method avoids the unit group term present in earlier Kummer-theoretic approaches, making it applicable even when $K \neq \mathbb{Q}$.