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A family $\mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}\in \mathcal{F}$ can be made uniformly bounded after removing from $\mathcal{F}$ those whose number field degrees lie in a subset of $\mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $\mathcal{E}_F$ of elliptic curves defined over number fields and with $F$-rational $j$-invariant is typically bounded in torsion. For any integer $d\in\mathbb{Z}^+$, we also strengthen a result on typically bounding torsion for the family $\mathcal{E}_d$ of elliptic curves defined over number fields and with degree $d$ $j$-invariant., 17 pages, to appear in Journal of Number Theory

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

A number field is called a Polya field if the module of integer-valued polynomials over its ring of integers has a regular basis. Let L be a field which is a compositum of two quadratic Polya fields. Some questions were raised on Polyaness of L in [7] . Part was solved in [3] and [8] . Here we develop a general strategy allowing us to treat the remaining cases but also to find all these previous results.

We introduce a tensor decomposition of the $\ell$-adic Tate module of an abelian variety $A_0$ defined over a number field which is geometrically isotypic. If $A_0$ is potentially of $\GL_2$-type and defined over a totally real number field, we use this decomposition to describe its Sato--Tate group and to prove the Sato--Tate conjecture in certain cases., Comment: Minor changes. To appear in Math. Zeit

By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular., Comment: 14 pages

Let O be the ring of integers of a number field, and let n ≥ 3 . This paper studies bi-interpretability of the ring of integers Z with the special linear group SL n ( O ) , the general linear group GL n ( O ) and the subgroup T n ( O ) of GL n ( O ) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

Let (X, L) be a polarized variety over a number field K. We suppose that L is an hermitian line bundle. Let M be a non compact Riemann Surface and $$U\subset M$$ be a relatively compact open set. Let $$\varphi :M\rightarrow X(\mathbf{C})$$ be a holomorphic map. For every positive real number T, let $$A_U(T)$$ be the cardinality of the set of $$z\in U$$ such that $$\varphi (z)\in X(K)$$ and $$h_L(\varphi (z))\le T$$ . After a revisitation of the proof of the sub exponential bound for $$A_U(T)$$ , obtained by Bombieri and Pila, we show that there are intervals of the reals such that for T in these intervals, $$A_U(T)$$ is upper bounded by a polynomial in T. We then introduce subsets of type S with respect of $$\varphi $$ . These are compact subsets of M for which an inequality similar to Liouville inequality on algebraic points holds. We show that, if M contains a subset of type S, then, for every value of T the number $$A_U(T)$$ is bounded by a polynomial in T. As a consequence, we show that if M is a smooth leaf of an algebraic foliation in curves defined over K then $$A_U(T)$$ is bounded by a polynomial in T. Let S(X) be the subset (full for the Lebesgue measure) of points which verify some kind of Liouville inequalities. In the second part we prove that $$\varphi ^{-1}(S(X))\ne \emptyset $$ if and only if $$\varphi ^{-1}(S(X))$$ is full for the Lebesgue measure on M.

Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.

The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $\chi$ of $K$, the $n$-Selmer groups of $E_1^\chi $ and $E_2^\chi$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogue of this for Bloch-Kato $p^r$-Selmer group of modular forms. We compare the Bloch-Kato Selmer groups of a modular form respectively with the Greenberg Selmer group when the modular form is $p$-ordinary and with the signed Selmer group of Lei-Loeffler-Zerbes when the modular form is non-ordinary at $p$. We also indicate the corresponding results over $\Q_\cyc$ and its relation with the well known congruence results of the special values of the corresponding $L$-functions due to Vatsal.

