In this article, we slightly refine the mean value theorem for the class number of quadratic extensions obtained by Goldfeld-Hoffstein and Datskovsky. We determine all the proportional constants of the mean value with respect to the local conditions including dyadic places.
The goal is to obtain an asymptotic formula for the number of quadratic extensions with bounded discriminant of a some quadratic number field with odd class number. This extends an already known result for Q.
The function $h^0$ for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. In this paper, we prove the conjecture of van der Geer and Schoof about the maximality of $h^0$ at the trivial Arakelov divisor for quadratic extensions of complex quadratic fields.
In this paper, we establish a version of the large sieve with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.
We consider infinite parametric families of octic fields, that are quartic extensions of quadratic fields. We describe all relative power integral bases of the octic fields over the quadratic subfields.
We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields with imaginary quadratic fields. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.
Let $K/F$ be a quadratic tamely ramified extension of a non-Archimedean local field $F$ of characteristic zero. In this paper, we give an explicit formula for Langlands' lambda function $λ_{K/F}$.
We prove that all elliptic curves defined over the cyclotomic $\mathbb{Z}_p$-extension of a real quadratic field are modular under the assumption that the algebraic part of the central value of a twisted $L$-function is a $p$-adic unit. Our result is a generalization of a result of X. Zhang, which is a real quadratic analogue of a result of Thorne.
We generalize Sczech's Eisenstein cocycle for $\mathrm{GL}(n)$ over totally real extensions of $\mathbb{Q}$ to finite extensions of imaginary quadratic fields. By evaluating the cocycle on certain cycles, we parametrize complex values of Hecke L-functions previously considered by Colmez, giving a cohomological interpretation of his algebraicity result on special values of the L-functions.
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…
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…
The conjecture due to Bertrand and Rodriguez Villegas asserts that the 1-norm of the nonzero element in an exterior power of the units of a number field has a certain lower bound. We prove this conjecture for the exterior square case when the number field is a real multi-quadratic Galois extension of any degree of the rationals.
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 $K$.
Investigations of monogenity and power integral bases were recently extended from the absolute case (over Q) to the relative case (over algebraic number fields). Formerly, in the relative case we only succeeded to calculate generators of power integral bases when the ground field is an imaginary quadratic field. This is the first case when we consider monogenity in the more difficult case, in extensions of real quadratic fields. We give efficient algorithms for calculating generators of power integral bases in cubic and quartic extensions of real quadratic fields, more exactly in composites of…
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 $SO(1,3)$.
The aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.
We prove the Iwasawa-theoretic version of a Conjecture of Mazur--Rubin and Sano in the case of elliptic units. This allows us to derive the $p$-part of the equivariant Tamagawa number conjecture at $s = 0$ for abelian extensions of imaginary quadratic fields in the semi-simple case and, provided that a standard $μ$-vanishing hypothesis is satisfied, also in the general case.
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 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 $2$.
We determine the minimal number of variables $Γ^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=4$ over the four ramified quadratic extensions $\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10}) $ of $\mathbb{Q}_2$. In all four fields, we prove that $Γ^*(4,K) = 11$. This is the first example of such a computation for a proper extension of $\mathbb{Q}_p$ where the degree is a power of $p$ greater than $p$.
We show that we can develop from scratch and using only classical language a theory of relative quadratic extensions of a given number field $K$ which is as explicit and easy as for the well-known case that $K$ is the field of rational numbers. As an application we prove a reciprocity law which expresses the number of solutions of a given quadratic equation modulo an integral ideal $\mathfrak{a}$ of $K$ in terms of $\mathfrak{a}$ modulo the discriminant of the equation. We study various $L$-functions associated to relative quadratic extensions. In particular, we define, for totally negative al…
Let $\mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal{A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $\mathcal{A}_n$'s stabilizes for the real fields $F=\mathbb{Q}(\sqrt{f})$ for any integer $0<f<10000$. Equivalently Greenberg's conjecture holds for those fields.
Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx C_2 \oplus C_2$ we determine how a given torsion structure can grow in a quadratic extension and the maximum number of quadratic extensions in which it grows. We also classify the torsion structures which occur as the quadratic twist of a given torsion structure.
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 determine the minimal number of variables $Γ^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=2m$, $m$ odd, $m \ge 3$ over the six ramified quadratic extensions of $\mathbb{Q}_2$. We prove that if $K$ is one of $\{\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10})\}$, $Γ^*(d,K) = \frac{3}{2}d$, and if $K$ is one of $\{\mathbb{Q}_2(\sqrt{-1}), \mathbb{Q}_2(\sqrt{-5})\}$, $Γ^*(d,K) = d+1$. The case $d=6$ was previously known.
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…
We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels--Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen--Stoll. We establish the local formula in many ins…
Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for $p$ and $k$, which states that the non-trivial Iwasawa module of the maximal multiple $\mathbb{Z}_p$-extension field over $k$ has a non-trivial pseudo-null submodule. We prove this conjecture for $p$ and $k$ under the assumption that the Iwasawa $λ$-invariant for a certain $\mathbb{Z}_p$-extension over a finite abelian extension of $k$ vanishes and that the characteristic ideal of the Iwasawa module associated to the cyclotomi…
Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx C_1$ we investigate how a given torsion structure can grow in a quadratic extension and the maximum number of extensions in which it grows. We also study the torsion structures which occur as the quadratic twist of a given torsion structure. In order to achieve this we examine $N$-isogenies defined over $K$ for $N=15,20,21,24,27,30,35$.
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 $…
D. Rudenko proved the homotopy invariance of the truncated polylogarithmic complexes. It follows that on these complexes there is the norm map with good proprieties. We apply his result and get the explicit formula for the norm map in the case of quadratic extensions.
Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.
