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We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely D N (t) : = D f , N (t) = 1 N ∑ n ⩽ N f (n) n i t , where f is a completely multiplicative function with | f (n) | = 1 for all n ∈ N. We use Soundararajan's resonance method to produce large values of | D N (t) | uniformly for all such f. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds and only for almost all such f. [ABSTRACT FROM AUTHOR]

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For example, we show for odd k ≤ 10 5 and even k ≤ 2 ⋅ 10 5 that any even integer n ≥ 40 can be expressed as the sum of a prime and a square-free number coprime to k. We also discuss applications to other Goldbach-like problems. [ABSTRACT FROM AUTHOR]

Given two non-CM newforms with integral Fourier coefficients, in this paper we study the number of distinct prime divisors of their Fourier coefficients in a probability way. Based on a multivariate version of the Erdős-Kac theorem, using the Galois representations attached to newforms and the effective Chebotarev density theorem, and assuming the generalized Riemann hypothesis, we show that the distribution of the number of distinct primes dividing the Fourier coefficients behaves like the standard multivariate normal distribution if these newforms are not twists of each other. As a consequence, we prove a multiplicity one result for modular forms under the generalized Riemann hypothesis. [ABSTRACT FROM AUTHOR]

The main theorem of the paper [2] requires a restriction on the prime p. This statement is being modified based on a certain conjecture on the p -adic zeta functions. [ABSTRACT FROM AUTHOR]

In this work we develop a framework that enables the use of Shimura curve parametrizations of elliptic curves to approach the abc conjecture, leading to a number of new unconditional applications over Q and, more generally, totally real number fields. Several results of independent interest are obtained along the way, such as bounds for the Manin constant, a study of the congruence number, extensions of the Ribet-Takahashi formula, and lower bounds for the L 2 -norm of integral quaternionic modular forms. The methods require a number of tools from Arakelov geometry, analytic number theory, Galois representations, complex-analytic estimates on Shimura curves, automorphic forms, known cases of the Colmez conjecture, and results on generalized Fermat equations. [ABSTRACT FROM AUTHOR]

We point out an error in [2]. [ABSTRACT FROM AUTHOR]

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by x over which there exist elliptic curves with good reduction everywhere and rational j -invariant is ≫ x log − 1 2 ⁡ (x). In this paper, we assume the abc -conjecture to show the sharp asymptotic ∼ c x log − 1 2 ⁡ (x) for this number, obtaining formulae for c in both the real and imaginary cases. Our method has three ingredients: (1) We make progress towards a conjecture of Granville: Given a fixed elliptic curve E / Q with short Weierstrass equation y 2 = f (x) for reducible f ∈ Z [ x ] , we show that the number of integers d , | d | ≤ D , for which the quadratic twist d y 2 = f (x) has an integral non-2-torsion point is at most D 2 3 + o (1) , assuming the abc -conjecture. (2) We apply the Selberg–Delange method to obtain a Tauberian theorem which allows us to count integers satisfying certain congruences while also being divisible only by certain primes. (3) We show that for a polynomially sparse subset of the natural numbers, the number of pairs of elements with least common multiple at most x is O (x 1 − ϵ) for some ϵ > 0. We also exhibit a matching lower bound. If instead of the abc -conjecture we assume a particular tail bound, we can prove all the aforementioned results and that the coefficient c above is greater in the real quadratic case than in the imaginary quadratic case, in agreement with an experimentally observed bias. [ABSTRACT FROM AUTHOR]

The property of Sheats compositions stated in Corollary 4.7 is false. As it enters into the proof of Theorem 5.7, one of the main results, we show that this proof is in fact independent of Corollary 4.7 and its derivatives Corollaries 4.9 and 4.10. Some other less important inaccuracies and typos are corrected. [ABSTRACT FROM AUTHOR]

