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Corvaja and Zannier conjectured that an abelian variety over a number field satisfies a modified version of the Hilbert property. We investigate their conjecture for products of elliptic curves using Kawamata's structure result for ramified covers of abelian varieties, and Faltings's finiteness theorem for rational points on higher genus curves., Comment: 11 pages. Changed title. Minor corrections and updated bibliography. Final version

Given a number field $K$ and a polynomial $f(z) \in K[z]$ of degree at least 2, one can construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, with an edge $\alpha \to \beta$ if and only if $f(\alpha) = \beta$. Restricting to quadratic polynomials, the dynamical uniform boundedness conjecture of Morton and Silverman suggests that for a given number field $K$, there should only be finitely many isomorphism classes of directed graphs that arise in this way. Poonen has given a conjecturally complete classification of all such directed graphs over $\mathbb{Q}$, while recent work of the author, Faber, and Krumm has provided a detailed study of this question for all quadratic extensions of $\mathbb{Q}$. In this article, we give a conjecturally complete classification like Poonen's, but over the cyclotomic quadratic fields $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. The main tools we use are dynamical modular curves and results concerning quadratic points on curves., Comment: v4 includes a few more details, especially toward the end of Section 3 and in Appendix A. Other minor changes have been made. An additional Magma file (main.txt), containing calculations for the main body of the article, have been added as an ancillary file

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Let $k$ be a number field and let ${\mathcal{A}}$ be a ${\rm GL}_2$-type variety defined over $k$ of dimension $d$. We show that for every prime number $p$ satisfying certain conditions (see Theorem 2), if the local-global divisibility principle by a power of $p$ does not hold for ${\mathcal{A}}$ over $k$, then there exists a cyclic extension $\widetilde{k}$ of $k$ of degree bounded by a constant depending on $d$ such that ${\mathcal{A}}$ is $\widetilde{k}$-isogenous to a ${\rm GL}_2$-type variety defined over $\widetilde{k}$ that admits a $\widetilde{k}$-rational point of order $p$. Moreover, we explain how our result is related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Ciperiani and Stix and Creutz.

The Duffin-Schaeffer theorem is a well-known result from metric number theory, which generalises Khinchin's theorem from monotonic functions to a wider class of approximating functions. In recent years, there has been some interest in proving versions of classical theorems from Diophantine approximation in various generalised settings. In the case of number fields, there has been a version of Khinchin's theorem proven which holds for all number fields, and a version of the Duffin-Schaeffer theorem proven only in imaginary quadratic fields. In this paper, we prove a version of the Duffin-Schaeffer theorem for all number fields.

Given a map $\varphi:\mathbb{P}^1\rightarrow \mathbb{P}^1$ of degree greater than 1 defined over a number field $k$, one can define a map $\varphi_\mathfrak{p}:\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})\rightarrow \mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})$ for each prime $\mathfrak{p}$ of good reduction, induced by reduction modulo $\mathfrak{p}$. It has been shown that for a typical $\varphi$ the proportion of periodic points of $\varphi_\mathfrak{p}$ should tend to $0$ as $|\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})|$ grows. In this paper, we extend previous results to include a weaker set of sufficient conditions under which this property holds. We are also able to show that these conditions are necessary for certain families of functions, for example, functions of the form $\varphi(x)=x^d+c$, where $0$ is not a preperiodic point of this map. We study the proportion of periodic points by looking at the fixed point proportion of the Galois groups of certain extensions associated to iterates of the map., Comment: This article draws heavily from arXiv:1410.3378

Consider an algebraic number field, $K$, and its ring of integers, $\mathcal{O}_K$. There exists a smallest $B_K>1$ such that for any $x>1$ we can find a prime ideal, $\mathfrak{p}$, in $\mathcal{O}_K$ with norm $N(\mathfrak{p})$ in the interval $[x,B_Kx]$. This is a generalization of Bertrand's postulate to number fields, and in this paper we produce bounds on $B_K$ in terms of the invariants of $K$ from an effective prime ideal theorem due to Lagarias and Odlyzko. We also show that a bound on $B_K$ can be obtained from an asymptotic estimate for the number of ideals in $\mathcal{O}_K$ less than $x$.

