Imaginary Quadratic Number Field Research Articles

ℓ-Adic Digits and Class Number of Imaginary Quadratic Fields

ℓ-Adic Digits and Class Number of Imaginary Quadratic Fields

Tempered perfect lattices in the binary case

A new algorithm for computing Hecke operators for SLn was introduced in [14]. The algorithm uses tempered perfect lattices, which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi [17]. The present paper is the first step in characterizing tempered perfect lattices. We obtain a complete classification in the plane, where the Hecke operators are for SL2(Z) and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields.

Sectorial Mertens and Mirsky formulae for imaginary quadratic number fields

We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the usual Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sectors and with congruences.

Secure Groups for Threshold Cryptography and Number-Theoretic Multiparty Computation

In this paper, we introduce secure groups as a cryptographic scheme representing finite groups together with a range of operations, including the group operation, inversion, random sampling, and encoding/decoding maps. We construct secure groups from oblivious group representations combined with cryptographic protocols, implementing the operations securely. We present both generic and specific constructions, in the latter case specifically for number-theoretic groups commonly used in cryptography. These include Schnorr groups (with quadratic residues as a special case), Weierstrass and Edwards elliptic curve groups, and class groups of imaginary quadratic number fields. For concreteness, we develop our protocols in the setting of secure multiparty computation based on Shamir secret sharing over a finite field, abstracted away by formulating our solutions in terms of an arithmetic black box for secure finite field arithmetic or for secure integer arithmetic. Secure finite field arithmetic suffices for many groups, including Schnorr groups and elliptic curve groups. For class groups, we need secure integer arithmetic to implement Shanks’ classical algorithms for the composition of binary quadratic forms, which we will combine with our adaptation of a particular form reduction algorithm due to Agarwal and Frandsen. As a main result of independent interest, we also present an efficient protocol for the secure computation of the extended greatest common divisor. The protocol is based on Bernstein and Yang’s constant-time 2-adic algorithm, which we adapt to work purely over the integers. This yields a much better approach for multiparty computation but raises a new concern about the growth of the Bézout coefficients. By a careful analysis, we are able to prove that the Bézout coefficients in our protocol will never exceed 3max(a,b) in absolute value for inputs a and b. We have integrated secure groups in the Python package MPyC and have implemented threshold ElGamal and threshold DSA in terms of secure groups. We also mention how our results support verifiable multiparty computation, allowing parties to jointly create a publicly verifiable proof of correctness for the results accompanying the results of a secure computation.

Ideal solutions in the Prouhet–Tarry–Escott problem

For given positive integers m m and n n with m > n m>n , the Prouhet–Tarry–Escott problem asks if there exist two disjoint multisets of integers of size n n having identical k k th moments for 1 ≤ k ≤ m 1\leq k\leq m ; in the ideal case one requires m = n − 1 m=n-1 , which is maximal. We describe some searches for ideal solutions to the Prouhet–Tarry–Escott problem, especially solutions possessing a particular symmetry, both over Z \mathbb {Z} and over the ring of integers of several imaginary quadratic number fields. Over Z \mathbb {Z} , we significantly extend searches for symmetric ideal solutions at sizes 9 9 , 10 10 , 11 11 , and 12 12 , and we conduct extensive searches for the first time at larger sizes up to 16 16 . For the quadratic number field case, we find new ideal solutions of sizes 10 10 and 12 12 in the Gaussian integers, of size 9 9 in Z [ i 2 ] \mathbb {Z}[i\sqrt {2}] , and of sizes 9 9 and 12 12 in the Eisenstein integers.

Third homology of SL2 over number fields: The norm-Euclidean quadratic imaginary case

In the article The third homology ofSL2(Q) ([7]), Hutchinson determined the structure of H3(SL2(Q),Z[12]) by expressing it in terms of K3ind(Q)≅Z/24 and the scissor congruence group of the residue field Fp with p a prime number. In this paper, we develop further the properties of the refined scissors congruence group in order to extend this result to the case of imaginary quadratic number fields whose ring of integers is a Euclidean domain with respect to the norm.

