Kiran S. Kedlaya
We exhibit, for n at least 5, infinitely many quadratic number fields admitting unramified degree n extensions with prescribed signature whose normal closures have Galois group A_n. This generalizes a result of Uchida and Yamamoto, which did not include the ability to restrict the signature, and a result of Yamamura, which was the case n=5.
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We show how to construct unramified qoaternion extensions of quadratic number fields.
Local densities of quadratic polynomials over non-dyadic fields and unramified dyadic fields
本文为非二进域和非分歧二进域上的二次多项式建立了局部密度的精确公式。
Continuing the line of thought of an earlier work, we provide the first infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(5)$, the (unique) smallest nonsolvable group for which this problem was previously open. Our approach also improves upon previous work by yielding the first infinite family of real-quadratic fields possessing an unramified Galois extension whose Galois group is perfect and not generated by involutions. Our result also amounts to a new existence result on quintic number fields with squarefree discriminant and additi…
Extensions with Galois group Hol(C8) unramified over a complex quadratic number field
研究Galois群为Hol(C8)且在复二次域上非分歧的正规扩张，确定其Galois群为半二面体群或16阶模群，并给出显式构造。
Unramified extensions of quadratic number fields with Galois group $2.A_n$
本文首次实现了无穷多个覆盖群 $2.A_n$ 作为无穷多个二次数域上处处非分歧伽罗瓦扩张的伽罗瓦群，解决了此前仅能处理特殊情形或条件性结果的问题。
Unramified Motivic Alternating Multiple Mixed Values
本文研究多重混合值的交错版本，利用Brown和Glanois的下降理论识别五族非分歧交错多重混合值，并猜想所有此类值均已被给出。
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.
Let $k$ be a positive integer and let $F$ be a finite unramified extension of $\mathbb{Q}_2$ with ring of integers $\mathcal{O}_F$. An integral (resp. classic) quadratic form over $\mathcal{O}_F$ is called $k$-universal (resp. classically $k$-universal) if it represents all integral (resp. classic) quadratic forms of dimension $k$. In this paper, we provide a complete classification of $k$-universal and classically $k$-universal quadratic forms over $\mathcal{O}_F$. The results are stated in terms of the fundamental invariants associated to Jordan splittings of quadratic lattices.
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.
Unramified Iwasawa module of $\mathbb{Z}_2$-extension of certain quadratic fields with a bounded quotient
针对判别式含三个奇素因子且 2 分裂的实二次域，本文证明其 $\mathbb{Z}_2$-扩张的非分歧 Iwasawa 模具有有界商，从而验证 Greenberg 猜想并细化模结构。
Unramified extensions of quadratic number fields with Galois group $SL_2(7)$
首次给出具有伽罗瓦群$SL_2(7)$的无限类二次数域处处非分歧扩张，解决了非对合生成完美群在逆伽罗瓦理论中局部-整体问题的实现难题。
Let $f\left(K\right)$ be the number of unramified extensions $L/K$ of a quadratic number field $K$ with $\mathrm{Gal}\left(L/K\right)=H$ and $\mathrm{Gal}\left(L/\mathbb{Q}\right)=G$ where $G$ is a central extension of $\mathbb{F}_{2}^{n}$ by $\mathbb{F}_{2}$. We find a function $g\left(K\right)$ such that $f/g$ has finite moments and a distribution on its values. We show this distribution is a point mass when $H$ is non-abelian and the Cohen-Lenstra distribution when $H$ is abelian, despite the fact that the set of values of $f/g$ do not form a discrete set. We prove an explicit formula for $…
We compute all the moments of a normalization of the function which counts unramified $H_{8}$-extensions of quadratic fields, where $H_{8}$ is the quaternion group of order 8, and show that the values of this function determine a constant distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified G-extensions of quadratic fields for G in a large class of 2-groups, which we conjecture will give finite moments which determine a distribution. Our method additionally can be used to determine the asymptotics of the unnormalized counting fu…
Twisted Gan-Gross-Prasad conjecture for unramified quadratic extensions
本文采用相对迹公式，在非分歧性与局部条件下证明了酉群嵌入一般线性群情形的扭曲全局 Gan-Gross-Prasad 猜想及其精细化。
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.
Let $K$ be a number field of degree $n$ over ${\mathbb Q}$. Then the 4-rank of the strict class group of $K$ is at least ${\text{rank}_2 \, } ({ E_{K}^{+} } / E_K^2) - \lfloor n /2 \rfloor$ where $E_K$ and ${ E_{K}^{+} }$ denote the units and the totally positive units of $K$, respectively, and $\text{rank}_2$ is the dimension as an elementary abelian 2-group. In particular, the strict class group of a totally real field $K$ with a totally positive system of fundamental units contains at least$(n-1)/2$ ($n$ odd) or $n/2 -1$ ($n$ even) independent elements of order 4. We also investigate when u…
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…
We construct étale generalized Heisenberg group covers of hyperelliptic curves over number fields. We use these to produce infinite families of quadratic extensions of cyclotomic fields that admit everywhere unramified generalized Heisenberg Galois extensions.