Let p be a prime, and let f ∈ Z [ x ] be a non-constant polynomial whose leading coefficient is prime to p and that f has no repeated root modulo p. Suppose the mod- p irreducible factors of f all have the same degree. Then there are infinitely many Abelian extensions of number fields L / K such that O L / p O L ≃ (Z / p) [ x ] / f , where O L denotes the ring of integers of L and p ⊂ O K is a maximal ideal with O K / p O K ≃ Z / p. Moreover, under the generalized Riemann hypothesis for the Dedekind zeta functions of number fields, for fixed deg ⁡ (f) we can construct Abelian lifts in polynomial time in log ⁡ p. This confirms in a stronger form a question raised by Mihăilescu and Vuletescu concerning polynomial cyclic algebras. The proofs make use of a weak form of Artin's conjecture on primitive roots and polynomial time polynomial factoring algorithms over number fields. [ABSTRACT FROM AUTHOR]

In this paper, we compute the q -adic slopes of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and Wan's decomposition theorems. This paper is a generalization of our previous paper. [ABSTRACT FROM AUTHOR]

The Katz-Sarnak Density Conjecture states that on a large scale, central point vanishing of suitably indexed L -functions is well-modeled by eigenvalues of random matrices drawn from the classical compact groups. In particular, over suitably restricted eigenvalues, an average order vanishing can bounded using only a chosen test function ϕ. For each type of symmetry, we find the optimal ϕ , and we give very explicit descriptions for small support. We compute the optimal bound on average rank, an improvement on the bounds in the existing literature. In general, we show that, for a given support, the optimal test function lies in a finite-dimensional family. We prove this using a new method of solving certain integral equations on finite intervals. We differentiate under the integral sign and solve the resulting system of delay differential equations using a priori symmetries of the solution. This method may prove useful in a variety of applications. [ABSTRACT FROM AUTHOR]

We show that, on the Riemann hypothesis, lim sup X → ∞ I (X) / X 2 ⩽ 0.8603 , where I (X) = ∫ X 2 X (ψ (x) − x) 2 d x. This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that 1.86 ⋅ 10 − 4 ⩽ I (X) / X 2 for sufficiently large X , and that the I (X) / X 2 has no limit as X → ∞. [ABSTRACT FROM AUTHOR]

• Integer lattice. • Discrepancy. • Annuli. • Gauss circle problem. Given a domain Ω ⊆ R 2 , let D (Ω , x , R) be the number of lattice points from Z 2 in R Ω − x , for R ≥ 1 and x : = (x 1 , x 2) ∈ T 2 , minus the area of R Ω: D (Ω , x , R) = # { (j , k) ∈ Z 2 : (j − x 1 , k − x 2) ∈ R Ω } − R 2 | Ω |. We call ∫ T 2 | D (Ω , x , R) | p d x the p -th moment of the discrepancy function D. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of D is bounded by O (R 2 log ⁡ R) , and in 2019, Colzani, Gariboldi, and Gigante extended this result to higher dimensions. In this paper, our contribution is twofold: first, we present a simple direct proof of Huxley's 2014 result; second, we establish new estimates for the p -th moments of lattice point discrepancy of annuli of radius R , and any fixed thickness 0 < t < 1 for p ≥ 2. For a video summary of this paper, please visit https://youtu.be/YWIe1IBIi9Q. [ABSTRACT FROM AUTHOR]

We define a new metric between natural numbers induced by the ℓ ∞ norm of their unique prime signatures. In this space, we look at the natural analog of the number line and study the arithmetic function L ∞ (N) , which tabulates the cumulative sum of distances between consecutive natural numbers up to N in this new metric. Our main result is to identify the positive and finite limit of the sequence L ∞ (N) / N as the expectation of a certain random variable. The main technical contribution is to show with elementary probability that for K = 1 , 2 or 3 and ω 0 , ... , ω K ≥ 2 the following asymptotic density holds lim n → ∞ ⁡ | { M ≤ n : ‖ M − j ‖ ∞ < ω j for j = 0 , ... , K } | n = ∏ p : prime (1 − ∑ j = 0 K 1 p ω j ). This is a generalization of the formula for k -free numbers, i.e. when ω 0 = ... = ω K = k. The random variable is derived from the joint distribution when K = 1. As an application, we obtain a modified version of the prime number theorem. Our computations up to N = 10 12 have also revealed that prime gaps show a considerably richer structure than on the traditional number line. Moreover, we raise additional open problems, which could be of independent interest. [ABSTRACT FROM AUTHOR]

