The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways. In this paper we construct a family of finite 3-groups satisfying these restrictions. We prove several results about this family and characterize them as finite extensions of certain quotients of a Sylow pro-3 subgroup of SL_2(Z_3). We verify that the first group in the family does indeed arise as such a Galois group and provide a small amount of evidence that this may hold for the other members. If this were the case then it would imply that there is no upper bound on the possible lengths of a finite p-class tower.
Let $p$ be a prime number. We show that, there exists an infinite family of imaginary abelian fields such that, the Iwasawa module of the maximal multiple ${\Bbb Z}_p$-extension is non trivial and pseudo-null for each field in the family. We also discuss on an application to non-abelian Iwasawa theory in the sense of Ozaki.
- Let K be a totally imaginary number field. Denote by G ur K (2) the Galois group of the maximal unramified pro-2 extension of K. By comparing cup-products in {é}tale cohomology of SpecO K and cohomology of uniform pro-2 groups, we obtain situations where G ur K (2) has no non-trivial uniform analytic quotient, proving some new special cases of the unramified Fontaine-Mazur conjecture. For example, in the family of imaginary quadratic fields K for which the 2-rank of the class group is equal to 5, we obtain that for at least 33.12% of such K, the group G ur K (2) has no non-trivial uniform an…
For an odd prime number $p$, we study the number of generators of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg's generalized conjecture holds.
We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that $\text{Aut}(\text{PSL}(2,q^2))$ for $q$ a…
In this note we study the supersingular locus of the GU(2,2) Shimura variety modulo a prime which is unramified in the imaginary quadratic extension. The supersingular locus of this Shimura variety can be related to the basic Rapoport-Zink space whose special fibre is described by the Bruhat-Tits stratification. The description for this supersingular locus in the case where the prime is inert in imaginary quadratic field is already known to Howard and Pappas by exploiting the exceptional isomorphism. Our method is more direct without using the exceptional isomorphism.
Boston, Bush, and Hajir have developed heuristics, extending the Cohen-Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-p extensions of imaginary quadratic number fields for p an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of F_q(t), the Galois groups of the maximal unramified pro-p extensions, as q goes to infinity, have the moments predicted by the Bo…
Let $p$ be a prime number. Mazur proved that a geometrically maximal unramified abelian covering of $X_0(p)$ over $\mathbb Q$ is given by the Shimura covering $X_2(p) \to X_0(p)$, that is, a unique subcovering of $X_1(p) \to X_0(p)$ of degree $N_p := (p-1)/\gcd(p-1, 12)$. In this short paper, we show that a geometrically maximal abelian covering $X_2'(p) \to X_0(p)$ of $X_0(p)$ over $\mathbb Q$ unramified outside cusps is cyclic of degree $2N_p$. The main ingredient for the construction of $X_2'(p)$ is the generalized Dedelind eta functions.
In the present paper, we shall show that for any prime number p, every finite p-group occurs as the Galois Group of the maximal unramified p-extension over a certain number field of finite degree. We shall also show that for any given pro-p-group G with countably many generators, there exists a number field (not necessary of finite degree) whose maximal unramified p-extension has Galois group isomorphic to G. This means that the set of the isomorphism classes of the Galois groups of the maximal unramified p-extensions over the number fields (including of infinite degree) is precisely equal to …
Let $K$ be a number field and $K_{ur}$ be the maximal extension of $K$ that is unramified at all places. In a previous article, the first author found three real quadratic fields $K$ such that $Gal(K_{ur}/K)$ is finite and nonabelian simple under the assumption of the GRH(Generalized Riemann Hypothesis). In this article, we will identify more quadratic number fields $K$ such that $Gal(K_{ur}/K)$ is a finite nonsolvable group and also explicitly calculate their Galois groups under the assumption of the Generalized Riemann Hypothesis.
We consider the distribution of the Galois groups $\operatorname{Gal}(K^{\operatorname{un}}/K)$ of maximal unramified extensions as $K$ ranges over $Γ$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We prove two properties of $\operatorname{Gal}(K^{\operatorname{un}}/K)$ coming from number theory, which we use as motivation to build a probability distribution on profinite groups with these properties. In Part I, we build such a distribution as a limit of distributions on $n$-generated profinite groups. In Part II, we prove as $q\rightarrow\infty$, agreement of $\operatorname{Gal}(K^{\operato…
We consider certain Massey products in the cohomology of a Galois extension of fields with coefficients in p-power roots of unity. We prove formulas for these products both in general and in the special case that the Galois extension in question is the maximal extension of a number field unramified outside a set of primes S including those above p and any archimedean places. We then consider those Z_p-Kummer extensions L of the maximal p-cyclotomic extension K of a number field that are unramified outside S. We show that Massey products describe the structure of a certain "decomposition-free" …
For an algebraic number field $K$ and a prime number $p$, let $\widetilde{K}/K$ be the maximal multiple $\mathbb{Z}_p$-extension. Greenberg's generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-$p$ extension of $\widetilde{K}$ is pseudo-null over the completed group ring $\mathbb{Z}_p[\![\mathop{\mathrm{Gal}}\nolimits(\widetilde{K}/K)]\!]$. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.
