Hilbert 90 for biquadratic extensions

Geometry of Biquadratic and Cyclic Cubic Log-Unit Lattices

By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular.

关于某些实双二次域的\(\mathbb{Z}_2\)-扩张的极大非分歧 pro - 2 - 扩张

On the maximal unramified pro-2-extension of $\mathbb{Z}_2$-extension of certain real biquadratic fields

研究实双二次域的\(\mathbb{Z}_2\)-扩张的极大非分歧 pro - 2 - 扩张，通过构造特定实数字域，证明其伽罗瓦群性质，构造出满足条件的实双二次域族，满足格林伯格猜想。

Principalization of $2$-class groups of type $(2,2,2)$ of biquadratic fields $\mathbb{Q}\left(\sqrt{\strut p_1p_2q},\sqrt{\strut -1}\right)$

Let $p_1\equiv p_2\equiv -q\equiv1 \pmod4$ be different primes such that $\displaystyle\left(\frac{2}{p_1}\right)= \displaystyle\left(\frac{2}{p_2}\right)=\displaystyle\left(\frac{p_1}{q}\right)=\displaystyle\left(\frac{p_2}{q}\right)=-1$. Put $d=p_1p_2q$ and $i=\sqrt{-1}$, then the bicyclic biquadratic field ${k}=\mathbb{Q}(\sqrt{d},i)$ has an elementary abelian $2$-class group, $\mathbf{C}l_2(k)$, of rank $3$. In this paper, we study the principalization of the $2$-classes of ${k}$ in its fourteen unramified abelian extensions $\mathbb{K}_j$ and $\mathbb{L}_j$ within ${k}_2^{(1)}$, that is t…

双二次扩张上的四元数代数与平方幂类

Quaternion algebras and square power classes over biquadratic extensions

针对 Klein 4-群双二次扩张，本文利用 Br(F) 的特定子空间确定平方幂类 Galois 模分解中各求和项类型的重数，并证明所有非例外类型均可实现。

Construction of class fields over imaginary biquadratic fields

Let $K$ be an imaginary biquadratic field and $K_1$, $K_2$ be its imaginary quadratic subfields. For integers $N>0$, $μ\geq 0$ and an odd prime $p$ with $\gcd(N,p)=1$, let $K_{(Np^μ)}$ and $(K_i)_{(Np^μ)}$ for $i=1,2$ be the ray class fields of $K$ and $K_i$, respectively, modulo $Np^μ$. We first present certain class fields $\widetilde{K_{N,p,μ}^{1,2}}$ of $K$, in the sense of Hilbert, which are generated by Siegel-Ramachandra invariants of $(K_i)_{(Np^{μ+1})}$ for $i=1,2$ over $K_{(Np^μ)}$ and show that $K_{(Np^{μ+1})}=\widetilde{K_{N,p,μ}^{1,2}}$ for almost all $μ$.

格林伯格猜想与实双二次域的Iwasawa模 II

Greenberg's conjecture and Iwasawa module of Real biquadratic fields II

本文研究实双二次域的$2$-类群秩稳定性，验证了部分数域的格林伯格猜想，并确定了具有平凡$2$-Iwasawa模的实双二次域列表。

An arithmetic intersection formula on Hilbert modular surfaces

In this paper, we obtain an explicit arithmetic intersection formula on a Hilbert modular surface between the diagonal embedding of the modular curve and a CM cycle associated to a non-biquadratic CM quartic field. This confirms a special case of the author's conjecture with J. Bruinier in \cite{BY}, and is a generalization of the beautiful factorization formula of Gross and Zagier on singular moduli. As an application, we proved the first non-trivial non-abelian Chowla-Selberg formula, a special case of Colmez conjecture.

Values at s=-1 of L-functions for multi-quadratic extensions of number fields, and the fitting ideal of the tame kernel

Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadra…

Structure of $Gal(k_2^{(2)}/k)$ for some fields $k=Q(\sqrt{ 2p_1p_2}, \sqrt{-1})$

Let $p_1 \equiv p_2 \equiv5\pmod8$ be different primes. Put $i=\sqrt{-1}$ and $d=2p_1p_2$, then the bicyclic biquadratic field $k=Q(\sqrt{d}, \sqrt{-1})$ has an elementary abelian 2-class group of rank $3$. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group $\mathrm{Gal}(k_2^{(2)}/k)$ of the second Hilbert 2-class field $k_2^{(2)}$ of $k$. We study the capitulation problem of the 2-classes of $k$ in its seven unramified quadratic extensions $K_i$ and in its seven unramified bicyclic biquadratic extensions $L_i$.

The mean vcalue of the product of class numbers of paired quadratic fields I

Let $k$ be a number field and $\tilde k$ a fixed quadratic extension of $k$. In this paper and its companion, we find the mean value of the product of class numbers and regulators of two quadratic extensions $F,F^*\not=\tilde k$ contained in the biquadratic extensions of $k$ containing $\tilde k$.