We study Tamagawa numbers of elliptic curves with torsion Z / 2 Z ⊕ Z / 14 Z over cubic fields and of elliptic curves with an n -isogeny over Q , for n ∈ { 6 , 8 , 10 , 12 , 14 , 16 , 17 , 18 , 19 , 37 , 43 , 67 , 163 }. Bruin and Najman [3] proved that every elliptic curve with torsion Z / 2 Z ⊕ Z / 14 Z over a cubic field is a base change of an elliptic curve defined over Q. We find that Tamagawa numbers of elliptic curves defined over Q with torsion Z / 2 Z ⊕ Z / 14 Z over a cubic field are always divisible by 142, with each factor 14 coming from a rational prime with split multiplicative reduction of type I 14 k , one of which is always p = 2. The only exception is the curve 1922.e2 , with c E = c 2 = 14. The same curves defined over cubic fields over which they have torsion subgroup Z / 2 Z ⊕ Z / 14 Z turn out to have the Tamagawa number divisible by 143. As for n -isogenies, Tamagawa numbers of elliptic curves with an 18-isogeny must be divisible by 4, while elliptic curves with an n -isogeny for the remaining n from the mentioned set must have Tamagawa numbers divisible by 2, except for finite sets of specified curves. [ABSTRACT FROM AUTHOR]

We correct an error found in Section 3.4.1 of Linear operators, the Hurwitz zeta function and Dirichlet L-functions published in JNT 217 (2020) 422–442. The error is related to the convergence of the inverse operator G − 1 defined in Section 3.3 and affects the statement and proof of Proposition 17. We provide a revised statement and proof. [ABSTRACT FROM AUTHOR]

The Birch and Swinnerton-Dyer conjecture – which is one of the seven million-dollar Clay Mathematics Institute Millennium Prize Problems – and its generalizations are a significant focus of number theory research. A 2017 article of Jetchev, Skinner and Wan proved a Birch and Swinnerton-Dyer formula at a prime p for certain rational elliptic curves of rank 1. We generalize and adapt that article's arguments to prove an analogous formula for certain modular forms. For newforms f of even weight higher than 2 with Galois representation V containing a Galois-stable lattice T , let W = V / T and let K be an imaginary quadratic field in which the prime p splits. Our main result is that under some conditions, the p -index of the size of the Shafarevich-Tate group of W with respect to the Galois group of K is twice the p -index of a logarithm of the Abel-Jacobi map of a Heegner cycle defined by Bertolini, Darmon and Prasanna. Significant original adaptations we make to the 2017 arguments are (1) a generalized version of a previous calculation of the size of the cokernel of a localization-modulo-torsion map, and (2) a comparison of different Heegner cycles. [ABSTRACT FROM AUTHOR]

Let F be a finite field, μ be a fixed additive character and s be an integer coprime to | F × |. For any a ∈ F , the corresponding Weil sum is defined to be W F , s (a) = ∑ x ∈ F μ (x s − a x). The Weil spectrum counts distinct values of the Weil sum as a runs through the invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in coding and information theory to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we examine the Weil sum, its bounds and its spectrum. As a consequence, we give a new proof to the Vanishing Conjecture of Helleseth (1971) on the presence of zero in the Weil spectrum in the case of Niho exponents. We also state a conjecture for when the Weil spectrum contains at least five elements, and prove it for a certain class of Weil sums. [ABSTRACT FROM AUTHOR]

