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It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a finite extension $F$ of $K$ where $p$ is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell-Weil ranks of $E$ over a subextension of the cyclotomic $\mathbb{Z}_p$-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions., Comment: 20 pages

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [Formula: see text]. We prove that if [Formula: see text], then [Formula: see text] is monogenic. But if [Formula: see text] and [Formula: see text], then [Formula: see text] is not monogenic. Some illustrating examples are given.

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is [Formula: see text], but negative if the rank is [Formula: see text]. Moreover, for rank [Formula: see text] Drinfeld modules we classify those cases where the answer is positive.

Let [Formula: see text] be a Galois extension of [Formula: see text]-adic number fields and let [Formula: see text] be a de Rham representation of the absolute Galois group [Formula: see text] of [Formula: see text]. In the case [Formula: see text], the equivariant local [Formula: see text]-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin [Formula: see text]-functions and it can be formulated as the vanishing of a certain element [Formula: see text] in [Formula: see text]; a similar approach can be followed also in the case of unramified twists [Formula: see text] of [Formula: see text] (see [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383; D. Izychev and O. Venjakob, Equivariant epsilon conjecture for 1-dimensional Lubin–Tate groups, J. Théor. Nr. Bordx. 28(2) (2016) 485–521]). One of the main technical difficulties in the computation of [Formula: see text] arises from the so-called cohomological term [Formula: see text], which requires the construction of a bounded complex [Formula: see text] of cohomologically trivial modules which represents [Formula: see text] for a full [Formula: see text]-stable [Formula: see text]-sublattice [Formula: see text] of [Formula: see text]. In this paper, we generalize the construction of [Formula: see text] in Theorem 2 of [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383] to the case of a higher dimensional [Formula: see text].

Let [Formula: see text] be a number field, [Formula: see text] its ring of integers, [Formula: see text] its classgroup and [Formula: see text] the class number of [Formula: see text]. Let [Formula: see text] be a finite group. Let [Formula: see text] be a maximal [Formula: see text]-order in the semi-simple algebra [Formula: see text] containing [Formula: see text], and [Formula: see text] its locally free classgroup. Let [Formula: see text] and [Formula: see text]. We define the set [Formula: see text] of Galois module classes realizable by the [Formula: see text]th power of the different to be the set of classes [Formula: see text] such that there exists a Galois extension [Formula: see text] with Galois group isomorphic to [Formula: see text] ([Formula: see text]-extension), which is tamely ramified, and for which the class of [Formula: see text] is equal to [Formula: see text], where we clarify that if [Formula: see text], where [Formula: see text], [Formula: see text] is the [Formula: see text]th root of the inverse different [Formula: see text] (respectively, the different [Formula: see text]) if [Formula: see text] (respectively, [Formula: see text]) when it exists. Let [Formula: see text] be a prime number and [Formula: see text] be a primitive [Formula: see text]th root of unity. In this article, we suppose that [Formula: see text] is cyclic of order [Formula: see text] and [Formula: see text] and [Formula: see text] are linearly disjoint. We prove, sometimes under an assumption on [Formula: see text], that [Formula: see text] is a subgroup of [Formula: see text], by an explicit description using a Stickelberger ideal. In addition, for each [Formula: see text], we determine the set of the Steinitz classes of [Formula: see text], [Formula: see text] runs through the tame [Formula: see text]-extensions of [Formula: see text], and prove that it is a subgroup of [Formula: see text], also sometimes under an hypothesis on [Formula: see text].

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

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.

Given a pair of distinct unitary cuspidal automorphic representations for GL(n) over a number field, let S denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues differ. In this note, we demonstrate how conjectures on the automorphy and possible cuspidality of adjoint lifts and Rankin-Selberg products imply lower bounds on the size of S. We also obtain further results for GL(3)., Comment: To appear in IJNT

For an algebraic number field [Formula: see text] and a prime number [Formula: see text], let [Formula: see text] be the maximal multiple [Formula: see text]-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-[Formula: see text] extension of [Formula: see text] is pseudo-null over the completed group ring [Formula: see text]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.

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].

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text]. Suppose that [Formula: see text] has complex multiplication over [Formula: see text], i.e. [Formula: see text] is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number.

The computation of a maximal order of an order in a semisimple algebra over a global field is a classical well-studied problem in algorithmic number theory. In this paper, we consider the related problems of computing all minimal overorders as well as all overorders of a given order. We use techniques from algorithmic representation theory and the theory of minimal integral ring extensions to obtain efficient and practical algorithms, whose implementation is publicly available.