Jordan, Rotger and de Vera-Piquero proved that Shimura curves have no points rational over imaginary quadratic fields under a certain assumption. In this article, we expand their results to the case of number fields of higher degree. We also give counterexamples to the Hasse principle on Shimura curves., Comment: 8 pages

A commutative order in a quaternion algebra is called selective if it is embeds into some, but not all, the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one indefinite quaternion algebra. Here we prove that the order generated by a cubic root of unity is selective for any definite quaternion algebra over the rationals with a type number 3 or larger. The proof extends to a few other closely related orders.

Let $k$ be a number field and $S$ a finite set of places of $k$ containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of $S$-integers of $k$. Moreover, we give an asymptotic formula for the number of $\bar{S}$-integers of bounded height and fixed degree over $k$, where $\bar{S}$ is the set of places of $\bar{k}$ lying above the ones in $S$., arXiv admin note: text overlap with arXiv:1305.0482, accepted for publication on Acta Arithmetica

Let $X$ be a proper smooth variety having an affine open subset defined by the normic equation $N_{k(\sqrt{a},\sqrt{b})/k}({x})=Q(t_{1},...,t_{m})^{2}$ over a number field $k$. We prove that : (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of $X$; (2) the analogue for rational points is also valid assuming Schinzel's hypothesis., comments are welcome

For a prime p, the well-known Wilson congruence says that (p−1)! ≡ −1 modulo p. A prime p is called a Wilson prime, if the congruence above holds modulo p. We now quote from [R95, Pa. 346, 350]: ‘It is not known whether there are infinitely many Wilson primes. In this respect, Vandiver wrote: This question seems to be of such a character that if I should come to life any time after my death and some mathematician were to tell me it had been definitely settled, I think I would immediately drop dead again.’ He also mentions that search (by Crandall, Dilcher, Pomerance [CDP]) up to 5× 10 produced the only known Wilson primes, namely 5, 13, and 563, which was discovered by Goldberg in 1953 (one of the first successful computer searches involving very large numbers). (See [R95, Dic19] for historical references.) Many strong analogies [Gos96, Ro02, Tha04] between number fields and function fields over finite fields have been used to benefit the study of both. These analogies are even stronger in the base case Q,Z ↔ F (t), F [t], where F is a finite field. We will study the concept of Wilson prime in this function field context, and in contrast to the Z case, we will exhibit infinitely many of them, at least for many F . For example, ℘ = t3∗13 n − t13n − 1 are Wilson primes for F3[t]. We also show strong connections between Wilson’s and Fermat’s quotients; between refined Wilson residues and discriminants, and introduce analogs of Bell numbers in F [t] setting.

In Dokchitser (2007) it is shown that given an elliptic curve $E$ defined over a number field $K$ then there are infinitely many degree 3 extensions $L/K$ for which the rank of $E(L)$ is larger than $E(K)$. In the present paper we show that the same is true if we replace 3 by any prime number. This result follows from a more general result establishing a similar property for the Jacobian varieties associated with curves defined by an equation of the shape $g(y) = f(x)$ where $f$ and $g$ are polynomials of coprime degree.

Let K be an algebraic number field with non-trivial class group G and OK be its ring of integers. For k 2 N and some real x 1, let Fk(x) denote the number of non-zero principal ideals aOK with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that Fk(x) behaves, for x ! 1, asymptotically like x(logx) 1/|G| 1 (loglogx) Nk(G) . In this article, it is proved that for every prime p, N1(Cp Cp) = 2p, and it is also proved that N1(Cmp Cmp) = 2mp if N1(Cm Cm) = 2m and m is large enough. In particular, it is shown that for each positive integer n there is a positive integer m such that N1(Cmn Cmn) = 2mn. Our results confirm a conjecture given by W. Narkiewicz thirty years ago partly and improve the known results substantially.

The problem of expressing an element of K_2(F) in a more explicit form gives rise to many works. To avoid a restrictive condition in a work of Tate, Browkin considered cyclotomic elements as the candidate for the element with an explicit form. In this paper, we modify and change Browkin's conjecture about cyclotomic elements into more precise forms, in particular we introduce the conception of cyclotomic subgroup. In the rational function field cases, we determine completely the exact numbers of cyclotomic elements and cyclotomic subgroups contained in a subgroup generated by finitely many different cyclotomic elements, while in the number field cases, using Faltings' theorem on Mordell conjecture we prove that there exist subgroups generated by an infinite number of cyclotomic elements to the power of some prime, which contain no nontrivial cyclotomic elements.