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 …
Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N, and quasi-linear or quasi-quadratic time in log p, for most basic operations on these fields, including Galois conjugation, Teichmuller lifting and computing minimal polynomials.
For a few quadratic fields, the non-existence is proved of continuous irreducible mod 2 Galois representations of degree 2 unramified outside 2.
- 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 various nonsolvable groups $G$, we prove the existence of extensions of the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set for $G$. For these groups $G$, this gives the existence of number fields of degree $ge(G)$ with an unramified $G$-extension. The existence of such extensions over $\mathbb{Q}$ for all finite groups would imply that, for every finite group $G$, there exists a quadratic number field admitting an unramified $G$-extension, as was recently conjectured. We also provide furt…
On the Finiteness and Structure of Galois Groups of Tamely ramified pro-p Extensions of Imaginary Quadratic Fields
本文研究虚二次域在有限集合外非分歧的最大 pro-p 扩张的 Galois 群，给出 p 为奇数时的显式表示，并确定 Q(i) 的最大 pro-3 扩张的结构。
We give an explicit description of L-packets and quadratic base change for depth-zero representations of unramified unitary groups in two and three variables. We show that this base change is compatible with unrefined minimal K-types.
The celebrated Smith-Minkowski-Siegel mass formula expresses the mass of a quadratic lattice (L, Q) as a product of local factors, called the local densities of (L,Q). This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly, by constructing a smooth integral group scheme model for an appropriate orthogonal group. Our method works for any unramified finite extension of Q_2. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generaliz…
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…
On $\mathbb{Z}_2$-extensions of real quadratic fields with class group of $2$-rank three
本文给出非分歧Iwasawa模有限性判据，构造无穷族实二次域，其类群$2$-秩为三且分圆$\mathbb{Z}_2$-扩张完全分歧并满足Greenberg猜想。
Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$ with $n$-th layer $K_n$. For certain quadratic extensions $F/K$, we prove a simple exact formula for the $λ$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $F_\infty = FK_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer gr…
We study solutions to the Brauer embedding problem with restricted ramification. Suppose $G$ and $A$ are a abelian groups, $E$ is a central extension of $G$ by $A$, and $f:\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow G$ a continuous homomorphism. We determine conditions on the discriminant of $f$ that are equivalent to the existence of an unramified lift $\widetilde{f}:\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow E$ of $f$. As a consequence of this result, we use conditions on the discriminant of $K$ for $K/\mathbf{Q}$ abelian to classify and count unramified nonabelian e…
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 …
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 short and purely bilinear proof of the fact that two chains of $p$-elementary lattices with quadratic form or alternating bilinear form over the $p$-adic integers ore more generally over a complete discrete valuation ring have common hyperbolic bases. This fact, which is useful for the study of Bruhat-Tits buildings, has been proven before with different methods by Abramenko and Nebe and by Frisch.
We give a construction of unramified cyclic octic extensions of certain complex quadratic number fields. The binary quadratic form used in this construction also shows up in the theory of 2-descents on Pell conics and elliptic curves, as well as in the explicit description of cyclic quartic extensions.
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 $2$.
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.
In this paper, we give a formula for the extended Gross-Keating datum of a quadratic form defined over a finite extension of $\mathbb{Z}_p$ (for $p>2$) or a finite unramified extension of $\mathbb{Z}_2$. As an application, we describe an explicit formula for the Siegel series for $\mathbb{Z}_p$. We also present the details of algorithms implemented in a Mathematica package to compute the extended Gross-Keating datum and the Siegel series.
Let $p$ and $q$ be two positive primes. Let $\ell$ be an odd positive prime integer and $F$ a quadratic number field. Let $K$ be an extension of $F$ such that $K$ is a dihedral extension of $\Q$ of degree $\ell$ over $F$ or $K$ is an abelian $\ell$-extension unramified over $F$ assuming $\ell$ divides the class number of $F$. In this paper, we obtain a complete characterization of division quaternion algebras $H_{K}(p, q)$ over $K$.
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…
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…
On the Existence of the Maximal Unramified Pro-$2$-Extension over the Cyclotomic $\mathbb{Z}_2$-Extension with Prescribed Metacyclic Galois Group
研究特定metacyclic非模群作为数域最大无分歧2-扩展的Galois群的可能性，并提出新方法探讨Greenberg猜想。
Let $F$ be the function field of an elliptic curve $X$ over $\F_q$. In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over $F$. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of $¶^1$-bundles on $X$. This allows a purely geometric approach, which involves, amongst others, a classification of the $¶^1$-bundles on $X$. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspid…
We produce infinitely many finite 2-groups that do not embed with index 2 in any group generated by involutions. This disproves a conjecture of Lemmermeyer and restricts the possible Galois groups of unramified 2-extensions, Galois over the rationals, of quadratic number fields.