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat’s Last Theorem (FLT) holds for many totally real fields. This result was extended by Deconinck to the generalized Fermat equation of the form [Formula: see text], where [Formula: see text] are odd integers belonging to a totally real field. Later Şengün and Siksek showed that the asymptotic FLT holds over number fields assuming two standard modularity conjectures. In this work, combining their techniques, we show that the generalized Fermat’s Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6.

This paper was motivated by a recent paper by Krumm and Pollack investigating modulo-$p$ behaviour of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the Malle conjecture: our results show in a precise sense the non-existence of "local obstructions" to such heuristics, in many cases essentially only under the assumption that $G$ occurs as the Galois group of a Galois cover defined over $\mathbb{Q}$. This complements and generalizes a similar result in the direction of the Malle conjecture by D\`ebes., Comment: Theorem 3.1 had been distorted due to several typos/omissions. This has been fixed

Given a non-isotrivial elliptic curve over [Formula: see text] with large Mordell–Weil rank, we explain how one can build, for suitable small primes [Formula: see text], infinitely many fields of degree [Formula: see text] whose ideal class group has a large [Formula: see text]-torsion subgroup. As an example, we show the existence of infinitely many cubic fields whose ideal class group contains a subgroup isomorphic to [Formula: see text].

We study the behavior under twisting of the Selmer rank parities of a self-dual prime-degree isogeny on a principally polarized abelian variety defined over a number field, subject to compatibility relations between the twists and the isogeny. In particular, we study isogenies on abelian varieties whose Selmer rank parities are related to the rank parities of elliptic curves with $j$-invariant 0 or 1728, assuming the Shafarevich-Tate conjecture. Using these results, we show how to classify the conjectural rank parities of all quartic or sextic twists of an elliptic curve defined over a number field, after a finite calculation. This generalizes previous results of Hadian and Weidner on the behavior of $p$-Selmer ranks under $p$-twists., 20 pages, 2 figures

Let [Formula: see text] be a square-free positive integer, [Formula: see text] if [Formula: see text] and [Formula: see text] otherwise. Let [Formula: see text] and [Formula: see text] be integers, where [Formula: see text] is a prime. Suppose that [Formula: see text] for some integer [Formula: see text]. Suppose that there exist integers [Formula: see text] and [Formula: see text] such that [Formula: see text]. We prove that if [Formula: see text] is [Formula: see text] or a prime for all integers [Formula: see text] with [Formula: see text], then the class number of the field [Formula: see text] is [Formula: see text].

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if [Formula: see text] is a finitely generated and torsion-free multiplicative subgroup of a number field [Formula: see text] having rank [Formula: see text], then the ratio between [Formula: see text] and the Kummer degree [Formula: see text] is bounded independently of [Formula: see text]. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

In this paper, we investigate the cyclicity of the [Formula: see text]-class group of the first Hilbert [Formula: see text]-class field of some quadratic number field whose discriminant is not a sum of two squares. For this, let [Formula: see text] be different prime integers. Put [Formula: see text], and denote by [Formula: see text] its [Formula: see text]-class group and by [Formula: see text] (respectively [Formula: see text]) its first (respectively second) Hilbert [Formula: see text]-class field. Then, we are interested in studying the metacyclicity of [Formula: see text] and the cyclicity of [Formula: see text] whenever the [Formula: see text]-rank of [Formula: see text] is [Formula: see text].

For an algebraic number field [Formula: see text], let [Formula: see text] denote the discriminant of an algebraic number field [Formula: see text]. It is well known that if [Formula: see text] are algebraic number fields with coprime discriminants, then [Formula: see text] are linearly disjoint over the field [Formula: see text] of rational numbers and [Formula: see text], [Formula: see text] being the degree of [Formula: see text] over [Formula: see text]. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields [Formula: see text] linearly disjoint over [Formula: see text].

Over the ring of algebraic integers [Formula: see text] of a number field [Formula: see text], the quadratic congruence [Formula: see text] modulo a nonzero ideal [Formula: see text] is considered. We prove explicit formulas for [Formula: see text] and [Formula: see text], the number of incongruent solutions [Formula: see text] and the number of incongruent solutions [Formula: see text] with [Formula: see text] coprime to [Formula: see text], respectively. If [Formula: see text] is contained in a prime ideal [Formula: see text] containing the rational prime [Formula: see text], it is assumed that [Formula: see text] is unramified over [Formula: see text]. Moreover, some interesting identities for exponential sums are proved.