A basic problem in transcendental number theory is to determine the arithmetic properties of analytic functions of the form f ( z ) = ∑ k = 0 ∞ a k z k where the coefficients a k ∈ K belong to an algebraic number field. In particular, one of the most basic problems is to determine if f ( α ) is algebraic or transcendental for non-zero algebraic arguments α. For example, if f ( z ) is a transcendental Mahler function, then under generic conditions f ( α ) is transcendental for all non-zero algebraic numbers with | α | 1 . Also, if f ( z ) is an E-function, then there exist algorithms which completely determine the arithmetic properties of f ( n ) ( α ) for non-zero algebraic numbers α. In contrast to these and other algorithmic results, we construct three functions f ( z ) , g ( z ) , and h ( z ) with computable rational coefficients for which no algorithms exist that determine if f ( n ) ∈ Q , g ( n ) ( 1 ) ∈ Q , or ∫ 0 1 h ( z ) z n d z ∈ Q for integral n ≥ 0 . Our results are an application of an undecidable variant of the Collatz Problem due to Lehtonen [9] .

It is an open problem whether the Iwasawa λ-invariants of the Z p -extensions of a fixed number field are bounded. Using the class-field theoretic tool of logarithmic class groups, we obtain bounds for the λ-invariants of Z p -extensions of suitable field extensions of imaginary quadratic number fields. We also prove the Gross-Kuz'min Conjecture for certain families of non-cyclotomic Z p -extensions.

We describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group. When presenting more modern developments, we can only hint at the much broader context and the very powerful general techniques that are involved, but we endeavour to give complete statements or at least examples where feasible. The timeline goes from a classical result proved in 1890 (Stickelberger’s Theorem) to a recent (2020) breakthrough: the proof of the Brumer-Stark conjecture by Dasgupta and Kakde.

We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukuda's theorem by Li, Ouyang, Xu and Zhang. As an application, we give an example of pseudo-null Iwasawa module over a certain $2$-adic Lie extension., 7 pages. Archiv der Mathematik (2021)

The present paper is devoted to methods of the automatic verifying of the Kronecker-Vahlen theorem on the shortest length of the Euclidean algorithm in algebraic number fields with the infinite group of units. We provide explicit methods to prove that the Kronecker-Vahlen theorem fails in certain algebraic number field. In particular, we give a complete solution of the problem for quadratic norm-Euclidean number fields.

Let $ K $ be a number field, $ S $ a finite set of places of $ K $, and $ \mathcal{O}_S $ be the ring of $ S $-integers. Moreover, let $$ G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)} $$ be a polynomial in $ Z $ having simple linear recurrences of integers evaluated at $ n $ as coefficients. Assuming some technical conditions we give a description of the zeros $ (n,z) \in \mathbb{N} \times \mathcal{O}_S $ of the above polynomial. We also give a result in the spirit of Hilbert irreducibility for such polynomials., Comment: 13 pages

LetK/kbe an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle forK/kand the defect of weak approximation for the norm one torus\[R_{K/k}^1{\mathbb{G}_m}\]. We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure ofK/khas symmetric or alternating Galois group.

Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would follow if it were known for abc sums that are not highly divisible. We prove both theorems for general number fields., 22 pages

Let K be a number field defined by an irreducible polynomial F(X) ∈ ℤ[X] and ℤK its ring of integers. For every prime integer p, we give sufficient and necessary conditions on F(X) that guarantee the existence of exactly r prime ideals of ℤK lying above p, where $$\overline F \left( X \right)$$ factors into powers of r monic irreducible polynomials in $${\mathbb{F}_p}\left[ X \right]$$ . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly r prime ideals of ℤK lying above p. We further specify for every prime ideal of ℤK lying above p, the ramification index, the residue degree, and a p-generator.

Let $d$ be a positive square-free integer. In this paper we shall investigate the structure of the $2$-class group of the cyclotomic $\mathbb{Z}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$. Furthermore, we deduce the structure of the $2$-class group of cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{-d})$., Comment: We corrected some typos. 13 pages

Let K=Q(α) be a pure number field generated by a root α of a monic irreducible polynomial F(x)=x2r·5s−m, with m≠∓1 is a square free integer, r and s are two positive integers. In this article, we s...

For an algebraic number field [Formula: see text], let [Formula: see text] be the associated Dedekind zeta-function. It is conjectured that [Formula: see text] is transcendental for any positive integer [Formula: see text]. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when [Formula: see text] is a totally real number field, [Formula: see text] is an algebraic multiple of [Formula: see text] and hence, is transcendental. If [Formula: see text] is not totally real, the question of whether [Formula: see text] is irrational or not remains open. In this paper, we prove that for a fixed integer [Formula: see text], at most one of [Formula: see text] is rational, as [Formula: see text] varies over all imaginary quadratic fields. We also discuss a generalization of this theorem to CM-extensions of number fields.