We study the normal number of prime factors of a f (p) + a g (p) with p prime and f , g distinct Hecke eigenforms of weight two. Assuming a quasi-generalized Riemann hypothesis, we show that the normal order is log log p. We also obtain an estimate for the number of primes p for which a f (p) + a g (p) = 0. [ABSTRACT FROM AUTHOR]

We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld modules are isogenous, again both over their field of definition and over a separable or algebraic closure thereof. Questions of efficiency are left completely out of consideration. [ABSTRACT FROM AUTHOR]

We close a gap in the literature by showing that the full covering groups of the Mathieu group M 22 and of its automorphism group Aut (M 22) occur as the Galois groups of Q -regular Galois extensions of Q (t). [ABSTRACT FROM AUTHOR]

A folklore conjecture in number theory states that the only integers whose expansions in base 3 , 4 and 5 contain solely binary digits are 0 , 1 and 82000. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base 3 or 4 expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in [ 0 , 1 ] who do not contain some digit in their b -expansion for all b ≥ 3 has zero Hausdorff dimension. [ABSTRACT FROM AUTHOR]

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose graded traces are weight 3 2 weakly holomorphic modular forms satisfying certain special properties. We then use these modules to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups of quadratic twists of certain elliptic curves. [ABSTRACT FROM AUTHOR]

In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function ψ : N → (0 , 1 / 2) with ∑ q ψ (q) = ∞ and any number γ , the following set W (ψ , γ) = { x ∈ [ 0 , 1 ] : | q x − p − γ | < ψ (q) for infinitely many q , p ∈ N } has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if γ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced by the upper bound condition ψ (q) = O ((q (log ⁡ log ⁡ q) 2) − 1). In particular, this covers the case when γ is not Liouville, for example π , e , ln ⁡ 2 , 2. In general, if γ is irrational, ψ (q) = O (q − 1 (log ⁡ log ⁡ q) − 2) and in addition, ( lim inf Q → ∞ ∑ q = Q Q (log ⁡ Q) 1 / 8 ψ (q)) = ∞ , then W (ψ , γ) has full Lebesgue measure. Our proof is based on a quantitative study of the discrepancy for irrational rotations. [ABSTRACT FROM AUTHOR]

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]. [ABSTRACT FROM AUTHOR]

A pair of non-empty subsets (W , W ′) in an abelian group G is an additive complement pair if W + W ′ = G. The set W ′ is said to be minimal to W if W + (W ′ ∖ { w ′ }) ≠ G , ∀ w ′ ∈ W ′. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. Additive complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others. The notion of minimal complements is due to Nathanson. We study tightness property of complement pairs (W , W ′) such that both W and W ′ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also study infinite sets forming co-minimal pairs. At the other extreme, motivated by unbounded arithmetic progressions in the integers, we look at sets which can never be a part of any minimal pair. This leads to a discussion on co-minimality, subgroups, approximate subgroups and asymptotic approximate subgroups of G. [ABSTRACT FROM AUTHOR]

The Erdős multiplication table problem asks how many distinct integers appear in the N × N multiplication table. The order of magnitude of this quantity was determined by Ford [8]. In this paper we study the number of y -smooth entries of the N × N multiplication table, that is to say entries with no prime factors greater than y. [ABSTRACT FROM AUTHOR]

Let A = (A 1 , ... , A q) be a q -tuple of finite sets of integers. Associated to every q -tuple of nonnegative integers h = (h 1 , ... , h q) is the linear form h ⋅ A = h 1 A 1 + ⋯ + h q A q. The set (h ⋅ A) (t) consists of all elements of this sumset with at least t representations. The structure of the set (h ⋅ A) (t) is computed for all sufficiently large h i. [ABSTRACT FROM AUTHOR]

