Imaginary Quadratic Research Articles

Nonadjacent Radix-tau Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

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8-ranks of Class Groups of Some Imaginary Quadratic Number Fields

Let \( F = \mathbb{Q}{\left( {{\sqrt { - p_{1} p_{2} } }} \right)} \) be an imaginary quadratic field with distinct primes p1 ≡ p2 ≡ 1 mod 8 and the Legendre symbol \( {\left( {\frac{{p_{1} }} {{p_{2} }}} \right)} = 1 \). Then the 8-rank of the class group of F is equal to 2 if and only if the following conditions hold: (1) The quartic residue symbols \( {\left( {\frac{{p_{1} }} {{p_{2} }}} \right)}_{4} = {\left( {\frac{{p_{2} }} {{p_{1} }}} \right)}_{4} = 1 \); (2) Either both p1 and p2 are represented by the form a2 +32b2 over ℤ and \( p^{{h_{ + } {\left( {2p_{1} } \right)}/4}}_{2} = x^{2} - 2p_{1} y^{2} ,x,y \in \mathbb{Z} \), or both p1 and p2 are not represented by the form a2 +32b2 over ℤ and \( p^{{h_{ + } {\left( {2p_{1} } \right)}/4}}_{2} = \varepsilon {\left( {2x^{2} - p_{1} y^{2} } \right)},\;x,y \in \mathbb{Z},\;\varepsilon \in {\left\{ { \pm 1} \right\}} \), where h+(2p1) is the narrow class number of \( \mathbb{Q}{\left( {{\sqrt {2p_{1} } }} \right)} \). Moreover, we also generalize these results.

ℓ-adic Representations Associated to Modular Forms over Imaginary Quadratic Fields

Let p be a regular algebraic cuspidal automorphic representation of GL(2) over an imaginary quadratic number field K, and let l be a prime number. Assuming the central character of p is invariant under the nontrivial automorphism of K, it is shown that there is a continuous irreducible l-adic representation rho of Gal((K) over bar /K) such that L(s, rho(v)) = L(s, pi(v)) whenever v is a prime of K outside an explicit finite set.

Towards Practical Non-interactive Public Key Cryptosystems Using Non-maximal Imaginary Quadratic Orders

We present a new non-interactive public-key distribution system based on the class group of a non-maximal imaginary quadratic order Cl( Δp). The main advantage of our system over earlier proposals based on (Z}/nZ)* [25,27] is that embedding id information into group elements in a cyclic subgroup of the class group is easy (straight-forward embedding into prime ideals suffices) and secure, since the entire class group is cyclic with very high probability. Computational results demonstrate that a key generation center (KGC) with modest computational resources can set up a key distribution system using reasonably secure public system parameters. In order to compute discrete logarithms in the class group, the KGC needs to know the prime factorization of ΔpeΔ1 p2. We present an algorithm for computing discrete logarithms in Cl(Δp) by reducing the problem to computing discrete logarithms in Cl(Δ1) and either F*p or F*p2. Our algorithm is a specific case of the more general algorithm used in the setting of ray class groups [5]. We prove—for arbitrary non-maximal orders—that this reduction to discrete logarithms in the maximal order and a small number of finite fields has polynomial complexity if the factorization of the conductor is known.

Thomas’ Family of Thue Equations Over Imaginary Quadratic Fields

We consider the family of relative Thue equations x3− (t− 1)x2y− (t+ 2)xy2−y3= μ,where the parameter t, the root of unityμ and the solutions x and y are integers in the same imaginary quadratic number field.We prove that there are only trivial solutions (with x, y≤ 1), if t is large enough or if the discriminant of the quadratic number field is large enough or ifRet=−1/2 (there are a few more solutions in this case which are explicitly listed). In the caseRet=−1/2, an algebraic method is used, in the general case, Baker’s method yields the result.

A Formula for the Dedekind ξ-Function of an Imaginary Quadratic Field

In this paper a formula is given for the Dedekind ξ-function, which is similar to that for the Riemann ξ-function. As an application, a lower bound is obtained for the value of the Dedekind ξ-function at the central point.

