15 pages, section 6.2 eliminated, major simplifications in former Proposition 4.3 (now Proposition 4.2), corrections in former Proposition 4.4 (now Proposition 4.3), many small changes in notations and language
Suppose G is a semi-direct product of the form Z/p^n \rtimes Z/m where p is prime and m is relatively prime to p. Suppose K is a local field of characteristic p > 0. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration whose dimension depends only on the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p^3.
Consider tuples of separable algebras over a common local or global number field, related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.
This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic mo…
Let $F$ be the function field of a smooth curve over the $p$-adic number field $\Q_p$. We show that for each prime-to-$p$ number $n$ the $n$-torsion subgroup $\H^2(F,μ_n)={}_n\Br(F)$ is generated by $\Z/n$-cyclic classes; in fact the $\Z/n$-length is equal to two. It follows that the Brauer dimension of $F$ is two (first proved in \cite{Sa97}), and any $F$-division algebra of period $n$ and index $n^2$ is decomposable.
Let k be a number field. For an odd prime p and an integer i>1, the i-th étale wild kernel is contained in the second cohomology group of o'_k with coefficients in Zp(i), where o'_k is the ring of p-integers of k. Using Iwasawa theory, we give conditions for this inclusion to split. In particular we relate this splitting problem to the triviality of two invariants, namely the asymptotic kernels of the Galois descent and codescent for class groups along the cyclotomic tower of k. We illustrate our results in both split and non-split cases for quadratic number fields.
We compute the ramification filtration on wildly ramified $p^2$-cyclic extensions of local fields of characteristic $p$. The ramification filtration on the compositum of two $p$-cyclic and $p^2$-cyclic extensions are also computed. As an application, some partial results towards Abhyankar's Inertia conjecture has been proved.
There are two types of Belyi's Theorem for curves defined over finite fields of characteristic p, namely the Wild and the Tame p-Belyi Theorems. In this paper, we discuss them in the language of function fields. We provide a self-contained proof for the Wild p-Belyi Theorem for any prime p and the Tame 2-Belyi Theorem.
Let $k$ be an algebraically closed field of characteristic $p > 0$. We consider the problem of lifting $p$-cyclic covers of $\Proj_k$ as $p$-cyclic covers of the projective line over some DVR under the condition that the wild monodromy is maximal. We answer positively the question for covers birational to $w^ p-w=tR(t)$ for some additive polynomial $R(t)$.
We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras over such fields, which are given {\it a priori} as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or {\it period-index bound}) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.
- 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 …
Let F be a field of transcendence degree one over a p-adic field, and let l be a prime not equal to p. Results of Merkurjev and Saltman show that H^2(F,μ_l) is generated by Z/l-cyclic classes. We prove the "Z/l-length" in H^2(F,μ_l) equals the l-Brauer dimension, which Saltman showed to be two. It follows that all F-division algebras of period l are crossed products, either cyclic (by Saltman's cyclicity result) or tensor products of two cyclic division algebras. Our result was originally proved by Suresh assuming F contains μ_l.
Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $\Sha_S$ of maps $H^2(G_S,\fq_p) \rightarrow \oplus_{v\in S} H^2(\G_v,\fq_p)$. There is a natural injection $\Sha_S \hookrightarrow \CyB_S$, into the dual $\CyB_S$ of a certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $\Sha_{S\cup X} \stackrel{\simeq}{\hookrightarrow} \CyB_{S\cup X} \stackrel{\simeq}{\t…
We give an intrinsic parametrisation of the set of tamely ramified extensions of a local field with finite residue field and bring to the fore the role played by group cohomology. We show that two natural definitions of the cohomology class of a tamely ramified finite galoisian extension coincide, and can be recovered from the parameter. We also give an elementary proof of Serre's mass formula in the tame case and in the simplest wild case, and we classify tame galoisian extensions of degree the cube of a prime.
