The Franke filtration is a finite filtration of the space of automorphic forms on a connected reductive linear algebraic group defined over an algebraic number field. The main feature of the filtration is that its quotients can be described in terms of parabolically induced representations using the main values of derivatives of Eisenstein series and the residues of these. The goal of this paper is to provide a complete explicit description of the Franke filtration of the space of automorphic forms on the symplectic group of rank two. The approach is different from the original approach of Franke, and takes into account the full cuspidal support of automorphic forms, that is, the cuspidal automorphic representation from which the Eisenstein series is built and the evaluation point at which it is evaluated. This considerably simplifies the exposition and makes it possible to obtain very explicit results and reveal the phenomena present in the filtration. The considered low rank case exhibits many of the properties and phenomena present in the cases of arbitrary rank. The explicit description of the Franke filtration in this case has important implications and applications in cohomology of congruence subgroups related to the Hilbert–Siegel modular forms of degree two, the Hilbert–Siegel modular variety of degree two and the corresponding Shimura variety.

Abstract Using trace formulas for Hecke operators, Eichler first provided a positive solution about basis problems of elliptic cusp forms by quadratic forms. J-L. Waldspurger established that elliptic cusp forms of arbitrary level are spanned by theta series by means of different and interesting ideas and methods. This result is given by Zagier’s analytic theorems, the Siegel main theorem of quadratic forms and the theory of Hecke operators. We intend to generalize Waldspurger’s results and determine theta series which span the space of Hilbert new forms over arbitrary totally real algebraic number fields following Waldspurger’s methods.

In the paper, we extend the ADC property to the representation of quadratic lattices by quadratic lattices, which we define as n -ADC-ness. We explore the relationship between n -ADC-ness, n -regularity, and n -universality for integral quadratic lattices. Also, for n\ge 2 , we give necessary and sufficient conditions for an integral quadratic lattice over arbitrary non-archimedean local fields to be n -ADC. Moreover, we show that over any algebraic number field F , an integral \mathcal{O}_{F} -lattice with rank n+1 is n -ADC if and only if it is \mathcal{O}_{F} -maximal of class number one.

Let [Formula: see text] denote the ring of algebraic integers of an algebraic number field [Formula: see text], where [Formula: see text] is a root of an irreducible polynomial [Formula: see text], [Formula: see text]. We say [Formula: see text] is monogenic if [Formula: see text] is a basis for [Formula: see text]. In this paper, we explicitly compute the discriminant of [Formula: see text] and give necessary and sufficient conditions involving only [Formula: see text] for [Formula: see text] to be monogenic. Moreover, we characterize all the primes dividing the index of the subgroup [Formula: see text] in [Formula: see text]. As an application, we also provide a class of monogenic polynomials having nonsquare-free discriminant and Galois group [Formula: see text], the symmetric group on [Formula: see text] letters. In the end, we state a conjecture and show its implications.

Let [Formula: see text] be the cyclotomic [Formula: see text]-extension field of an algebraic number field [Formula: see text]. Moreover, we take a [Formula: see text]-extension [Formula: see text] over [Formula: see text]. In this paper, we study the behavior of the [Formula: see text]-part of the class number of intermediate fields of [Formula: see text], and the structure of the unramified Iwasawa module [Formula: see text] as a module over [Formula: see text]. We show that a formula on the [Formula: see text]-part of the class number of certain intermediate fields of [Formula: see text] holds under several assumptions (one of these assumptions is that [Formula: see text] is a finitely generated torsion [Formula: see text]-module). We also consider the structure of [Formula: see text] for the case that every prime lying above [Formula: see text] is unramified in [Formula: see text]. In particular, we give sufficient conditions and examples such that the above assumptions are satisfied.

Let R be a ring of algebraic integers of an algebraic number field F and let G ≤ GLn(R) be a finite group. In this paper we show that the R-span of G is just the matrix ring Mn(R) of the n X n-matrices over R if and only if G/Opi(G) is absolutely simple for all pi ∈ π, where π is the set of the positive prime divisors of |G| and Opi(G) is the largest normal pi-subgroup.

Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent vectors in its set of minimal vectors. Both classes of lattices have been applied for signal transmission in some channels, such as wiretap channels. Recently, some advances have been made in the search for well-rounded lattices that can be realized as algebraic lattices. Moreover, some works have been published studying algebraic lattices obtained from modules in cyclic number fields of odd prime degree $p$. In this work, we generalize some results of a recent work of Tran et al. and we provide new constructions of well-rounded algebraic lattices from a certain family of modules in the ring of integers of each of these fields when $p$ is ramified in its extension over the field of rational numbers.Comment: 11 pages

Abstract The condition number of a generator matrix of an ideal lattice derived from the ring of integers of an algebraic number field is an important quantity associated with the equivalence between two computational problems in lattice-based cryptography, the “Ring Learning With Errors (RLWE)” and the “Polynomial Learning With Errors (PLWE)”. In this work, we compute the condition number of a generator matrix of the ideal lattice from the whole ring of integers of any odd prime degree cyclic number field using canonical embedding.

Fields of Algebraic Numbers Whose Multiplicative Groups are Free Modulo the Group of Units

For an arbitrary odd-degree polynomial 𝑓 over an arbitrary field of algebraic numbers K, the class of always quasiperiodic elements in K((𝑥)) of the form 𝑣+𝑤√𝑓/𝑢 , where 𝑣,𝑤, 𝑢 ∈ K[𝑥], in the hyperelliptic field K(𝑥)(√𝑓), has been determined. This class is characterized by certain relationships involving the polynomials 𝑢, 𝑣,𝑤, and 𝑓, as well as their degrees. The class is guaranteed to be nonempty if at least one quasiperiodic element exists in the hyperelliptic field.Furthermore, a specific subclass of always periodic elements has been identified within this broader class.