具有循环类群的低次扩张

Low degree extensions with Cyclic class group

本文证明了阿贝尔四次域及某些类数为2的双二次域中存在欧几里得理想类，并探讨了特定三次和二次扩张中的相关性质。

某些双二次域的类数公式

Class Number Formulas for Certain Biquadratic Fields

本文利用Shintani的工作，针对类数为1的全实二次域的虚二次扩张，提出了两个初等类数公式，分别基于有理函数幂级数展开系数的交错和与$1/p$展开式的周期长度。

Extensions in the cohomology of Hilbert modular varieties

We describe extension classes arising in the $\ell$-adic and Hodge cohomology of Hilbert modular varieties, generalising results of Caspar to arbitrary dimensions. We show that this description is consistent with the "plectic conjectures" of Nekovář and the second author.

Hopf-Galois module structure of quartic Galois extensions of $\mathbb{Q}$

Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and a Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness of the ring of integers $\mathcal{O}_L$ as module over the associated order $\mathfrak{A}_H$ in $H$. For the classical Galois structure $H_c$, we know by Leopoldt's theorem that $\mathcal{O}_L$ is $\mathfrak{A}_{H_c}$-free. If $L/\mathbb{Q}$ is cyclic, it admits a unique non-classical Hopf-Galois structure, whereas if it is biquadratic, it admits three such Hopf-Galois structures. In both cases, we obtain that freeness depends on the solvability in $\mathbb{…

Iwasawa Theory of Hilbert modular forms for anticyclotomic extension

Following Bertolini and Darmon's method, with "Ihara's lemma" among other conditions Longo and Wang proved one divisibility of Iwasawa main conjecture for Hilbert modular forms of weight $2$ and general low parallel weight respectively. In this paper, we remove the "Ihara's lemma" condition in their results.

Hilbert性质在小扩张中的基变换下的保持

Hilbert properties under base change in small extensions

研究域扩张基变换下Hilbert性质与弱Hilbert性质的保持性，证明了有限生成或伽罗瓦群有限生成的伽罗瓦扩张保持这些性质，并给出否定结果。

某些实二次域的割圆$\mathbb{Z}_{2}$-扩张的岩泽模

Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields

本文针对实二次域的割圆$\mathbb{Z}_{2}$-扩张，在岩泽模循环的情形下构造了Greenberg猜想成立的新例子，并利用双二次域的基本单位系给出了计算岩泽模阶数的方法。

Formes modulaires de Hilbert modulo p et valeurs d'extensions galoisiennes

Let F be a totally real field, v an unramified place of F dividing p and rho a continuous irreducible two-dimensional mod p representation of G_F such that the restriction of rho to G_{F_v} is reducible and sufficiently generic. If rho is modular (and satisfies some weak technical assumptions), we show how to recover the corresponding extension between the two characters of G_{F_v} in terms of the action of GL_2(F_v) on the cohomology mod p.

The Hilbert-Grunwald specialization property over number fields

Given a finite group $G$ and a number field $K$, we investigate the following question: Does there exist a Galois extension $E/K(t)$ with group $G$ whose set of specializations yields solutions to all Grunwald problems for the group $G$, outside a finite set of primes? Following previous work, such a Galois extension would be said to have the "Hilbert-Grunwald property". In this paper we reach a complete classification of groups $G$ which admit an extension with the Hilbert-Grunwald property over fields such as $K=\mathbb{Q}$. We thereby also complete the determination of the ``local dimension…

Hilbert 2-Class Fields and 2-Descent

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.

关于某些分圆 $\mathbb{Z}_2$-扩张的 Hilbert $2$-类域塔

On the Hilbert $2$-class field towers of some cyclotomic $\mathbb{Z}_2$-extensions

本文研究特殊 Dirichlet 域分圆 $\mathbb{Z}_2$-扩张各层的 $2$-类域塔长度、Galois 群结构及 capitulation 问题。

The tame Hilbert symbol via K-theory and central extensions

Let $K$ be a mixed characteristic local field whose residue field has cardinality $q$, and let $n$ be an integer dividing $q-1$. In the first part of this document we construct a $K$-theoretic enhancement of the $n$-th power residue symbol $K^\times \times K^\times \rightarrow μ_n$. In the second part we construct central extensions of $GL_m(K)$ by $μ_n$ and we express the $n$-th power residue symbol in terms of a symbol defined using the extension obtained for $m=1$. Our constructions both rely on the study of finite free pointed $μ_n$-sets.