Let [Formula: see text] be a number field of degree [Formula: see text] over [Formula: see text] and [Formula: see text] its ring of integers. For a prime number [Formula: see text], we determine the types of splittings of [Formula: see text] in [Formula: see text] for which the set [Formula: see text] is of cardinality a power of [Formula: see text]. We prove that this necessary condition is also sufficient for [Formula: see text] to be a subgroup of the additive group [Formula: see text]. Consequently, we show that, in this case, the subset of [Formula: see text], [Formula: see text] is an order of the number field.

In this paper, we consider number fields containing quadratic subfields with minimal index that is large relative to the discriminant of the number field. We give new upper bounds on the minimal index, and construct families with the largest possible minimal index.

Let [Formula: see text]. We study the [Formula: see text]-part of the narrow class group in thin families of quadratic number fields of the form [Formula: see text], where [Formula: see text] are prime numbers, and we prove new lower bounds for the proportion of narrow class groups in these families that have an element of order [Formula: see text]. In the course of our proof, we prove a general double-oscillation estimate for the quadratic residue symbol in quadratic number fields.

We establish new cases of quadratic number fields [Formula: see text] unramified away from a prime [Formula: see text] and [Formula: see text] whose absolute Galois group has no irreducible two-dimensional continuous Galois representations in [Formula: see text]. Our work builds on methods of Moon–Taguchi and Şengün and the usual analytic techniques of Odlyzko and Poitou where we note one of the new conditional cases arises via a correction of Poitou’s estimate. The results here seem optimal in that it seems these methods alone will yield no further cases either due to prohibitive computational issues or a failure of the analytic obstructions.

We give a list of [Formula: see text] number fields for [Formula: see text] which have rational companion forms. Our list has 53 fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group.

We prove Cadoret and Tamagawa’s open image theorem for curves defined over number fields using their arguments and the machinery of Ellenberg–Hall–Kowalski employed in their paper on expander graphs.

The geometric embedding of an ideal in the algebraic integer ring of some number field is called an ideal lattice. Ideal lattices and the shortest vector problem (SVP) are at the core of many recent developments in lattice-based cryptography. We utilize the matrix of the linear transformation that relates two commonly used geometric embeddings to provide novel results concerning the equivalence of the SVP in these ideal lattices arising from rings of cyclotomic integers.

Resume Soient p un nombre premier impair et K un corps de nombres. Soient K∞ une ℤp-extension de K et L∞ une extension finie de K∞, abelienne sur K. Le but de cet article est d’ameliorer les resultats bien connus, e.g. [16], sur la comparaison de l’ideal caracteristique du χ-quotient de la limite projective du groupe de classes d’ideaux et celui du χ-quotient de la limite projective des unites modulo les unites de Stark, en traitant le cas non semi-simple, pour certains caracteres ℚ¯p-irreductibles χ de Gal(L∞/K∞). La preuve utilise essentiellement la theorie des systemes d’Euler. Abstract Let K be a number field and let p be an odd rational prime. Let K∞ be a ℤp-extension of K and let L∞ be a finite extension of K∞, abelian over K. In this paper we extend the classical results, e.g. [16], relating characteristic ideal of the χ-quotient of the projective limit of the ideal class groups to the χ-quotient of the projective limit of units modulo Stark units, in the non-semi-simple case, for some ℚp¯-irreductible characters χ of Gal(L∞/K∞). The proof essentially uses the theory of Euler systems.

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and let $G$ be a finite abelian group of odd order. Given a $G$-Galois $K$-algebra $K_h$, let $A_h$ denote its square root of the inverse different, which exists by Hilbert's formula. If $K_h/K$ is weakly ramified, then the pair $(A_h,Tr_h)$ is locally $G$-isometric to $(\mathcal{O}_KG,t_K)$ and hence defines a class in the unitary class group $\mbox{UCl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. Here $Tr_h$ denotes the trace of $K_h/K$ and $t_K$ the symmetric bilinear form on $\mathcal{O}_KG$ for which $t_K(s,t)=\delta_{st}$ for all $s,t\in G$. We study the collection of all such classes and show that a subset of them is in fact a subgroup of $\mbox{UCl}(\mathcal{O}_KG)$., Comment: Version 2; fixed a few errors and added more details in throughout the paper

Let [Formula: see text] be a number field, let [Formula: see text] denote a fixed rational prime. We study the local behavior of Iwasawa’s invariants as functions on the set [Formula: see text] of all [Formula: see text]-extensions of [Formula: see text]. With respect to a certain topology on [Formula: see text] that takes care of ramification, we prove that for each [Formula: see text] the [Formula: see text]-invariant of [Formula: see text] is locally maximal among the [Formula: see text]-invariants, and we give sufficient conditions for the [Formula: see text]-invariant to be locally maximal (e.g., a vanishing [Formula: see text]-invariant). This concerns a question raised by R. Greenberg in 1973. Our main result also provides information about [Formula: see text]- (and even [Formula: see text]-) invariants in the case of a nonvanishing [Formula: see text]-invariant. The main tool used in the proof is a new result based on the stabilization of certain ranks, which considerably generalizes a theorem of T. Fukuda.