In this paper, we study Ramanujan sums $$c_{\mathcal {J}}({\mathcal {I}})$$ , where $${\mathcal {I}}$$ and $${\mathcal {J}}$$ are integral ideals in an arbitrary quadratic number field. In particular, the asymptotic behaviour of sums of $$c_{\mathcal {J}}({\mathcal {I}})$$ over both $${\mathcal {I}}$$ and $${\mathcal {J}}$$ is investigated.

We generalize the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over $\mathbb Q$ and its balanced version introduced recently by Beli-Chan-Icaza-Liu to positive definite quadratic forms over a totally real number field $K$. We apply the balanced HKZ-reduction theory to study the growth of the {\em $g$-invariants} of the ring of integers of $K$. More precisely, for each positive integer $n$, let $\mathcal O$ be the ring of integers of $K$ and $g_{\mathcal O}(n)$ be the smallest integer such that every sum of squares of $n$-ary $\mathcal O$-linear forms must be a sum of $g_{\mathcal O}(n)$ squares of $n$-ary $\mathcal O$-linear forms. We show that when $K$ has class number 1, the growth of $g_{\mathcal O}(n)$ is at most an exponential of $\sqrt{n}$. This extends the recent result obtained by Beli-Chan-Icaza-Liu on the growth of $g_{\mathbb Z}(n)$ and gives the first sub-exponential upper bound for $g_{\mathcal O}(n)$ for rings of integers $\mathcal O$ other than $\mathbb Z$.

We consider a G-function F ( z ) = ∑ k = 0 ∞ A k z k ∈ K 〚 z 〛 , where K is a number field, of radius of convergence R and annihilated by the G-operator L ∈ K ( z ) [ d / d z ] , and a parameter β ∈ Q ∖ Z ⩽ 0 . We define a family of G-functions F β , n [ s ] ( z ) = ∑ k = 0 ∞ A k ( k + β + n ) s z k + n indexed by the integers s and n. Fix α ∈ K ⁎ ∩ D ( 0 , R ) . Let Φ α , β , S be the K -vector space generated by the values F β , n [ s ] ( α ) , n ∈ N , 0 ⩽ s ⩽ S . We show that there exist some positive constants u K , F , β and v F , β such that u K , F , β log ⁡ ( S ) ⩽ dim K ⁡ Φ α , β , S ⩽ v F , β S . This generalizes a previous theorem of Fischler and Rivoal (2017), which is the case β = 0 . Our proof is an adaptation of their article [6] , making use of the Andre-Chudnovsky-Katz Theorem on the structure of the G-operators and of the saddle point method. Moreover, we rely on Dwork and Andre's quantitative results on the size of G-operators to obtain an explicit formula for the constant u K , F , β , which was not given in [6] in the case β = 0 .

Let L be a number field. For a given prime p, we define integers α p L $ \alpha_{p}^{L} $ and β p L $ \beta_{p}^{L} $ with some interesting arithmetic properties. For instance, β p L $ \beta_{p}^{L} $ is equal to 1 whenever p does not ramify in L and α p L $ \alpha_{p}^{L} $ is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α p L $ \alpha_{p}^{L} $ is not zero for all p then such residues determine the genus of the integral trace.