International audience; The aim of this paper is to apply the work of Morris on Eisenstein series over global function fields to the study of the asymptotic behavior of the points of bounded height on a generalized flag variety defined as the quotient of a semi-simple algebraic group by a reduced parabolic subgroup over such a field. The formula obtained for the height zeta function has an interpretation similar to the one known over a number field.

Let K be a number field, let S be a finite set of places of K containing the archimedean places and let μ, α1, α2, α3 be non–zero elements in K. Denote by OS the ring of S–integers in K and by O × S the group of S–units. Then the set of equivalence classes (namely, up to multiplication by S–units) of the solutions (x, y, z, e1, e2, e3, e) ∈ O S × (O× S ) 4 of the diophantine equation (X − α1E1Y )(X − α2E2Y )(X − α3E3Y )Z = μE, satisfying Card{α1e1, α2e2, α3e3} = 3, is finite. With the help of this last result, we exhibit, for every integer n > 2, new families of Thue-Mahler equations of degreee n having only trivial solutions. Furthermore, we produce an effective upper bound for the number of these solutions. The proofs of this paper rest heavily on Schmidt’s subspace theorem. Resume Soit K un corps de nombres, soit S un ensemble fini de places de K contenant les places archimediennes et soient μ, α1, α2, α3 des elements non nuls de K. Notons OS l’anneau des S– entiers de K et O× S le groupe des S–unites. Alors l’ensemble des classes d’equivalence (c’est-a-dire, a multiplication pres par des S–unites) des solutions (x, y, z, e1, e2, e3, e) ∈ O S × (O× S ) 4 de l’equation diophantienne (X − α1E1Y )(X − α2E2Y )(X − α3E3Y )Z = μE, verifiant Card{α1e1, α2e2, α3e3} = 3, est fini. Grâce a ce dernier resultat, nous pouvons exhiber pour chaque entier n ≥ 3 de nouvelles familles d’equations de Thue-Mahler de degre n ne possedant que des solutions triviales. De plus, nous donnons une borne superieure explicite pour le nombre de ces solutions. Les demonstrations de cet article reposent sur le theoreme du sous–espace de Schmidt. 2010 Mathematics Subject Classification: Primary 11D59 ; Secondary 11D45 11D61 11D25 11J87

Let π be an irreducible unitary cuspidal representation of GLd(QA). Let L(π, s) be the L-function attached to π. For real k ≥ 0 let Ik(π, T ) = Z T 1 |L (π, 1/2 + it)| dt be the k-th (power) moment of L(π, s). Let απ(p, j) ∈ C (1 ≤ j ≤ d) be the local parameters at prime p. We prove that if |απ(p, j)| ≤ p for unramified primes and for a fixed 0 ≤ γ < 1/4, then Ik(π, T ) T (log T ) 2 for any rational k ≥ 0. As a corollary of this result we establish unconditional lower bounds of the conjectured order of magnitude for the fractional k-th moments of Dirichlet L-functions, modular L-functions, twisted modular L-functions, and Maass L-functions. We derive our results as corollaries of more general theorems related to the lower bounds for fractional moments of analytic functions which have Dirichlet series representations on a complex half-plane. We also establish lower bounds for the k-th moments of Artin L-functions and Dedekind zeta functions of number fields.

Let p be an odd prime and let L/k be a Galois extension of number fields whose Galois group is isomorphic to the dihedral group of order 2p. Let S be a finite set of primes of L which is stable under the action of Gal(L/k). The Lichtenbaum conjecture on special values of the Dedekind zeta function at negative integers, together with Brauer formalism for Artin's L-functions, gives a (conjectural) formula relating orders of motivic cohomology groups of rings of S-integers and higher regulators of S-units of the subextensions of L/k. In analogy with the classical case of special values at 0, we give an algebraic proof of this formula, i.e. without using the Lichtenbaum conjecture nor Brauer formalism. Our method also gives an interpretation of the regulator term as a higher unit index.

Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x from X to the projective line over K and a place v of K. We introduce a local canonical height on the set of K_v-valued points of X associated to x as an integral with logarithmic integrand, generalizing Tate's local Neron function on an elliptic curve. The resulting global height can be viewed as a 'Mahler measure' associated to x. We prove that the local canonical height can be obtained by averaging, and taking a limit, over divisors of higher order Weierstrass points on X. This generalizes previous results by Everest-ni Fhlathuin and Szpiro-Tucker. Our construction of the local canonical height is an application of potential theory on Berkovich curves in the presence of a canonical measure., Comment: 18 pages

Let E be an elliptic curve defined over an abelian number field K of degree m. For a prime ideal p of OK of good reduction we consider E over the finite field OK/p and let ap(E) be the trace of the Frobenius morphism. If E does not have complex multiplication, a generalization of the Lang-Trotter Conjecture asserts that given r, f ∈ Z with f > 0 and f | m, there exists a constant CE,r,f ≥ 0 such that #{p : N(p) ≤ x, degK(p) = f and ap(E) = r} ∼ CE,r,f ·  √ x log x if f = 1, log log x if f = 2, 1 if f ≥ 3. We prove that this conjecture holds on average in certain families of elliptic curves defined over K.

We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the Brauer-Manin obstruction is not the only obstruction to the Hasse principle for K3 surfaces.

Let F be a Galois extension of a number field k with Galois group G. The Brauer-Kuroda theorem gives an expression of the Dedekind zeta function of the field F as a product of the zeta functions of some of its subfields containing k, provided the group G is not exceptional. In this paper, we investigate the exceptional groups. In particular, we determine all nilpotent exceptional groups and give a sufficient condition for a group to be exceptional. We give many examples of nonnilpotent solvable and nonsolvable exceptional groups.

Let k be a number field, let A be a commutative algebraic group defined over k and let n be a positive integer. We will prove that for almost all primes p ∈ k, the existence of a counterexample to the localglobal divisibility by p in A assures the existence of a counterexample to the local-global divisibility by p in A, for all positive integers s. In particular, we will prove the existence of counterexamples to the localglobal divisibility by 2 and 3, for every n ≥ 2, in the case when k = Q and A is an elliptic curve. Furthermore, we will show a method to find numerical examples. We will use this method to produce numerical counterexamples to the local-global divisibility by 2, for every n ≥ 2, in elliptic curves defined over Q.

In 2007, Jarden and Narkiewicz raised the following question: Is it true that each algebraic number field has a finite extension L such that the ring of integers of L is generated by its units (as a ring)? In this article, we answer the analogous question in the function field case. More precisely, it is shown that for every finite non-empty set S of places of an algebraic function field F | K over a perfect field K, there exists a finite extension F' | F, such that the integral closure of the ring of S-integers of F in F' is generated by its units (as a ring)., Comment: 12 pages

International audience; Let ε be a quartic algebraic unit. We give necessary and sufficient conditions for (i) the quartic number field K=ℚ(ε) to contain an imaginary quadratic subfield, and (ii) for the ring of algebraic integers of K to be equal to ℤ[ε]. We also prove that the class number of such K's goes to infinity effectively with the discriminant of K.

We define a group stucture on the primitive integer points (A,B,C) of the algebraic variety Q_0(B,C)=A^n, where Q_0 is the principal binary quadratic form of fundamental discriminant \Delta and n is a fixed integer greater than 1. A surjective homomorphism is given from this group to the $n$-torsion subgroup of the narrow ideal class group of the quadratic number field Q(\sqrt{\Delta}).

Öz bulunamadı.

In this paper, Thue's Fundamentaltheorem is analysed. We show that it includes, and often strengthens, known effective irrationality measures obtained via the so-called hypergeometric method as well as showing that it can be applied to previously unconsidered families of algebraic numbers. Furthermore, we extend the method to also cover approximation by algebraic numbers in imaginary quadratic number fields., Comment: accepted version (Acta Arithmetica)

We prove a new effective result for equations a1x1+a2x2 = 1 in (x1, x2) ∈ Γ, where a1, a2 are non-zero algebraic numbers and Γ is a finitely generated multiplicative subgroup of (Q)2. In our result, we give an explicit upper bound in terms of a1, a2 and Γ for the heights of the solutions (x1, x2). More generally, we prove effective results for solutions (x1, x2) in the division group of Γ and for solutions ‘close’ to this division group. Here, the solutions do not lie anymore in a given number field. Therefore, to achieve effectiveness we give for each solution (x1, x2), explicit upper bounds both for its height and for the degree of the number field it generates.