We codify some aspects of Deuring's zero-spacing phenomenon, namely that in the presence of an exceptional Landau/Siegel zero, various product L -functions have their low-height zeros lie in a very regular pattern, nearly an arithmetic progression on the central line. This piece of "analytic number theory folklore" seems to have had its actual proof escape the literature in generality, yet we still claim to provide a "different" proof in that we use Lavrik's general method for approximate functional equations, rather than (say, in the imaginary quadratic case) a decomposition into Epstein ζ -functions and analysis with K -Bessel functions and Kloosterman sums. [ABSTRACT FROM AUTHOR]

For positive integers α and β , we define an (α , β) -walk to be any sequence of positive integers satisfying w k + 2 = α w k + 1 + β w k. We say that an (α , β) -walk is n -slow if w s = n with s as large as possible. Slow (1 , 1) -walks have been investigated by several authors. In this paper we consider (α , β) -walks for arbitrary positive α , β. We derive a characterization theorem for these walks, and with this we prove several results concerning the total number of n -slow walks for a given n. In addition to this, we study the slowest n -slow walk for a given n amongst all possible α , β. [ABSTRACT FROM AUTHOR]

This corrigendum replaces the family g u (x) in Theorem 1 of the paper "Continued fractions for rational torsion" published in the Journal of Number Theory, vol. 189 (2018) 115–130. [ABSTRACT FROM AUTHOR]

New results towards the Duffin-Schaeffer conjecture, which is a fundamental problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function ψ : N → R we denote by W (ψ) the set of all x ∈ R such that | n x − a | < ψ (n) for infinitely many a , n. Analogously, denote W ′ (ψ) if we additionally require a , n to be coprime. Aistleitner et al. [2] proved that W ′ (ψ) is of full Lebesgue measure if there exist an ε > 0 such that ∑ n = 2 ∞ ψ (n) φ (n) / (n (log ⁡ n) ε) = ∞. This result seems to be the best one can expect from the method used. Assuming the extra divergence ∑ n = 2 ∞ ψ (n) / (log ⁡ n) ε = ∞ we prove that W (ψ) is of full measure. This could also be deduced from the result in [2] , but we believe that our proof is of independent interest, since its method is totally different from the one in [2]. As a further application of our method, we prove that a variant of Khintchine's theorem is true without monotonicity, subject to an additional condition on the set of divisors of the support of ψ. [ABSTRACT FROM AUTHOR]

We show that the natural density of positive integers n for which σ (2 n + 1) ≥ σ (2 n) is between 0.053 and 0.055. [ABSTRACT FROM AUTHOR]

Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point i. He then conjectured that the Taylor coefficients d (n) either vanish or are periodic modulo any prime p ; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, with the latter trio considering arbitrary half integral weight modular forms. We refine previous work for p ≡ 1 (mod 4) by displaying a concise algebraic relation between d (n + p − 1 2) and d (n) related to the p -adic factorial, from which we can deduce periodicity with an effective period. [ABSTRACT FROM AUTHOR]

In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find corresponding values in the Markov spectrum. Further we construct a counterexample to the generalised Markov uniqueness conjecture. The proposed generalisation is based on geometry of numbers. It substantively uses lattice trigonometry and geometric theory of continued numbers. [ABSTRACT FROM AUTHOR]

Let W be a nonempty subset of the set of integers Z. A nonempty subset C of Z is said to be an asymptotic complement to W if W + C contains almost all the integers except a set of finite size. The set C is said to be a minimal asymptotic complement to W if C is an asymptotic complement to W , but C ∖ { c } is not an asymptotic complement to W for every c ∈ C. Asymptotic complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathanson. In this article, we study minimal asymptotic complements in Z and deal with a problem of Nathanson on their existence and their inexistence. [ABSTRACT FROM AUTHOR]