We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVert\alpha p\rVert_\omega < \mathrm{N}(p)^{-1/8+\epsilon}$ is satisfied for infinitely many $p$, where $\lVert\varrho\rVert_\omega$ measures the distance of $\varrho\in\mathbb{C}$ to $\mathscr{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$. The proof is based on Harman's sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural map $\text{CH}^1(E_1)_0\otimes\text{CH}^1(E_2)_0\to\text{CH}^2(E_1\times E_2)$, and a conjecture due to Beilinson predicts that the image of this map is finite when $k$ is a number field. We construct a $2$-parameter family of elliptic curves that can be used to produce examples of pairs $E_1,E_2$ where this image is finite. The family is constructed to guarantee the existence of a rational curve passing through a specified point in the Kummer surface of $E_1\times E_2$., 12 pages, 1 figure, 2 tables. Organizational changes, expanded some comments into lemmas. Content is identical to the published version, aside from the organization of information in the Appendix

Let p and q be two primes and $$\ell$$ be an odd positive square-free integer such that $$\ell \not \in \{1,p, q, pq\}$$ . In this paper we give the unit group of some number fields of the form $${\mathbb {Q}}(\sqrt{2},\sqrt{p}, \sqrt{q}, \sqrt{-\ell } )$$ .

Let F be a number field and let i ≥ 2 be an integer. In this paper, we study the positive etale wild kernel WK 2 i − 2 et , + F , which is the twisted analogue of the 2-primary part of the narrow class group. If E / F is a Galois extension of number fields with Galois group G, we prove a genus formula relating the order of the groups ( WK 2 i − 2 et , + E ) G and WK 2 i − 2 et , + F .

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The present paper gives explicit construction of an integral basis of $K$ along with applications. This construction of an integral basis of $K$ extends a result proved in [J. Number Theory, {173} (2017), 129-146] regarding periodicity of integral bases of $\mathbb{Q}(\sqrt[n]{a})$ when $a$ is squarefree.

In this article we establish the arithmetic purity of strong approximation for certain semi-simple simply connected $k$-simple linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with codim$(G\setminus U, G)\geq 2$, we prove that (i) if $G$ is $k$-isotropic, then $U$ satisfies strong approximation off any one (hence any finitely many) place; (ii) if $G$ is the spin group of a non-degenerate quadratic form which is non-compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine combinatorial sieve which allows to produce integral points with almost prime polynomial values., V4 major revision (notably in Section 3). To appear in Compositio Mathematica

Let $$K$$ be an imaginary quadratic number field and $$\mathcal{O}_K$$ be its ring of integers. We show that, if the arithmetic functions $$f, g\colon\mathcal{O}_K\rightarrow \mathbb{C}$$ both have level of distribution $$\vartheta$$ for some $$0

Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond(Vn ) of the Vandermonde matrix Vn associated to the nth cyclotomic polynomial. We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn ) for n = 2 kpℓ , where k, ℓ ≥ 0 are integers and p is an odd prime number.

In this paper, we establish a necessary and sufficient criterion for a finite metabelian 2-group G whose abelianized $$G^{ab}$$ is of type $$(2, 2^m)$$ , with $$m\ge 2$$ , to be metacyclic. This criterion is based on the rank of the maximal subgroup of G which contains the three normal subgroups of G of index 4. Then, we apply this result to study the structure of the Galois group of the maximal unramified pro-2-extension of the cyclotomic $$\mathbb {Z}_2$$ -extension of certain number fields. Illustration is given by some real quadratic fields.

We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776 ) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070 ). In particular, we note that morphisms of motivic dga’s can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga’s is a certain limit of such spectra; we give an explicit formula for this limit—a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call “motivic Chow coalgebras”. We discuss the relationship between motivic Chow coalgebras and motivic dga’s of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite etale over a number field, morphisms of associated motivic dga’s are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim’s relative unipotent section conjecture, hence as an ounce of evidence for the latter.