Fix a number field $k$ and a rational prime $\ell$. We consider abelian varieties whose $\ell$-power torsion generates a pro-$\ell$ extension of $k(\mu_{\ell^\infty})$ which is unramified away from $\ell$. It is a necessary, but not generally sufficient, condition that such varieties have good reduction away from $\ell$. In the special case of $\ell = 2$, we demonstrate that for abelian surfaces $A/\mathbb{Q}$, good reduction away from $\ell$ does suffice. The result is extended to elliptic curves and abelian surfaces over certain number fields unramified away from $\{2,\infty\}$. An explicit example is constructed to demonstrate that good reduction is not sufficient, at $\ell = 2$, for abelian varieties of sufficiently high dimension., Comment: 9 pages, 1 table

Let [Formula: see text] be a prime number. For a number field [Formula: see text], let [Formula: see text] be the compositum of all [Formula: see text]-extensions of [Formula: see text]. Then Greenberg’s generalized conjecture (GGC) claims that the unramified Iwasawa module [Formula: see text] is pseudo-null over the Iwasawa algebra associated to the Galois group of [Formula: see text]. In this paper, we establish sufficient conditions of GGC when [Formula: see text] is a complex cubic field and give many examples which satisfy the conditions with the help of computer programs.

We prove the Hasse principle and weak approximation for varieties defined by the smooth complete intersection of three quadratics in at least 19 variables, over arbitrary number fields.

The function [Formula: see text] for a number field is analogous to the dimension of the Riemann–Roch spaces at divisors on an algebraic curve. We provide a method to compute this function for number fields with unit group of rank at most 2, even with large discriminant. This method is based on using LLL-reduced bases, the “jump algorithm” and Poisson summation formula.

Given a quintic number field K/ℚ, we study the set of irreducible trinomials, polynomials of the form x5 + ax + b, that have a root in K. We show that there is a genus 4 curve CK whose rational points are in bijection with such trinomials. This curve CK maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on CK using elliptic curve Chabauty.

Consider a non-split one-dimensional torus defined over a number field [Formula: see text]. For a finitely generated group [Formula: see text] of rational points and for a prime number [Formula: see text], we investigate for how many primes [Formula: see text] of [Formula: see text] the size of the reduction of [Formula: see text] modulo [Formula: see text] is coprime to [Formula: see text]. We provide closed formulas for the corresponding Dirichlet density in terms of finitely many computable parameters. To achieve this, we determine in general which torsion fields and Kummer extensions contain the splitting field.

Let [Formula: see text] be a number field of a finite degree and let [Formula: see text] be the Rankin–Selberg [Formula: see text]-function associated to unitary cuspidal automorphic representations [Formula: see text] and [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. The main result of the paper is an asymptotic formula for evaluation of coefficients appearing in the Laurent (Taylor) series expansion of the logarithmic derivative of the function [Formula: see text] at [Formula: see text]. As a corollary, we derive orthogonality and weighted orthogonality relations.

Let [Formula: see text] be an algebraic number field with [Formula: see text] in the ring [Formula: see text] of algebraic integers of [Formula: see text] and [Formula: see text] be the minimal polynomial of [Formula: see text] over the field [Formula: see text] of rational numbers. In 1977, Uchida proved that [Formula: see text] if and only if [Formula: see text] does not belong to [Formula: see text] for any maximal ideal [Formula: see text] of the polynomial ring [Formula: see text] (see [Osaka J. Math. 14 (1977) 155–157]). In this paper, we apply the above result to obtain some necessary and sufficient conditions involving the coefficients of [Formula: see text] for [Formula: see text] to equal [Formula: see text] when [Formula: see text] is a trinomial of the type [Formula: see text]. In the particular case when [Formula: see text], it is deduced that [Formula: see text] is an integral basis of [Formula: see text] if and only if either (i) [Formula: see text] and [Formula: see text] or (ii) [Formula: see text] divides [Formula: see text] and [Formula: see text].