For every positive integer n and every δ ∈ 0 , 1 , let B (n , δ) denote the probabilistic model in which a random set A ⊆ { 1 , ... , n } is constructed by choosing independently every element of { 1 , ... , n } with probability δ. Moreover, let (u k) k ≥ 0 be an integer sequence satisfying u k = a 1 u k − 1 + a 2 u k − 2 , for every integer k ≥ 2 , where u 0 = 0 , u 1 ≠ 0 , and a 1 , a 2 are fixed nonzero integers; and let α and β , with | α | ≥ | β | , be the two roots of the polynomial X 2 − a 1 X − a 2. Also, assume that α / β is not a root of unity. We prove that, as δ n / log ⁡ n → + ∞ , for every A in B (n , δ) we have log ⁡ lcm (u a : a ∈ A) ∼ δ Li 2 (1 − δ) 1 − δ ⋅ 3 log ⁡ | α / (a 1 2 , a 2) | π 2 ⋅ n 2 with probability 1 − o (1) , where lcm denotes the lowest common multiple, Li 2 is the dilogarithm, and the factor involving δ is meant to be equal to 1 when δ = 1. This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Mátyás, who studied the deterministic case δ = 1 , and is motivated by an asymptotic formula for lcm (A) due to Cilleruelo, Rué, Šarka, and Zumalacárregui. [ABSTRACT FROM AUTHOR]

Abstract upper densities are monotone and subadditive functions from the power set of positive integers into the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density. At the open problem session of the Workshop "Densities and their application", held at St. Étienne in July 2013, G. Grekos asked a question whether there is a "nice" abstract upper density, whose the family of null sets is precisely a given ideal of subsets of N , where "nice" would mean the properties of the familiar densities consider in number theory. M. Di Nasso, R. Jin (2018) [3] showed that the answer is positive for the summable ideals (for instance, the family of finite sets and the family of sequences whose series of reciprocals converge) when "nice" density means translation invariant and rich density (i.e. density which is onto the unit interval). In this paper we extend their result to all ideals with the Baire property. [ABSTRACT FROM AUTHOR]

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K ×. We consider Kummer extensions of G of the form K (ζ 2 m , G 2 n ) / K (ζ 2 m ) , where n ⩽ m. In the paper by Debry and Perucca (2016) [1] , the degrees of those extensions have been evaluated in terms of divisibility parameters over K (ζ 4). We prove how properties of G over K explicitly determine the divisibility parameters over K (ζ 4). This result yields a clear computational advantage, since no field extension is required. [ABSTRACT FROM AUTHOR]

We construct an analogue of the ring of algebraic numbers, living in a quotient of the product of all finite fields of prime order. We use this ring to deduce some results about linear recurrent sequences. [ABSTRACT FROM AUTHOR]

We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative f ∈ L 1 (− 1 / 4 , 1 / 4) , max − 1 / 2 ≤ t ≤ 1 / 2 ⁡ ∫ R f (t − x) f (x) d x ≥ 1.28 (∫ − 1 / 4 1 / 4 f (x) d x) 2 which is related to g −Sidon sets (1.28 cannot be replaced by 1.52). We prove a dual statement, related to difference bases, and show that for f ∈ L 1 (R) , min 0 ≤ t ≤ 1 ⁡ ∫ R f (x) f (x + t) d x ≤ 0.42 ‖ f ‖ L 1 2 , where the constant 1/2 is trivial, 0.42 cannot be replaced by 0.37. This suggests a natural conjecture about the asymptotic structure of g −difference bases. Finally, we show for all functions f ∈ L 1 (R) ∩ L 2 (R) , ∫ − 1 2 1 2 ∫ R f (x) f (x + t) d x d t ≤ 0.91 ‖ f ‖ L 1 ‖ f ‖ L 2 . [ABSTRACT FROM AUTHOR]

We call an abelian variety over a finite field F q super-isolated if its (F q -rational) isogeny class contains a single isomorphism class. In this paper, we use the Honda-Tate theorem to characterize super-isolated ordinary simple abelian varieties by certain algebraic integers. Our main result is that for a fixed dimension g ≥ 3 , there are finitely many such varieties over all finite fields. [ABSTRACT FROM AUTHOR]