In this paper, we study the ring of integer-valued polynomials $${\text {Int}}(M_n(\mathcal {O}_K)):=\{f\in M_n(K)[x]~|~ f(M_n(\mathcal {O}_K))\subseteq M_n(\mathcal {O}_K)\}$$ , where K is a number field and $$\mathcal {O}_K$$ is the ring of algebraic integers of K. We show that for a prime number $$p\in {\mathbb {Z}}$$ , the polynomial $$f_{p,n}(x):=\frac{(x^{p^n}-x)(x^{p^{n-1}}-x)\ldots (x^p-x)}{p}$$ is an element of $${\text {Int}}(M_n(\mathcal {O}_K))$$ if and only if p is a totally split prime in $$\mathcal {O}_K$$ . Also, we consider the ring $${\text {Int}}_{M_n({\mathbb {Q}})}(M_n(\mathcal {O}_K)):={\text {Int}}(M_n(\mathcal {O}_K))\bigcap M_n( {\mathbb { Q}})[x]$$ . Then, we characterize finite Galois extensions K of $${\mathbb {Q}}$$ in terms of the ring $${\text {Int}}_{M_n({\mathbb { Q}})}(M_n(\mathcal {O}_K))$$ . In fact, we prove that $${\text {Int}}_{M_n({\mathbb { Q}})}(M_n(\mathcal {O}_K))={\text {Int}}_{M_n({\mathbb { Q}})}(M_n(\mathcal {O}_{K'}))$$ if and only if $$K=K'$$ , where $$K, K'$$ are two finite Galois extensions of $${\mathbb {Q}}$$ . Finally, we present some results on Noetherian property of the rings $${\text {Int}}_{M_n({\mathbb { Q}})}(M_n(\mathcal {O}_K))$$ . Then, we obtain many non-Noetherian integral domains, $${\text {Int}}_{\mathbb {Q}}(\mathcal {O}_K)$$ , between the ring $${\mathbb {Z}}[x]$$ and the classical ring of integer-valued polynomials $${\text {Int}}({\mathbb {Z}})$$ .

We present the basics of the local theory, which arises from global Rankin-Selberg integrals, attached to pairs of irreducible globally generic cuspidal automorphic representations of the quasi-split unitary group U 2 n + 1 and Res E / F GL m , for a quadratic extension of number fields E / F , when m > n .

Let l be a rational prime number. Assuming the Gross-Kuz'min conjecture along a Z l -extension K ∞ of a number field K, we show that there exist integers μ ˜ , λ ˜ and ν ˜ such that the exponent e ˜ n of the order l e ˜ n of the logarithmic class group C l ˜ n for the n-th layer K n of K ∞ is given by e ˜ n = μ ˜ l n + λ ˜ n + ν ˜ , for n big enough. We show some relations between the classical invariants μ and λ, and their logarithmic counterparts μ ˜ and λ ˜ for some class of Z l -extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case.

Let [Formula: see text] be a number field, and let [Formula: see text] be elements of [Formula: see text] which generate a subgroup of [Formula: see text] of rank [Formula: see text]. Consider the cyclotomic-Kummer extensions of [Formula: see text] given by [Formula: see text], where [Formula: see text] divides [Formula: see text] for all [Formula: see text]. There is an integer [Formula: see text] such that these extensions have maximal degree over [Formula: see text], where [Formula: see text] and [Formula: see text]. We prove that the constant [Formula: see text] is computable. This result reduces to finitely many cases the computation of the degrees of the extensions [Formula: see text] over [Formula: see text].

We classify all the number fields with signature (4,2), (6,1), (1,4) and (3,3) having discriminant lower than a specific upper bound. This completes the search for minimum discriminants for fields of degree 8 and continues it in the degree 9 case. We recall the theoretical tools and the algorithmic steps upon which our procedure is based, then we focus on the novelties due to a new implementation of this process on the computer algebra system PARI/GP; finally, we make some remarks about the final results, among which the existence of a number field with signature $(3,3)$ and small discriminant which was not previously known.

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke $L$-functions of imaginary quadratic number fields of class number one. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27... \%$ of the $L$-functions under consideration do not vanish at $1/2$., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1710.04909

The hull of a linear code over finite fields, the intersection of the code and its dual, has been of interest and extensively studied due to its wide applications. For example, it plays a vital role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and for computing the automorphism group of a linear code. People are interested in pursuing linear codes with small hulls since, for such codes, the aforementioned algorithms are very efficient. In this field, Carlet, Mesnager, Tang and Qi gave a systematic characterization of LCD codes, i.e, linear codes with null hull. In 2019, Carlet, Li and Mesnager presented some constructions of linear codes with small hulls. In the same year, Li and Zeng derived some constructions of linear codes with one-dimensional hull by using some specific Gaussian sums. In this paper, we use general Gaussian sums to construct linear codes with one-dimensional hull by utilizing number fields, which generalizes some results of Li and Zeng (IEEE Trans. Inf. Theory 65(3), 1668–1676, 2019) and also of those presented by Carlet et al. (Des. Codes Cryptogr. 87(12), 3063–3075, 2019). We give sufficient conditions to obtain such codes. Notably, some codes we obtained are optimal or almost optimal according to the Database. This is the first attempt on constructing linear codes by general Gaussian sums which have one-dimensional hull and are optimal. Moreover, we also develop a bound of on the minimum distances of linear codes we constructed.

