Jürgen Klüners, Toru Komatsu
In this paper we obtain a complete list of imaginary $n$-quadratic fields with class groups of exponent $3$ and $5$ under ERH for every positive integer $n$ where an $n$-quadratic field is a number field of degree $2^n$ represented as the composite of $n$ quadratic fields.
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In this paper, we present a complete classification of all imaginary $n$-quadratic fields of class number 1.
This paper gives a method to find all imaginary multiquadratic fields of class number dividing $2^{m},$ provided the list of all imaginary quadratic fields of class number dividing $2^{m+1}$ is known. We give a bound on the degree of such fields. As an application of this algorithm, we compute a complete list of imaginary multiquadratic fields with class number dividing $32.$
Mixed multiquadratic splitting fields
本文研究特征2下指数为2的中心单代数以混合多重二次扩张作为分裂域的问题，并构造出非优混合双二次域扩张的具体例子。
In his work about Galois representations, Greenberg conjectured the existence, for any odd prime p and any positive integer t, of a multiquadratic p-rational number field of degree 2 t. In this article, we prove that there exists infinitely many primes p such that the triquadratic field Q(p(p + 2), p(p -- 2), i) is p-rational. To do this, we use an analytic result, proved apart in section \S4, providing us with infinitely many prime numbers p such that p + 2 et p -- 2 have ''big'' square factors. Therefore the related imaginary quadratic subfields Q(i $\sqrt$ p + 2), Q(i $\sqrt$ p -- 2) and Q(…
Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a conjecture by H. Wada \cite{WA70} on the structure of ideal class groups.
By using a new formula of cubing ideals in imaginary quadratic number and function fields combined with Shank's NUCOMP algorithm, Imbert et al. presented a fast algorithms that compute a reduced output of cubing ideals and keep the sizes of the intermediate operands small. The authors asked whether there are specialized formulas to compute higher powers of ideals. In this note, we will derive such formulas.
Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor of $8$. We compute all $D$ with $|D|\leq 3.1\cdot 10^{20}$ such that $E(D)\leq 8$.
Let $a\geq 1$ and $n>1$ be odd integers. For a given prime $p$, we prove under certain conditions that the class groups of imaginary quadratic fields $\mathbb{Q}(\sqrt{a^2-4p^n})$ have a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$. We also show that this family of fields has infinitely many members with the property that their class groups have a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$. In addition, we deduce some unconditional results concerning the divisibility of the class numbers of certain imaginary quadratic fields. At the end, we provide some numerical examples to verify our…
Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat's Last Theorem over $\mathbb Q(i)$. Under the same assumption, we also prove that, for all prime exponents $p \geq 5$, Fermat's equation $a^p+b^p+c^p=0$ does not have non-trivial solutions over $\mathbb Q(\sqrt{-2})$ and $\mathbb Q(\sqrt{-7})$.
In this note we revisit classic work of Soundararajan on class groups of imaginary quadratic fields. Let $A,B,g \ge 3$ be positive integers such that $\gcd(A,B)$ is square-free. We refine Soundararajan's result to show that if $4 \nmid g$ or if $A$ and $B$ satisfy certain conditions, then the number of negative square-free $D \equiv A \pmod{B}$ down to $-X$ such that the ideal class group of $\mathbb{Q} (\sqrt{D})$ contains an element of order $g$ is bounded below by $X^{\frac{1}{2} + ε(g) - ε}$, where the exponent is the same as in Soundararajan's theorem. Combining this with a theorem of Fre…
This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number for all odd primes (both ordinary and supersingular), using Kato's explicit reciprocity law for one dimensional Lubin-Tate formal group. I also study p-part of Shafarevich-Tate group for motives associated to Hecke characters, using Rubin's Main Conjecture in Iwasawa theory. Interestingly a congruence property between p-adic periods of elliptic curve and weig…
Imaginary Non-CM Fields of $2$-Power Degree
本文提出研究任意2-幂次虚非CM域的新方法，证明次数大于16的二面体域结构，并完整分类了次数16类数1的104个虚非CM域，连同5个CM域完成该次数所有虚域分类。
Let $\mathbb{I}$ denote an imaginary quadratic field or the field $\mathbb{Q}$ of rational numbers and $\mathbb{Z}_{\mathbb{I}}$ its ring of intergers. We shall prove an explicit Baker type lower bound for $\mathbb{Z}_{\mathbb{I}}$-linear form of the numbers \begin{equation}\label{1} 1,\ e^{α_1},...,\ e^{α_m},\quad m\ge 2, \end{equation} where $α_0=0$, $α_1,...,α_m$, are $m+1$ different numbers from the field $\mathbb{I}$. Our work gives gives some improvements to the existing explicit versions of of Baker's work about exponential values at rational points. In particilar, dependences on $m$ ar…
We propose and study a generalized continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued fractions are shown to be retained, including exponential convergence, best-of-the-second-kind approximation quality (up to a constant), periodicity of quadratic irrational expansions, and polynomial time complexity.
In the thirties of the last century, I. M. Vinogradov proved that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$ using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1…
On the cohomology of GL_2 and SL_2 over imaginary quadratic fields
本文利用 Voronoi 复形的同调技术计算了虚二次域上 GL_2(O_D) 和 SL_2(O_D) 的上同调，恢复了远离素数 2、3 的整上同调，并发现 H^2 挠子群随 D 增加呈指数增长。
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