On Boston's Unramified Conjecture for $\GL_2$ and McLeman's $(3,3)$-Conjecture
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究院）
We show that $L$-packets of toral supercuspidal representations arising from unramified maximal tori of $p$-adic groups are realized by Deligne--Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra's characterization of real discrete series represe…
Let $k$ be an imaginary quadratic field and $p$ an odd prime number such that the $p$-rank of the class group of $k$ is one. Let $S$ be a finite set of places of $k$ distinct from $p$-adic places. We give sufficient conditions for the Galois group $G_S$, of the maximal pro-$p$-extension of $k$ which is unramified outside $S$, to be \textit{mild}, hence of cohomological dimension $2$.
For a CM-field $K$ and an odd prime number $p$, let $\widetilde K'$ be a certain multiple $\mathbb{Z}_p$-extension of $K$. In this paper, we study several basic properties of the unramified Iwasawa module $X_{\widetilde K'}$ of $\widetilde K'$ as a $\mathbb{Z}_p[[{\rm Gal}(\widetilde K'/K)]]$-module. Our first main result is a description of the order of a Galois coinvariant of $X_{\widetilde K'}$ in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic $\mathbb{Z}_p$-extension of $K$ under a certain assumption on the splitting of primes above $p$. Second …
We consider a mod 7 Galois representation attached to a genus 2 Siegel cuspforms of level 1 and weight 28 and using some of its Fourier coefficients and eigenvalues computed by N. Skoruppa and the classification of maximal subgroups of PGSp(4,p) we show that its image is as large as possible. This gives a realization of PGSp(4,7) as a Galois group over $\Q$ and the corresponding number field provides a non-solvable extension of $\Q$ which ramifies only at 7.
Fontaine and Mazur conjecture that a number field k has no infinite unramified Galois extension such that its Galois group is a p-adic analytic pro-p-group. We consider this conjecture for the maximal unramified p-extension of a CM-field k.
Let p be a prime number which is split in an imaginary quadratic field k. Let \mathfrak{p} be a place of k above p. Let k_\infty be the unique Z_p-extension of k which unramified outside of \mathfrak{p}, and let K_\intfy be a finite extension of k_\infty, abelian over k. In case p \notin {2,3}, we prove that the characteristic ideal of the projective limit of global units modulo elliptic units coincides with the characteristic ideal of the projective limit of the p-class groups. Our approach uses Euler systems, which were first used in this context by K.Rubin. If p \in {2,3}, we obtain a divis…
For given rational prime number $p$ consider the tower of finite extensions of fields $K_0/\mathbb{Q}_p,$ $K/K_0, L/K, M/L$, where $K/K_0$ is unramified and $M/L$ is a Galois extension with Galois group $G$. Suppose one dimensional Honda formal group over the ring $\mathcal{O}_K$, relative to the extension $K/K_0$ and uniformizer $π\in K_0$ is given. The operation $x\underset{F}+y=F(x,y)$ sets a new structure of abelian group on the maximal ideal $\mathfrak{p}_M$ of the ring $\mathcal{O}_M$ which we will denote by $F(\mathfrak{p}_M)$. In this paper the structure of $F(\mathfrak{p}_M)$ as $\mat…
Using the mixed Lie algebras of Lazard, we extend the results of the first author on mild groups to the case p=2. In particular, we show that for any finite set S_0 of odd rational primes we can find a finite set S of odd rational primes containing S_0 such that the Galois group of the maximal 2-extension of Q unramified outside S is mild. We thus produce a projective system of such Galois groups which converge to the maximal pro-2-quotient of the absolute Galois group of $\Q$ unramified at 2 and $\infty$. Our results also allow results of Alexander Schmidt on pro-p-fundamental groups of marke…
Let $\mathbb{K}$ be an imaginary quadratic field such that $2$ splits into two primes $\mathfrak{p}$ and $\bar{\mathfrak{p}}$. Let $\mathbb{K}_{\infty}$ be the unique $\mathbb{Z}_2$-extension of $\mathbb{K}$ unramified outside $\mathfrak{p}$. Let $\mathfrak{f}$ be an ideal coprime to $\mathfrak{p}$ and $\mathbb{L}$ be an arbitrary extension of $\mathbb{K}$ contained in the ray class field $\mathbb{K}(\mathfrak{p}^2\mathfrak{f})$. Let $\mathbb{L}_{\infty}=\mathbb{K}_{\infty}\mathbb{L}$ and let $\mathbb{M}$ be the maximal $p$-abelian, $\mathfrak{p}$-ramified extension of $\mathbb{L}_{\infty}$. W…
Cohen and Lenstra detailed a heuristic for the distribution of odd p-class groups for imaginary quadratic fields. One such formulation of this distribution is that the expected number of surjections from the class group of an imaginary quadratic field k to a fixed odd abelian group is 1. Class field theory tells us that the class group is also the Galois group of the Hilbert class field, the maximal unramified abelian extension of k, so we could equivalently say the expected number of unramified G-extensions of k is 1/#Aut(G) for a fixed abelian group G. We generalize this to asking for the ex…
Let D be a finite-dimensional central division algebra over a field K. We define the genus gen(D) of D to be the collection of classes in the Brauer group of K represented by central division K-algebras D' having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K.
