[Paper Review] Analytic lie extensions of number fields with cyclic fixed points and tame ramification

Farshid Hajir, Christian Maire|arXiv (Cornell University)|Oct 25, 2017

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper extends Boston's approach to the tame Fontaine-Mazur conjecture by analyzing uniform $p$-adic analytic Galois extensions $\mathrm{L}/\mathrm{K}$ where $\mathrm{K}/\mathrm{k}$ is a cyclic extension of prime degree $\ell \neq p$, focusing on the action of the Galois group $\mathrm{Gal}(\mathrm{K}/\mathrm{k})$ on the Galois group $\Gamma = \mathrm{Gal}(\mathrm{L}/\mathrm{K})$. It shows that the presence of fixed points under this action still imposes strong arithmetic constraints, enabling the deduction of non-existence of certain tamely ramified $p$-adic analytic extensions over $\mathrm{K}$ from their non-existence over $\mathrm{k}$, even when fixed points are non-trivial.

ABSTRACT

- Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group $Γ$ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) or ramified at an infinite number of primes. In this work, we take up a study (initiated by Boston) of this type of question under the assumption that L is Galois over some subfield k of K such that [K : k] is a prime = p. Letting $σ$ be a generator of Gal(K/k), we study the constraints posed on the arithmetic of L/K by the cyclic action of $σ$ on $Γ$, focusing on the critical role played by the fixed points of this action, and their relation to the ramification in L/K. The method of Boston works only when there are no non-trivial fixed points for this action. We show that even in the presence of arbitrarily many fixed points, the action of $σ$ places severe arithmetic conditions on the existence of finitely and tamely ramified uniform p-adic analytic extensions over K, which in some instances leads us to be able to deduce the non-existence of such extensions over K from their non-existence over k.

Motivation & Objective

• To investigate the arithmetic structure of $p$-adic analytic Galois extensions $\mathrm{L}/\mathrm{K}$ with tame ramification under the action of a cyclic Galois group $\mathrm{Gal}(\mathrm{K}/\mathrm{k})$ of prime order $\ell \neq p$.
• To understand how the fixed points of the Galois action on the uniform pro-$p$ group $\Gamma = \mathrm{Gal}(\mathrm{L}/\mathrm{K})$ constrain the existence of such extensions.
• To generalize Boston's fixed-point-free (FPF) method to cases with non-trivial fixed points, thereby extending the scope of non-existence results for tamely ramified $p$-adic analytic extensions.
• To establish conditions under which the non-existence of such extensions over a base field $\mathrm{k}$ implies their non-existence over a cyclic extension $\mathrm{K}$ of degree $\ell$.

Proposed method

• Use of the Schur-Zassenhaus theorem to analyze the structure of group extensions involving $\Gamma$ and the Galois group $\Delta = \mathrm{Gal}(\mathrm{K}/\mathrm{k})$.
• Application of the theory of uniform pro-$p$ groups and their associated Lie algebras to study the action of $\sigma \in \Delta$ on $\Gamma$.
• Employment of cohomological techniques, particularly on units and $T$-units, to analyze the freeness and structure of modules over group rings.
• Utilization of Kummer theory and the Gras-Munnier theorem to relate ramification data to Galois module structure.
• Application of Chebotarev's density theorem to study the distribution of primes with prescribed splitting behavior in the extensions.
• Analysis of the fixed-point subgroups $\Gamma^\sigma$ and their role in constraining the existence of tamely ramified $\Gamma$-extensions.

Experimental results

Research questions

• RQ1 How does the presence of non-trivial fixed points under a cyclic Galois action affect the existence of tamely ramified, uniform $p$-adic analytic extensions over a number field $\mathrm{K}$?
• RQ2 Can non-existence results for such extensions over a base field $\mathrm{k}$ be lifted to non-existence over a cyclic extension $\mathrm{K}/\mathrm{k}$ of degree $\ell \neq p$?
• RQ3 What structural constraints does the action of $\sigma \in \mathrm{Gal}(\mathrm{K}/\mathrm{k})$ impose on the Galois group $\Gamma = \mathrm{Gal}(\mathrm{L}/\mathrm{K})$ when $\Gamma$ is uniform and $p$-adic analytic?
• RQ4 To what extent can the FPF (fixed-point-free) assumption in Boston’s original method be relaxed while still obtaining non-existence theorems for $p$-adic analytic extensions?
• RQ5 How do the fixed points of the Galois action interact with the ramification behavior of $\mathrm{L}/\mathrm{K}$, particularly in the context of tame ramification?

Key findings

• Even in the presence of arbitrarily many non-trivial fixed points under the Galois action, the structure of the action of $\sigma$ on $\Gamma$ still imposes severe arithmetic constraints on the existence of finitely and tamely ramified uniform $p$-adic analytic extensions over $\mathrm{K}$.
• The paper establishes a general principle: if there are no such extensions over the base field $\mathrm{k}$, then under certain conditions, there are also no such extensions over $\mathrm{K}$, even when the Galois action has fixed points.
• The existence of fixed points does not invalidate the method; instead, it leads to refined cohomological and module-theoretic conditions that can still be used to rule out extensions.
• The analysis shows that the fixed-point subgroup $\Gamma^\sigma$ plays a critical role in determining the ramification and structure of $\Gamma$, particularly in relation to the $T$-units and their freeness.
• For the case where $\sigma$ has order 2 or divides $p-1$, the paper derives explicit non-existence results based on the interplay between the Galois action and the uniformity of $\Gamma$.
• In the context of the cyclotomic $\mathbb{Z}_p$-extension, the method yields non-existence theorems when the base field $\mathrm{k}$ satisfies certain class number and unit group conditions.