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…
In this paper we intend to show that certain integers do not occur as the norms of principal ideals in a family of cubic fields studied by Cohn, Shanks, and Ennola. These results will simplify the construction of certain unramified quadratic extensions of such fields.
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…
In this paper we describe an algorithm that quickly computes a maximal a-valued lattice in an F-vector space equipped with a non-degenerate bilinear form, where a is a fractional ideal in a number field F. We then apply this construction to give an algorithm to compute an a-maximal lattice in a quadratic space over any number field F where the prime 2 is unramified. We also develop the theory of p-neighbors for a-valued quadratic lattices at an arbitrary prime p of O_F (including when p | 2) and prove its close connection to the residual geometry of certain quadrics mod p. Finally we give a we…
Let $K/F$ be an unramified quadratic extension of non-Archimedian local fields with residue character not equals to 2. We prove the linear Arithmetic Fundamental Lemma for GL$_4$ with the unit element in the spherical Hecke Algebra. In this article, all measures are normalized by its hyperspecial subgroup.
In this article we will describe a surprising observation that occurred in the construction of quadratic unramified extensions of a family of pure cubic number fields. Attempting to find an explanation will lead us on a magical mystery tour through the land of pure cubic number fields, Hilbert class fields, and elliptic curves.
For a prime $\ell$ and an abelian variety $A$ over a global field $K$, the $\ell$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the $\mathbb{Z}_{\ell}$-corank of the $\ell^{\infty}$-Selmer group and the analytic rank agree modulo $2$. Assuming that $\mathrm{char} K > 0$, we prove that the $\ell$-parity conjecture holds for the base change of $A$ to the constant quadratic extension if $\ell$ is odd, coprime to $\mathrm{char} K$, and does not divide the degree of every polarization of $A$. The techniques involved in the proof include the étale cohomo…
Explicit formula in a imaginary quadratic number field
本文将Haran基于Riesz位势、平等处理所有位的Riemann zeta函数显式公式推广到Q的虚二次扩张，通过定义完全分歧与复情形的Riesz核，揭示其与Dedekind zeta函数各位贡献的自然联系。
A recent paper of A. Sofo proves some results about sums of products of quadratic alternating harmonic numbers and reciprocal binomial coefficients. In this paper, we extend his result to cubic alternating harmonic number sums and develop new closed form representations of sums of cubic alternating harmonic numbers and reciprocal binomial coefficients. Some inter- esting (known or new) illustrative special cases as well as immediate consequences of the main results are also considered.
Maximal Unramified $p$-Extensions with Prescribed Galois Groups: A Quantitative Refinement of Ozaki's Theorem
本文改进了 Ozaki 定理，通过利用 Frattini 结构将实现有限 p-群为 p-类塔群所需的基域次数从阶尺度降至二次尺度，并给出初等交换群的精确估计。
Let p be a rational prime. Let F be a totally real number field such that F is unramified over p and the residue degree of any prime ideal of F dividing p is 1 or 2. In this paper, we show that the eigenvariety for Res_{F/Q}(GL_2), constructed by Andreatta-Iovita-Pilloni, is proper at integral weights for p>=3. We also prove a weaker result for p=2.
Let K = Q(ω) with ω^3 = m be a pure cubic number field. We show that the elementsα\in K^\times whose squares have the form a - ωform a group isomorphic to the group of rational points on the elliptic curve E_m: y^2= x^3 - m. We also show how to apply these results to the construction of unramified quadratic extensions of pure cubic number fields.
Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_\mathfrak{n}$ be the ray class field of $K$ modulo $\mathfrak{n}=N\mathcal{O}_K$. By using the congruence subgroup $\pmΓ_1(N)$, we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group $\mathrm{Gal}(K_\mathfrak{n}/K)$. We also present algorithms to find all form classes and show how to multiply two form classes. As an application, we describe $…
In this paper we give a conjecture for the average number of unramified $G$-extensions of a quadratic field for any finite group $G$. The Cohen-Lenstra heuristics are the specialization of our conjecture to the case that $G$ is abelian of odd order. We prove a theorem towards the function field analog of our conjecture, and give additional motivations for the conjecture including the construction of a lifting invariant for the unramified $G$-extensions that takes the same number of values as the predicted average and an argument using the Malle-Bhargava principle. We note that for even $|G|$, …
In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.
On Boston's Unramified Conjecture for $\GL_2$ and McLeman's $(3,3)$-Conjecture
本文证明 Boston 非分歧猜想在二维奇情形及 McLeman $(3,3)$-猜想的充分性，即虚二次域 $p$-类秩为二且 Galois 群具 Zassenhaus 型 $(3,3)$ 时，$p$-类域塔有限。
Some new infinite families of non-$p$-rational real quadratic fields
受量子信息SIC-POVMs问题研究启发，提出构造同时非p_j-有理、p_j上方非分歧的实二次域无限族的简单方法，可得到p幂循环分量任意大的相关域。
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