Tate first proposed a method to determine $K_2\mathcal{O}_F,$ the tame kernel of $F,$ and gave the concrete computations for some special quadratic fields with small discriminant. After that, many examples for quadratic fields with larger discriminants are given, and similar works also have been done for cubic fields and for some special quartic fields with discriminants not large. In the present paper, we investigate the case of more general imaginary cyclic quartic field $F=\mathbb{Q}\Big(\sqrt{-(D+B\sqrt{D})}\Big)$ with class number one and large discriminants. The key problem is how to dec…
Usually the boundary map in K-theory localization only gives the tame symbol at $K_{2}$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at p=2. Our idea is to pinch singularities near the ramification locus. This fattens up K-theory and makes the wild symbol vi…
We consider p-extensions of number fields such that the filtration of the Galois group by higher ramification groups is of prescribed finite length. We extend well-known properties of tame extensions to this more general setting; for instance, we show that these towers, when infinite, are ``asymptotically good'' (an explicit bound for the root discriminant is given). We study the difficult problem of bounding the relation-rank of the Galois groups in question. Results of Gordeev and Wingberg imply that the relation-rank can tend to infinity when the set of ramified primes is fixed but the leng…
Let us consider an abelian variety defined over $\mathbb{Q_{\ell}}$ with good supersingular reduction. In this paper we give explicit conditions that ensure that the action of the wild inertia group on the $\ell$-torsion points of the variety is trivial. Furthermore we give a family of curves of genus 2 such that their Jacobian surfaces have good supersingular reduction and satisfy these conditions. We address this question by means of a detailed study of the formal group law attached to abelian varieties.
Congruence Classes of Supporting the Erdös-Straus Conjecture II: Wild Solutions
HLM, Institute of Mathematics, Academy of Mathematics & System Sciences Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematics, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Let $m>1$ and $\mathfrak{d} \neq 0$ be integers such that $v_{p}(\mathfrak{d}) \neq m$ for any prime $p$. We construct a matrix $A(\mathfrak{d})$ of size $(m-1) \times (m-1)$ depending on only of $\mathfrak{d}$ with the following property: For any tame $\mathbb{Z}/m\mathbb{Z}$-number field $K$ of discriminant $\mathfrak{d}$ the matrix $A(\mathfrak{d})$ represents the Gram matrix of the integral trace zero form of $K$. In particular, we have that the integral trace zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to th…
Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $σ$ be an irreducible smooth representation of the absolute Weil group $\Cal W_F$ of $F$ and $\sw(σ)$ the Swan exponent of $σ$. Assume $\sw(σ) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and $\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\varSigma$, of order prime to $p$, so that $σ(\Cal I_F) = σ(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $σ$ of any character of $\varSigma$ is bounded by $\sw(σ)$.
Given a prime $p$, a number field $\K$ and a finite set of places $S$ of $\K$, let $\K_S$ be the maximal pro-$p$ extension of $\K$ unramified outside $S$. Using the Golod-Shafarevich criterion one can often show that $\K_S/\K$ is infinite. In both the tame and wild cases we construct infinite subextensions with bounded ramification using the refined Golod-Shafarevich criterion. In the tame setting we achieve new records on Martinet constants (root discriminant bounds) in the totally real and totally complex cases. We are also able to answer a question of Ihara by producing infinite asymptotica…
Let $N/K$ be a finite Galois extension of $p$-adic number fields and let $ρ^\mathrm{nr} : G_K \to \mathrm{Gl}_r(\mathbb Z_p)$ be an $r$-dimensional unramified representation of the absolute Galois group $G_K$ which is the restriction of an unramified representation $ρ^\mathrm{nr}_{\mathbb Q_p} : G_{\mathbb Q_p} \to \mathrm{Gl}_r(\mathbb Z_p)$. In this paper we consider the $\mathrm{Gal}(N/K)$-equivariant local $ε$-conjecture for the $p$-adic representation $T = \mathbb Z_p^r(1)(ρ^\mathrm{nr})$. For example, if $A$ is an abelian variety of dimension $r$ defined over $\mathbb Q_p$ with good ordi…
Let $ K $ be a number field and let $ L/K $ be a tamely ramified radical extension of prime degree $ p $. If $ K $ contains a primitive $ p^{th} $ root of unity then $ L/K $ is a cyclic Kummer extension; in this case the group algebra $ K[G] $ (with $ G=\mbox{Gal}(L/K) $) gives the unique Hopf-Galois structure on $ L/K $, the ring of algebraic integers $ \mathfrak{O}_L $ is locally free over $ \mathfrak{O}_{K}[G] $ by Noether's theorem, and Gómez Ayala has determined a criterion for $ \mathfrak{O}_L $ to be a free $ \mathfrak{O}_{K}[G] $-module. If $ K $ does not contain a primitive $ p^{th} $…
Given an elliptic curve $E$ over a local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is that of the $\ell$-adic Galois representation attached to an elliptic curve such that the image of inertia is non-cyclic, and isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation explicitly.