On Hermitian Eisenstein series of degree $2$

We consider the Hermitian Eisenstein series $E^{(\mathbb{K})}_k$ of degree $2$ and weight $k$ associated with an imaginary-quadratic number field $\mathbb{K}$ and determine the influence of $\mathbb{K}$ on the arithmetic and the growth of its Fourier coefficients. We find that they satisfy the identity $E^{{(\mathbb{K})}^2}_4 = E^{{(\mathbb{K})}}_8$, which is well-known for Siegel modular forms of degree $2$, if and only if $\mathbb{K} = \mathbb{Q}(\sqrt{-3})$. As an application, we show that the Eisenstein series $E^{(\mathbb{K})}_k$, $k=4,6,8,10,12$ are algebraically independent whenever $\mathbb{K}\neq \mathbb{Q}(\sqrt{-3})$. The difference between the Siegel and the restriction of the Hermitian to the Siegel half-space is a cusp form in the Maa{\ss} space that does not vanish identically for sufficiently large weight; however, when the weight is fixed, we will see that it tends to $0$ as the discriminant tends to $-\infty$. Finally, we show that these forms generate the space of cusp forms in the Maaß Spezialschar as a module over the Hecke algebra as $\K$ varies over imaginary-quadratic number fields.

Quaternion lattices and quaternion fields

Let Q_8 be the quaternion group of order 8 and {chi } its faithful irreducible character. Then {chi } can be realized over certain imaginary quadratic number fields K={mathbb Q}bigl (sqrt{-N}bigr ) but not over their rings of integers (Feit, Serre); here N is a positive square-free integer. We show that this happens precisely when {mathbb Q}bigl (sqrt{N}bigr ) but not {mathbb Q}bigl (sqrt{2}, sqrt{N}bigr ) can be embedded into a Q_8-field over the rationals (Galois with group Q_8) and N is not a sum of two integer squares. In particular, we get that {chi } cannot be integrally realized if N is (properly) divisible by some prime qequiv 7,({textrm{mod},}8).

Bounded gaps between product of two primes in imaginary quadratic number fields

We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston–Graham–Pintz–Yildirim (Proc Lond Math Soc 98:741–774, 2009), and Maynard (Ann Math 181:383–413, 2015). An important consequence of our main theorem is existence of infinitely many pairs alpha _1, alpha _2 which are product of two primes in the imaginary quadratic field K such that |sigma (alpha _1-alpha _2)|le 2 for all embeddings sigma of K if the class number of K is one and |sigma (alpha _1-alpha _2)|le 8 for all embeddings sigma of K if the class number of K is two.

On the p-ranks of the ideal class groups of imaginary quadratic fields

For a prime number $$p \ge 5$$ , we explicitly construct a family of imaginary quadratic fields K with ideal class groups $$Cl_{K}$$ having p-rank $${\mathrm{{rk}}_{p}(Cl_{K})}$$ at least 2. We also quantitatively prove, under the abc-conjecture that for sufficiently large positive real numbers X and any real number $$\varepsilon $$ with $$0< \varepsilon < \frac{1}{p - 1}$$ , the number of imaginary quadratic fields K with the absolute value of the discriminant $$d_{K}\le X$$ and $${\mathrm{{rk}}_{p}(Cl_{K})} \ge 2$$ is $$\gg X^{\frac{1}{p - 1} - \varepsilon }$$ . This conditionally improves the previously known lower bound of $$X^{\frac{1}{p} - \varepsilon }$$ due to Byeon and the recent bound $$X^{\frac{1}{p}}/(\log X)^{2}$$ due to Kulkarni and Levin.

Infinite families of class groups of quadratic fields with 3-rank at least one: quantitative bounds

We improve an effective lower bound on the number of imaginary quadratic fields whose absolute discriminants are less than or equal to [Formula: see text] and whose ideal class groups have 3-rank at least one, which is [Formula: see text]. We also obtain a better bound on the number of imaginary quadratic fields with 3-rank at least two, which is [Formula: see text]; the best-known lower bound so far is [Formula: see text]. For finding these effective lower bounds, we use the Scholz criteria and the parametric families of quadratic fields [Formula: see text] and [Formula: see text] (defined as follows) with escalatory case. We find new infinite families of quadratic fields [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are integers subject to certain conditions for [Formula: see text]. More specifically, we find a complete criterion for the 3-rank difference between [Formula: see text] and its associated quadratic field [Formula: see text] to be one; this is the escalatory case. We also obtain a sufficient condition for the family [Formula: see text] and its associated family [Formula: see text] to have escalatory case. We illustrate some selective implementation results on the 3-class group ranks of [Formula: see text] and [Formula: see text] for [Formula: see text].