Refined Lower Bounds on the 2-Class Number of the Hilbert 2-Class Field of Imaginary Quadratic Number Fields with Elementary 2-Class Group of Rank 3

Let k be an imaginary quadratic number field withCk,2, the 2-Sylow subgroup of its ideal class group, isomorphic toZ/2Z×Z/2Z×Z/2Z. By the use of various versions of the Kuroda class number formula, we improve significantly upon our previous lower bound for |Ck1, 2|, the 2-class number of the Hilbert 2-class field ofk.

On the Construction of Indecomposable Positive Definite Hermitian Forms over Imaginary Quadratic Fields

LetDmbe the ring of integers of an imgainary quadratic fieldQ(−m)withm≡3 (mod4). Then there are indecomposable positive definite hermitianDm-lattices of given ranknand given discriminantdwith exactly eight exceptions ifn≠2 and six exceptions ifn=2 and assume the generalized Riemann hypothesis. In these exceptional cases there are no lattices with the desired properties. In particular, this result holds without assuming the generalized Riemann hypothesis, if the square-freem≡−1 (mod8) orm≡−1 (mod12) or the class number ofQ(−m)is unity.

The Maximal Unramified Extensions of the Imaginary Quadratic Number Fields with Class Number Two

The maximal unramified extensions of the imaginary quadratic number fields with class number two are determined explicitly under the Generalized Riemann Hypothesis.

Periods of Cusp Forms and Elliptic Curves Over Imaginary Quadratic Fields

In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the other hand, rational newforms F of weight two for the congruence subgroups , where n is an ideal in the ring of integers R of K. This continues work of the first author and forms part of the Ph.D. thesis of the second author. In each case we compute numerically the value of the L-series at and compare with the value of which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that whenever has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.

A Sufficient Arithmetical Condition for the Ideal Class Group of an Imaginary Quadratic Field to be Cyclic

We give a sufficient arithmetical condition for the ideal class group of an imaginary quadratic field to be cyclic by using Ono’s number.

L-Functions and Class Numbers of Imaginary Quadratic Fields and of Quadratic Extensions of an Imaginary Quadratic Field

L-Functions and Class Numbers of Imaginary Quadratic Fields and of Quadratic Extensions of an Imaginary Quadratic Field

Abelian Varieties with Extra Twist, Cusp Forms, and Elliptic Curves Over Imaginary Quadratic Fields

Abelian Varieties with Extra Twist, Cusp Forms, and Elliptic Curves Over Imaginary Quadratic Fields

Automorphisms of SL 2 of Imaginary Quadratic Integers

Automorphisms of SL 2 of Imaginary Quadratic Integers

The Parity Distribution of Traces in Imaginary Quadratic Fields

AbstractComputations of the Iwasawa λ -invariant for imaginary quadratic fields showed a discrepancy in the proportion of even and odd traces of certain integers from these imaginary quadratic fields. This paper shows that such a discrepancy is in some sense to be expected and that the proportion of even and odd traces of principal generators of powers of prime ideals in imaginary quadratic fields is related to the 3-primary component of the class group.

Primes of the form x2 + ny2: Fermat, class field theory, and complex multiplication

FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Fields Theory and the Cebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j--Invariants. The Class Equation. Ellpitic Curves. References. Index.

On the Computation of all Imaginary Quadratic Fields of Class Number One

On the Computation of all Imaginary Quadratic Fields of Class Number One

Restricted Ramification for Imaginary Quadratic Number Fields and a Multiplicator Free Group

Restricted Ramification for Imaginary Quadratic Number Fields and a Multiplicator Free Group

The Field Generated by the Discriminant of the Class Invariants of an Imaginary Quadratic Field

AbstractThis note determines the quadratic field generated by the square root of the discriminant of the modular equation satisfied by the special value j(α) of the modular function α for a an integer in an imaginary quadratic field.

Imaginary Quadratic Fields with Class Number 2

Imaginary Quadratic Fields with Class Number 2