Let K be a number field of degree k ≥ 8 . In this article, we investigate a certain density theorem for zeros of the Dedekind zeta function attached to K. In this context, our theorem strengthens a result of Heath-Brown. Further, we investigate an upper-bound of the error term for an ideal counting function attached to K. When K is a cyclotomic field, we prove a stronger upper-bound for this error term.

We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Manin's conjecture for all nonsplit quartic del Pezzo surfaces of type $\mathbf A_3+\mathbf A_1$ over arbitrary number fields. The counting problem on the split torsor is solved in the framework of o-minimal structures.

We study the phase transition of KMS states for the C*-algebras of $ax+b$-semigroups of algebraic integers in which the multiplicative part is restricted to a congruence monoid, as in recent work of Bruce generalizing earlier work of Cuntz, Deninger, and Laca. Here we realize the extremal low-temperature KMS states as generalized Gibbs states by constructing concrete representations induced from extremal traces of certain group C*-algebras. We use these representations to compute the Murray--von Neumann type of extremal KMS states and we determine explicit partition functions for the type I factor states. The collection of partition functions that arise this way is an invariant under $\mathbb{R}$-equivariant isomorphism of C*-dynamical systems, which produces further invariants through the analysis of the topological structure of the KMS state space. As an application we characterize several features of the underlying number field and congruence monoid in terms of these invariants. In most cases our systems have infinitely many type I factor KMS states and at least one type II factor KMS state at the same inverse temperature and there are infinitely many partition functions. In order to deal with this multiplicity, we establish, in the context of general C*-dynamical systems, a precise way to associate partition functions to extremal KMS states that are of type I, and we then show that for our systems these partition functions depend only on connected components in the KMS simplex. The discussion of partition functions of general C*-dynamical systems may be of interest by itself and is likely to have applications in other contexts, so we include it in a self-contained initial section that is partly expository and is independent of the number-theoretic background and of the technical results about congruence monoids., Comment: final version, accepted for publication in Comm. Math. Phys.; 38 pages

In this paper, we use the regularized quaternionic Siegel–Weil formula to deal with the $$\mathrm {PGSp}_{1,1}({\mathbb {A}})$$ -period problem for a cuspidal representation $$\tau $$ of $$\mathrm {PGSp}_4({\mathbb {A}}_E)$$ , where E/F is a quadratic extension of number fields and E is contained in the 4-dimensional quaternion division algebra D of F with $$s_D=2$$ .

We prove the Artin conjecture asserting the finiteness of the Brauer group of an arithmetic model of a smooth projective variety V over a number field k under the assumptions that V admits a k-embedding V ↪ W to a smooth projective hyperkahler ksubvariety $$ W\to {\mathbb{P}}_k^N $$ such that V = H1 ∩ … ∩ Hr ∩ W in the sense of the theory of schemes for some k-hypersurfaces H1, . . .,Hr in the space $$ {\mathbb{P}}_k^N $$ ; moreover, for any i ≤ r the variety H1 ∩ . . . ∩ Hi ∩ W is smooth, has codimension i in the variety W, and dimkV ≥ 3.

We prove the Grothendieck conjecture on invariant cycles for an Abelian scheme over a smooth connected complex algebraic curve and the Tate conjecture for codimension r algebraic cycles on an Abelian variety over a number field. Bibliography: 18 titles.

We determine the Hilbert 2-class field tower for some quartic number fields k whose 2-class group Ck,2 is isomorphic to ℤ/2ℤ×ℤ/2ℤ.