Some Remarks on the Divisibility of the Class Numbers of Imaginary Quadratic Fields

For a given integer n, we provide some families of imaginary quadratic number fields of the form Q(4q2−pn), whose ideal class group has a subgroup isomorphic to Z/nZ.

Partitions associated to class groups of imaginary quadratic number fields

We investigate properties of attainable partitions of integers, where a partition $$(n_1,n_2, \dots , n_r)$$ of n is attainable if $$\sum (3-2i)n_i\ge 0$$ . Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian p-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes p. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.

English

For any number field K with non-elementary 3-class group Cl(3,K) = C(3^e) x C(3), e >= 2, the punctured capitulation type kappa(K) of K in its unramified cyclic cubic extensions Li, 1 <= i <= 4, is an orbit under the action of S3 x S3. By means of Artin's reciprocity law, the arithmetical invariant kappa(K) is translated to the punctured transfer kernel type kappa(G2) of the automorphism group G2 = Gal(F(3,2,K)/K) of the second Hilbert 3-class field of K. A classification of finite 3-groups G with low order and bicyclic commutator quotient G/G' = C(3^e) x C(3), 2 <= e <= 6, according to the algebraic invariant kappa(G), admits conclusions concerning the length of the Hilbert 3-class field tower F(3,infty,K) of imaginary quadratic number fields K.

The large sieve with power moduli in imaginary quadratic number fields

In this paper, we establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal [S. Baier and A. Bansal, The large sieve with power moduli for [Formula: see text], Int. J. Number Theory 14 (10) (2018) 2737–2756; Large sieve with sparse sets of moduli for [Formula: see text], Acta Arith. 196 (1) (2020) 17–34] for the Gaussian field.

The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup constructed here is in fact the whole monodromy group. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field K, but only for polarizations with discriminant 1. The latter result is inspired by a recent observation of O'Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type. Finally, we prove the surjectivity of the Abel-Jacobi map from the Chow group of co-dimension two algebraic cycles homologous to zero on every projective irreducible holomorphic symplectic manifold Y of Kummer type onto the third intermediate Jacobian of Y, as predicted by the generalized Hodge Conjecture.

Chromatic Selmer groups and arithmetic invariants of elliptic curves

Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes p. We sketch their role in establishing the p-primary part of the Birch–Swinnerton-Dyer formula in Sections 2–5, and then study the growth of the Mordell–Weil rank along the ℤ p 2 -extension of a quadratic imaginary number field in which p splits in Section 6.

On the Distribution of αp Modulo One in Quadratic Number Fields

Abstract We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.

Determination of all imaginary quadratic fields for which their Hilbert 2-class fields have 2-class groups of rank 2

We determine those imaginary quadratic number fields whose 2-class groups have rank 3 and 4-rank ≤2 and such that the 2-class groups of their Hilbert 2-class fields have rank 2. This result, along with previous work, gives a complete determination of all complex quadratic number fields for which the 2-class groups of their 2-class fields have rank at most 2.

An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative

Let N N be one of the 38 38 distinct square-free integers such that the arithmetic group Γ 0 ( N ) + \Gamma _0(N)^+ has genus one. We constructed canonical generators x N x_N and y N y_N for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of x N x_N , which we express as a polynomial in y N y_N with coefficients that are rational functions in x N x_N . As a corollary, we prove that for any point e e in the upper half-plane which is fixed by an element of Γ 0 ( N ) + \Gamma _0(N)^+ , one can explicitly evaluate x N ( e ) x_N(e) and y N ( e ) y_N(e) . As it turns out, each value x N ( e ) x_N(e) and y N ( e ) y_N(e) is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of x N ( τ ) x_N(\tau ) and y N ( τ ) y_N(\tau ) at any CM point τ \tau , including elliptic points, for any genus one group Γ 0 ( N ) + \Gamma _0(N)^+ . Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.