Let $E/F$ be a unramified quadratic extension of non-archimedean local fields of odd characteristic $p$, and $G$ be the unramified unitary group $U(2, 1)(E/F)$. For an irreducible smooth representation $π$ of $G$ over $\overline{\mathbf{F}}_p$, with an underlying irreducible smooth representation $σ$ of a maximal compact open subgroup $K$, we prove that $π$ admits eigenvectors for an appropriate Hecke operator $T_σ$, and we classify those $π$ with non-zero eigenvalues for $T_σ$ by a tree argument; as a corollary, we show $π$ is supersingular if and only if it is supercuspidal.
Let $\mathcal{\scriptstyle{O}}_K$ be the ring of integers of an imaginary quadratic number field $K$. In this paper we give a new description of the maximal discrete extension of the group $SL_2(\mathcal{\scriptstyle{O}}_K)$ inside $SL_2(\mathbb{C})$, which uses generalized Atkin-Lehner involutions. Moreover we find a natural characterization of this group in $SO(1,3)$.
Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z_p-extensions of the pth cyclotomic field and the Galois group G of the unramified pro-p extension of the cyclotomic field of all p-power roots of unity. We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for G to be abelian. The bound and the condition in the two results are given in terms of…
In this paper the new techniques and results concerning the structure theory of modules over non-commutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions K of number fields k "up to pseudo-isomorphism". In particular, a close relationship is revealed between the Selmer group of abelian varieties, the Galois group of the maximal abelian unramified p-extension of K as well as the Galois group of the maximal abelian outside S unramified p-extension where S is a finite set of certain places of k. Moreover, we determine the Galois module structure …
In this paper we will give a computation of the étale fundamental group of an integral arithmetic scheme. For such a scheme, we will prove that the étale fundamental group is naturally isomorphic to the Galois group of the maximal formally unramified extension over the function field. It consists of the main theorem of the paper. Here, formally unramified will be proved to be arithmetically unramified which is defined in an evident manner and coincides with that in algebraic number theory. Hence, formally unramified has an arithmetic sense. At the same time, such a computation coincides with t…
Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of global points defined over the strict Hilbert class field $H^+_F$ of the real quadratic field $F$ and twisted by (unramified) quadratic characters of $Gal(H_c^+/F)$. We extend these results to the situation of ramified quadratic characters; more precisely, we show…
Let $\boldsymbol{G}$ be an unramified connected reductive group defined over a non-archemedian local field $k$ and let $\boldsymbol{T}$ be a maximal torus in $\boldsymbol{G}.$ Let $λ$ be an unramified character of $\boldsymbol{T}.$ Then the conjugacy classes of hyperspecial subgroups of $\boldsymbol{G}(k)$ is a principal homogenous space for a certain finite abelian group $\hatΩ$. Also, the $L$-packet $Π(φ_λ)$ associated to $λ$ is parametrized by an abelian group $\hat{R}$. We show that $\hat{R}$ is naturally a homogenous space for $\hatΩ$. Further, let $π_ρ\inΠ(φ_λ)$, where $ρ\in\hat{R}$ and …
Let F be a global field, G an unramified quasi-split reductive group over F and chi an everywhere unramified automorphic character of a maximal maximally split torus of G. Using Langlands-Shahidi theory, we compute the meromorphic function defined by the action of a global standard intertwining operator associated to chi on a spherical vector and show that the ratio of its poles in the positive Weyl chamber is well behaved.
