Imaginary Quadratic Research Articles

Explicit Formula in an Imaginary Quadratic Number Field

Explicit Formula in an Imaginary Quadratic Number Field

A Simpler Proof on the Existence of Good Nested Lattice Codes Over Imaginary Quadratic Integers for AWGN Channel

A Simpler Proof on the Existence of Good Nested Lattice Codes Over Imaginary Quadratic Integers for AWGN Channel

On the Irreducibility of Polynomials Associated with the Complete Residue Systems in any Imaginary Quadratic Fields

For a Gaussian prime π and a nonzero Gaussian integer β = a + b i ∈ ℤ i with a ≥ 1 and β ≥ 2 + 2 , it was proved that if π = α n β n + α n − 1 β n − 1 + ⋯ + α 1 β + α 0 ≕ f β where n ≥ 1 , α n ∈ ℤ i \ 0 , α 0 , … , α n − 1 belong to a complete residue system modulo β , and the digits α n − 1 and α n satisfy certain restrictions, then the polynomial f x is irreducible in ℤ i x . For any quadratic field K ≔ ℚ m , it is well known that there are explicit representations for a complete residue system in K , but those of the case m ≡ 1 mod 4 are inapplicable to this work. In this article, we establish a new complete residue system for such a case and then generalize the result mentioned above for the ring of integers of any imaginary quadratic field.

On the Standard Conjecture for Compactifications of Néron Models of Three-Dimensional Abelian Varieties with Multiplications in an Imaginary Quadratic Field

On the Standard Conjecture for Compactifications of Néron Models of Three-Dimensional Abelian Varieties with Multiplications in an Imaginary Quadratic Field

On the Standard Conjecture for Compactifications of Néron Models of Three-Dimensional Abelian Varieties with Multiplications in an Imaginary Quadratic Field

On the Standard Conjecture for Compactifications of Néron Models of Three-Dimensional Abelian Varieties with Multiplications in an Imaginary Quadratic Field

Lattice Reduction Over Imaginary Quadratic Fields

Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. In this work, we study such lattices and their reduction algorithms. Firstly, when the lattice is spanned over a two dimensional basis, we show that the algebraic variant of Gauss's algorithm returns a basis that corresponds to the successive minima of the lattice if the chosen ring is Euclidean. Secondly, we extend the celebrated Lenstra-Lenstra-Lovász (LLL) reduction from over real bases to over complex bases. Properties and implementations of the algorithm are examined. In particular, satisfying Lovász's condition requires the ring to be Euclidean. Lastly, we numerically show the time-advantage of using algebraic LLL by considering lattice bases generated from wireless communications and cryptography.

Iwasawa Theory of Elliptic Modular Forms Over Imaginary Quadratic Fields at Non-ordinary Primes

AbstractThis article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate integral Iwasawa main conjectures at non-ordinary primes $p$ for suitable twists of the base change of a newform $f$ to an imaginary quadratic field $K$ where $p$ splits, over the cyclotomic ${\mathbb{Z}}_p$-extension, the anticyclotomic ${\mathbb{Z}}_p$-extensions (in both the definite and the indefinite cases) as well as the ${\mathbb{Z}}_p^2$-extension of $K$. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed $p$-adic $L$-functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances).

Annihilators of the Ideal Class Group of a Cyclic Extension of an Imaginary Quadratic Field

AbstractThe 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 $L$.

On Lifting and Modularity of Reducible Residual Galois Representations Over Imaginary Quadratic Fields

In this paper we study deformations of mod $p$ Galois representations $\tau$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $\tau_1$ and $\tau_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_{\Sigma}(F, {\rm Hom}(\tau_2, \tau_1)) \subset {\rm Ext}^1(\tau_2, \tau_1)$. We establish that there exists a basis $\mathcal{B}$ of $H^1_{\Sigma}(F, {\rm Hom}(\tau_2, \tau_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among other things that the elements of $\mathcal{B}$ admit only finitely many crystalline characteristic 0 deformations we prove a modularity lifting theorem asserting that if $\tau$ itself is modular then so is its every crystalline characteristic zero deformation (Theorems 8.2 and 8.5).

Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields

Let l>3 be a prime such that l≡3 (mod 4) and Q(l) has class number 1. Then Hirzebruch and Zagier noticed that the class number of Q(-l) can be expressed as h(-l)=(1/3)(b1+b2+⋯+bm)-m where the bi are partial quotients in the “minus” continued fraction expansion l=[[b0;b1,b2,…,bm¯]]. For an odd prime p≠l, we prove an analogous formula using these bi which computes the sum of Iwasawa lambda invariants λp(-l)+λp(-4) of Q(-l) and Q(-1). In the case that p is inert in Q(-l), the formula pleasantly simplifies under some additional technical assumptions.

On Manin's Conjecture for a Certain Singular Cubic Surface Over Imaginary Quadratic Fields

We prove Manin's conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E6.

On Eisenstein Series of Half Integral Weight and Theta Series Over Imaginary Quadratic Fields

On Eisenstein Series of Half Integral Weight and Theta Series Over Imaginary Quadratic Fields

On the Gras Conjecture for Imaginary Quadratic Fields

AbstractIn this paper we extend K. Rubin's methods to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field k and prime numbers p that divide the number of roots of unity in k.

Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields Whose Discriminant Has Only Two Prime Factors

We will prove a theorem providing sufficient condition for the divisibility of class numbers of certain imaginary quadratic fields by 2g, where g > 1 is an integer and the discriminant of such fields has only two prime divisors.

The Unreasonable Slightness of E2 over Imaginary Quadratic Rings

It is almost always the case that the elementary matrices generate the special linear group SLn over a ring of integers in a number field. The only exceptions to this rule occur for SL2 over rings of integers in imaginary quadratic fields. The surprise is compounded by the fact that, in these cases when elementary generation fails, it actually fails rather badly: the group E2 generated by the elementary 2-by-2 matrices turns out to be an infinite-index, non-normal subgroup of SL2.We give an elementary proof of this strong failure of elementary generation for SL2 over imaginary quadratic rings.

On the Security of Cryptosystems Based on Imaginary Quadratic Class Semigroups

In this paper, we propose a new discrete logarithm problem(DLP) based on the class semigroups of imaginary quadratic non-maximal orders using the special character of non-invertible ideal and analysis its security. To do this, we first explain the mathematical background explicitly and prove some properties of Cls (O) which relate to constructing the DLP and guaranteeing the security. To test the security of the proposed DLP, we compare the class number of the maximal order with that of the non-maximal order and investigate the unique factorization problems of ideals between class groups of the maximal orders and class semigroups of non-maximal orders to ensure the security of the cryptosystem.

Periods of Modular Forms and Imaginary Quadratic Base Change

AbstractLet f be a classical newform of weight 2 on the upper half-plane , E the corresponding strong Weil curve, K a class number one imaginary quadratic field, and F the base change of f to K. Under a mild hypothesis on the pair (f, K), we prove that the period ratio is in ℚ. Here ΩF is the unique minimal positive period of F, and ΩE the area of E(ℂ). The claim is a specialization to base change forms of a conjecture proposed and numerically verified by Cremona and Whitley.

Imaginary Quadratic Fields with Ono Number 3

In this article, we prove that an imaginary quadratic field F has the ideal class group isomorphic to ℤ/2ℤ ⊕ ℤ/2ℤ if and only if the Ono number of F is 3 and F has exactly 3 ramified primes under the Extended Riemann Hypothesis (ERH). In addition, we give the list of all imaginary quadratic fields with Ono number 3.

Truncated Euler Systems over Imaginary Quadratic Fields

AbstractLet K be an imaginary quadratic field and let F be an abelian extension of K. It is known that the order of the class group ClF of F is equal to the order of the quotient UF/ElF of the group of global units UF by the group of elliptic units ElF of F. We introduce a filtration on UF/ElF made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.

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

AbstractIn his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-τ expansion of integers in the number fields and . The (window) nonadjacent form of τ -expansion of integers in was first investigated by Solinas. For integers in , the nonadjacent form and the window nonadjacent form of the τ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-τ expansions for integers in all Euclidean imaginary quadratic number fields.