Let $K$ be a number field, $\mathfrak{q}$ be an integral ideal, and $\mathrm{Cl}(\mathfrak{q})$ be the associated ray class group. Suppose $\mathrm{Cl}(\mathfrak{q})$ possesses a real exceptional character $\psi$, possibly principal, with a Siegel zero $\beta$. For $\mathcal{C} \in \mathrm{Cl}(\mathfrak{q})$ satisfying $\psi(\mathcal{C}) = 1$, we establish an effective $K$-uniform Linnik-type bound with explicit constants for the least norm of a prime ideal $\mathfrak{p} \in \mathcal{C}$. A special case of this result is related to rational primes represented by certain binary quadratic forms., Comment: 25 pages

Let $O$ be an order in a quadratic number field $K$ with ring of integers $D$, such that the conductor $\mathfrak F = f D$ is a prime ideal of $O$, where $f\in\mathbb Z$ is a prime. We give a complete description of the $\mathfrak F$-primary ideals of $O$. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those $\mathfrak F$-primary ideals not contained in $\mathfrak F^2$. We get three different cases, according to whether the prime number $f$ is split, inert or ramified in $D$., Comment: Keywords: Orders, Conductor, Primary ideal, Lattice of ideals. to appear in Int. J. Number Theory (2016)

In this work, we prove a converse theorem for metaplectic Eisenstein series on the [Formula: see text]th metaplectic cover of the group [Formula: see text], where [Formula: see text] is an imaginary quadratic number field containing the [Formula: see text]th roots of unity. This is an analog to previous converse theorems relating certain double Dirichlet series to the Mellin transforms of Eisenstein series of half-integer weight. We also propose a way to generalize this result to any number field.

Euler showed that there can be no more than three integer squares in arithmetic progression. In quadratic number fields, Xarles has shown that there can be arithmetic progressions of five squares, but not of six. Here, we prove that there are no cubic number fields which contain five squares in arithmetic progression.

For a field of definition [Formula: see text] of an abelian variety [Formula: see text] and prime ideal [Formula: see text] of [Formula: see text] which is of a good reduction for [Formula: see text], the structure of [Formula: see text] as abelian group is: [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. We are interested in finding an asymptotic formula for the number of prime ideals [Formula: see text] with [Formula: see text], [Formula: see text] has a good reduction at [Formula: see text], [Formula: see text]. We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field.

In 1940, Weil [Sur les fonctions algébriques à corps de constantes finis, C. R. Acad. Sci. Paris210 (1940) 592–594] proved the Riemann hypothesis for curves over finite fields. It follows from the Castelnuovo–Severi defect inequality concerning correspondences between algebraic curves (see [A. Mattuck and J. Tate, On the inequality of Castelnuovo–Severi, Abh. Math. Sem. Univ. Hamburg22 (1958) 295–299]). An important step in the proof of Castelnuovo–Severi's defect inequality is the invariance of the Castelnuovo–Severi defect under trivial correspondences, so that the degree of divisors can be modified by adding multiples of trivial correspondences. In the number field case, the Weil distribution Δ(h) (see [A. Weil, Sur les formules explicites de la théorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat.36 (1972) 3–18]) corresponds to the Castelnuovo–Severi defect. Functions of the form ∑ξ∈K* f(ξx) with f in the Schwartz–Bruhat space S(𝔸)0 correspond to trivial correspondences. In this paper, we show that the two terms [Formula: see text] and [Formula: see text] in the Weil distribution can be chosen to be zero by adding "trivial correspondences" to h while keeping the Weil distribution essentially unchanged. As an application of this result, the Weil distribution is expressed as the spectral trace of an operator on a Hilbert space.

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field k with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over k(ζp). In this paper, for p = 5 and 7, we study the ramification of such Kummer extensions using explicit Kummer generators directly computed by Verdure in 2006.

Let k be a global field. Let G be a connected linear algebraic k-group, assumed reductive when k is a function field. It follows from a result of a preprint by Bary-Soroker, Fehm and Petersen that when H is a smooth connected k-subgroup of G, the quotient space G/H is of Hilbert type. We prove a similar result for certain non-connected k-subgroups H of G. In particular, we prove that if G is a simply connected k-group over a number field k, and H is an abelian k-subgroup of G, not necessarily connected, then G/H is of Hilbert type., Comment: 7 pages, final version, to appear in the International Journal of Number Theory

In this paper, we define a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of $\mathbf{Q}_p$, over a Galois extension of $\mathbf{Q}_p$ whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localisation at the primes above p are extensions of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim., Final version, to appear in Int. J. Number Theory