Let F denote an unramified extension of the cyclotomic extension of Q_p by (p^n)th roots of unity, for an odd prime p. We determine the conductors of those Kummer extensions of F of degree dividing p^n which are Galois over the maximal unramified subfield of F. This follows from a comparison of the Galois module structure of the multiplicative group of F with the unit filtration of F.
Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u is a place of E dividing v, we show that any field embedding E_S \to \bar{E_u} has a dense image. The "unramified outside S" number fields we use are cut out from the l-adic cohomology of the "simple" Shimura varieties studied by Kottwitz and Harris-Taylor. The main ingredients of the proof are then the local Langlands correspondence for GL_n, the main glob…
We remove the assumption "let p be odd or k totally imaginary" from several well-known theorems in Galois cohomology of number fields. For example, we show that the Galois group of the maximal extension of a number field k which is unramified outside 2 has a finite cohomological 2-dimension (also if k has real places). (The dvi-file requires the xypic-package)
Let p be a prime number and F be a number field. We consider the Galois group G over the cyclotomic Z_p extension of F of the maximal unramified, p-decomposed, pro-p-extension of the cyclotomic Z_p extension of F. The question whether G is free pro-p was already asked by many authors. In this article, we highlight a link between the freeness of G and the Galois descent for some localisation kernels. Then we give explicit criterions to show that G is not a free pro-p-group.
We introduce a heuristic prediction for the distribution of the isomorphism class of the Galois group of the maximal pro-p extension of Q unramified outside a "random" set of primes. This is guided by reasoning similar to that governing the Cohen-Lenstra conjectures. We conclude by describing theoretical and experimental evidence for our heuristic.
Let p be an odd prime and S a finite set of primes = 1 mod p. We give an effective criterion for determining when the Galois group G=G_S(p) of the maximal p-extension of Q unramified outside of S is mild when |S|=4 and the cup product H^1(G,Z/pZ) \otimes H^1(G,Z/pZ) --> H^2(G,Z/pZ) is surjective.
Let $E$ be a CM number field and $F$ its maximal real subfield. We prove a level-raising result for regular algebraic conjugate self-dual automorphic representations of $GL_n(\mathbb{A}_E)$. This generalizes previously known results of Thorne by removing certain hypotheses occurring in that work. In particular, the level-raising prime $p$ is allowed to be unramified as opposed to inert in $F$, the field $E/F$ is not assumed to be everywhere unramified, and the field $E$ is allowed to be a general CM field.
Let F be a number field and p be a prime. In the Successive Approximation Theorem, we prove that, for each positive integer n, finitely many candidates for the Galois group G(p,n,F) of the n-th stage F(p,n) of the p-class tower F(p,infinity) over F are determined by abelian type invariants of p-class groups Cl(p,E) of unramified extensions E/F with degree [E:F]=$p^{n-1}$. Illustrated by the most extensive numerical results available currently, the transfer kernels ker(T(F,E)) of the p-class extensions T(F,E):Cl(p,F)-->Cl(p,E) from F to unramified cyclic degree-p extensions E/F are shown to be …
Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$, leads to many interesting predictions about the distribution of $p$-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field $K$, the Galois group of the $p$-class tower of $K$, i.e. $G_K:=\mathrm{Gal}(K_\infty/K)$ where $K_\infty$ is the maximal unramified $p$-extension of $K$. By class field theory, the maximal abelian quotient of $G_K$ is isomorphic to the $p$-class group of $K$. For integers $c\geq 1$, we give a heur…
Explicit solutions of the cubic Fermat equation are constructed in ring class fields $Ω_f$, with conductor $f$ prime to $3$, of any imaginary quadratic field $K$ whose discriminant satisfies $d_K \equiv 1$ (mod $3$), in terms of the Dedekind $η$-function. As $K$ and $f$ vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension $\textsf{K}_3$ of the $3$-adic field $\mathbb{Q}_3$. This is used to give a dynamical proof of a class number rela…
For a quadratic field K, we investigate continuous mod p representations of the absolute Galois groups of K that are unramified away from p and infinity. We prove that for certain pairs (K,p), there are no such irreducible representations. We also list some imaginary quadratic fields for which such irreducible representations exist. As an application, we look at elliptic curves with good reduction away from 2 over quadratic fields.
We give a description of the maximally unramified extension of completed second Real Johnson-Wilson theory using supersingular elliptic curves with $Γ_0(3)$-level structures.