关于具有指定metacyclic Galois群的 cyclotomic $\mathbb{Z}_2$-扩展的最大无分歧pro-$2$-扩展的存在性

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。

Irrational periods of Hilbert Eisenstein series via toroidal compactification

We show that the periods of the holomorphic Eisenstein series of level 1, parallel weight 2, on a Hilbert modular surface are not rational, even up to scaling. This is deduced from a study of the mixed Hodge structure on the cohomology of the Hilbert modular surface, where we find extension classes related to the units of the real quadratic field. We prove similar results for Hilbert modular varieties of all dimensions, and produce extensions of Galois representations in étale cohomology. The key point is to study the restriction of the canonical extension of the Hodge bundle to the boundary o…

On PAC Extensions and Scaled Trace Forms

Any non-degenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields. In particular, prosolvable and prime-to-p extensions of a Hilbertian field. The proofs are based on the theory of PAC extensions.

二次扩张中的秩稳定性及数域整数环的希尔伯特第十问题

Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

本文证明了任意数域二次扩张中存在基变换下秩不增长的正秩 Abel 簇，由此推出任意数域整数环上的希尔伯特第十问题具有否定解答。

There Exist Non-CM Hilbert Modular Forms of Partial Weight 1

In this note, we prove that there exists a classical Hilbert modular cusp form over Q(\sqrt{5}) of partial weight one which does not arise from the induction of a Grossencharacter from a CM extension of Q(\sqrt{5}).

Galois cohomology of certain field extensions and the divisible case of Milnor-Kato conjecture

We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's proof of this conjecture for arbitrary prime l) in a rather clear and elementary way. Assuming this conjecture, we construct a 6-term exact sequence of Galois cohomology with cyclotomic coefficients for any finite extension of fields whose Galois group has an exact quadruple of permutational representations over it. Examples include cyclic groups, dihedral groups, the biquadratic group Z/2\times Z/2, and the symmetric group S_4. Several exact sequences conjectured by Bloch-Kato, Merkurjev-…

Hilbert specialization results with local conditions

Given a field $k$ of characteristic zero and an indeterminate $T$, the main topic of the paper is the construction of specializations of any given finite extension of $k(T)$ of degree $n$ that are degree $n$ field extensions of $k$ with specified local behavior at any given finite set of primes of $k$. First, we give a full non-Galois analog of a result with a ramified type conclusion from a preceding paper and next we prove a unifying statement which combines our results and previous work devoted to the unramified part of the problem in the case $k$ is a number field.

Computing the Hilbert Class Fields of Quartic CM Fields Using Complex Multiplication

Let $K$ be a quartic CM field, that is, a totally imaginary quadratic extension of a real quadratic number field. In a 1962 article titled On the classfields obtained by complex multiplication of abelian varieties, Shimura considered a particular family $\{F_K(m) : m \in \mathbb{Z} >0 \}$ of abelian extensions of $K$, and showed that the Hilbert class field $H_K$ of $K$ is contained in $F_K(m)$ for some positive integer m. We make this m explicit. We then give an algorithm that computes a set of defining polynomials for the Hilbert class field using the field $F_K(m)$. Our proof-of-concept imp…

Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence t…

Quantum Drinfeld Modules I: Quantum Modular Invariant and Hilbert Class Fields

This is the first of a series of two papers in which we present a solution to Manin's Real Multiplication program -- an approach to Hilbert's 12th problem for real quadratic extensions of $\mathbb{Q}$ -- in positive characteristic, using quantum analogs of the exponential function and the modular invariant. In this first paper, we treat the problem of Hilbert class field generation. If $k=\mathbb{F}_{q}(T)$ and $k_{\infty}$ is the analytic completion of $k$, we introduce the quantum modular invariant \[ j^{\rm qt}: k_{\infty}\multimap k_{\infty}\] as a multivalued, modular invariant function. …

A Hilbert Irreducibility Theorem for Enriques surfaces

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0. Bounding the rank of this object, we prove that a conjecture by Campana and Corvaja--Zannier holds for Enriques surfaces, as well as K3 surfaces of Picard rank greater than 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 satisfy the weak Hilbert property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded.

Métriques de sous-quotient et théorème de Hilbert-Samuel arithmétique pour les faisceaux cohérents

The aim of this paper is twofold. First we prove a theorem of extension of sections of a coherent subquotient of a hermitian vector bundle on a complex analytic space with control of the norms, without any of the smoothness assumptions that were needed in previously known analogous results. Then we show how to associate an arithmetic Hilbert-Samuel function to a coherent sheaf on an arithmetic variety -- provided this coherent sheaf is a subquotient of a hermitian vector bundle -- and using the classical arithmetic Hilbert-Samuel theorem and our extension theorem, we give the leading term of t…

对“测度叶状结构与Hilbert第12问题”的补充

Addendum to "Measured foliations and Hilbert 12th problem"

本文基于前期研究结果，研究了实二次数域阿贝尔扩张的数值例子，是对“测度叶状结构与Hilbert第12问题”的补充研究。