The product formula for the measure of representations of a quadratic form over the rational number field was found by Siegel, and were generalized by Fractman to the higher-dimensional case over totally real algebraic number fields. We prove product formulas for quadratic forms over arbitrary algebraic number fields and generalize formulas given by Siegel and Fractman to the case of arbitrary algebraic number fields.

Consider a curve $\mathcal C$ defined over an algebraic number field $k$. This work is concerned with the number of generalized Jacobians $J_{\mathfrak{m}}$ of $\mathcal C$ associated with moduli $\mathfrak{m}$ defined over $k$ such that a fixed class of finite order in the Jacobian $J$ of $\mathcal C$ is lifted to a torsion class in the generalized Jacobian $J_{\mathfrak{m}}$. On the one hand it is shown that there are infinitely many generalized Jacobians with the above property, and on the other hand, under some additional constraints on the support of $\mathfrak{m}$ or the structure of $J_{\mathfrak{m}}$, it is shown that the set of generalized Jacobians of this type is finite. In addition, it is proved that there are finitely many generalized Jacobians which have a lift of two given divisors to classes of finite orders in $J_{\mathfrak{m}}$. These results are applied to the problem of the periodicity of continued fractions in the field of formal power series $k((1/x))$ constructed for special elements of the function field $k(\widetilde{\mathcal{C}})$ of a hyperelliptic curve $\widetilde{\mathcal{C}}\colon y^2=f(x)$. In particular, it is shown that for each $n \in \mathbb N$ there is a finite number of monic polynomials $\omega(x) \in k[x]$ of degree at most $n$ such that the element $\omega(x) \sqrt{f(x)}$ has a periodic expansion in a continued fraction. Bibliography: 14 titles.

A characterization of half-factorial orders in algebraic number fields

This paper considers the construction of the fundamental function and Abelian differentials of the third kind on a plane algebraic curve over the field of complex numbers that has no singular points. The algorithm for constructing differentials of the third kind was described in Weierstrass’s lectures. The paper discusses its implementation in the Sage computer algebra system. The specifics of this algorithm, as well as the very concept of the differential of the third kind, implies the use of both rational numbers and algebraic numbers, even when the equation of a curve has integer coefficients. Sage has a built-in tool for computations in algebraic number fields, which allows the Weierstrass algorithm to be implemented almost literally. The simplest example of an elliptic curve shows that it requires too many resources, far beyond the capabilities of an office computer. A symmetrization of the method is proposed and implemented, which makes it possible to solve the problem while saving a significant amount of computational resources.

Abstract The idea of extending the classical RSA protocol using algebraic number fields was introduced by Takagi and Naito (Construction of RSA cryptosystem over the algebraic field using ideal theory and investigation of its security. Electron Commun Japan Part III Fund Electr Sci. 2000;83:19–29). Recently, Zheng et al. proposed the use of the ring of algebraic integers of an algebraic number field and the lattice theory to present a high-dimensional form of RSA. The authors claim that their proposal is post-quantum and is significant both from the theoretical and practical point of view. In this article, we prove that the security of Zheng et al.’s scheme is still based on the factorization problem, and we present a practical quantum attack on this proposed scheme, our attack is a quantum polynomial time algorithm that employs Shor’s algorithm as a subroutine.

Accumulation points of normalized approximations

Abstract Let p be a prime number. Let $n\geq 2$ be an integer given by $n = p^{m_1} + p^{m_2} + \cdots + p^{m_r}$ , where $0\leq m_1 < m_2 < \cdots < m_r$ are integers. Let $a_0, a_1, \ldots , a_{n-1}$ be integers not divisible by p. Let $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta \in {\mathbb C}$ a root of an irreducible polynomial $f(x) = \sum _{i=0}^{n-1}a_i{x^i}/{i!} + {x^n}/{n!}$ over the field $\mathbb Q$ of rationals. We prove that p divides the common index divisor of K if and only if $r>p$ . In particular, if $r>p$ , then K is always nonmonogenic. As an application, we show that if $n \geq 3$ is an odd integer such that $n-1\neq 2^s$ for $s\in {\mathbb Z}$ and K is a number field generated by a root of a truncated exponential Taylor polynomial of degree n, then K is always nonmonogenic.

The article proves the following statement: in any hyperelliptic field L defined over the field of algebraic numbers K which having non-trivial units of the ring of integer elements of the field L, there is an element for which the period length of the continued fraction is greater any pre-given number.

A new result has been obtained generalizing A.G. Postnikov's formula on indices for the case of a power of 2. The multiplicative structure of the reduced residue systems modulo the degree of a prime ideal is investigated. Estimates of some sums of characters in the fields of algebraic numbers are established.

Let Q be the d-dimensional space of finite adeles over the algebraic number field K and let P=Q∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P=Q^*$$\\end{document} be its dual space. For a certain type of Vladimirov type time-dependent Hamiltonian HV(t):Q×P→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_V(t):Q\ imes P\\rightarrow {\\mathbb {C}}$$\\end{document} we construct the Feynman formulas for the solution of the Cauchy problem with the Schrödinger operator where the caret operator stands for the qp- or pq-quantization.