In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence ${\mathcal{D}}$, we obtain a sharp criterion such that for almost every $\unicode[STIX]{x1D6FC}$ the inequality

$$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$

$(n,p)\in \mathbb{N}\times \mathbb{Z}$

$\unicode[STIX]{x1D713}$

${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$

$\unicode[STIX]{x1D713}(n)$

$\unicode[STIX]{x1D6FC}$

$$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$

$(n,p)\in \mathbb{N}\times \mathbb{Z}$

We construct a random model to study the distribution of class numbers in special families of real quadratic fields ${\open Q}(\sqrt d )$ arising from continued fractions. These families are obtained by considering continued fraction expansions of the form $\sqrt {D(n)} = [f(n),\overline {u_1,u_2, \ldots ,u_{s-1} ,2f(n)]} $ with fixed coefficients u1, …, us−1 and generalize well-known families such as Chowla's 4n2 + 1, for which analogous results were recently proved by Dahl and Lamzouri [‘The distribution of class numbers in a special family of real quadratic fields’, Trans. Amer. Math. Soc. (2018), 6331–6356].

Let $G$ be a finite abelian group, $A$ a nonempty subset of $G$ and $h\geq 2$ an integer. For $g\in G$, let $R_{A,h}(g)$ denote the number of solutions of the equation $x_{1}+\cdots +x_{h}=g$ with $x_{i}\in A$ for $1\leq i\leq h$. Kiss et al. [‘Groups, partitions and representation functions’, Publ. Math. Debrecen85(3) (2014), 425–433] proved that (a) if $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $|G|=2|A|$, and (b) if $h$ is even and $|G|=2|A|$, then $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$. We prove that $R_{G\setminus A,h}(g)-(-1)^{h}R_{A,h}(g)$ does not depend on $g$. In particular, if $h$ is even and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for some $g\in G$, then $|G|=2|A|$. If $h>1$ is odd and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $R_{A,h}(g)=\frac{1}{2}|A|^{h-1}$ for all $g\in G$. If $h>1$ is odd and $|G|$ is even, then there exists a subset $A$ of $G$ with $|A|=\frac{1}{2}|G|$ such that $R_{A,h}(g)\not =R_{G\setminus A,h}(g)$ for all $g\in G$.

Denote by $\mathbb{P}$ the set of all prime numbers and by $P(n)$ the largest prime factor of positive integer $n\geq 1$ with the convention $P(1)=1$. In this paper, we prove that, for each $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant $c(\unicode[STIX]{x1D702})>1$ such that, for every fixed nonzero integer $a\in \mathbb{Z}^{\ast }$, the set

$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$

$\mathbb{P}$

$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$

$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$

Putnam [‘On the non-periodicity of the zeros of the Riemann zeta-function’, Amer. J. Math.76 (1954), 97–99] proved thatthe sequence of consecutive positive zeros of $\unicode[STIX]{x1D701}(\frac{1}{2}+it)$ does not contain any infinite arithmetic progression. Weextend this result to a certain class of zeta functions.

Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$.

The aim of this note is to give a simple topological proof of the well-knownresult concerning continuity of roots of polynomials. We also consider amore general case with polynomials of a higher degree approaching a givenpolynomial. We then examine the continuous dependence of solutions of lineardifferential equations with constant coefficients.

We consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan, and recently rediscovered in the theory of Laurent phenomenon algebras (a generalization of cluster algebras). All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences.

In 2014, Pila and Tsimerman gave a proof of the Ax–Schanuel conjecture for the$j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax–Schanuel conjecture. In this article, we show that the hyperbolic Ax–Schanuel conjecture can be used to reduce the Zilber–Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila–Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila–Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber–Pink conjecture for curves in abelian varieties.

For an optimal modular parametrization $J_{0}(n){\twoheadrightarrow}E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^{0}(E,\unicode[STIX]{x1D6FA}^{1})$. Multiple authors generalized his conjecture to higher-dimensional newform quotients. We prove the Manin conjecture for semistable $E$, give counterexamples to all the proposed generalizations, and prove several semistable special cases of these generalizations. The proofs establish general relations between the integral $p$-adic étale and de Rham cohomologies of abelian varieties over $p$-adic fields and exhibit a new exactness result for Néron models.

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined in [43], and extend them to include limit shapes with different scaling functions.

We give transcendence measures for $p$-adic numbers $\unicode[STIX]{x1D709}$, having good rational (respectively, integer) approximations, that force them to be either $p$-adic $S$-numbers or $p$-adic $T$-numbers.

In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight $-1$ .

We prove new integral representations of the approximation forms in zetavalues.

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate–Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.

For a positive integer $d$ and a nonnegative number $\unicode[STIX]{x1D709}$, let $N(d,\unicode[STIX]{x1D709})$ be the number of $\unicode[STIX]{x1D6FC}\in \overline{\mathbb{Q}}$ of degree at most $d$ and Weil height at most $\unicode[STIX]{x1D709}$. We prove upper and lower bounds on $N(d,\unicode[STIX]{x1D709})$. For each fixed $\unicode[STIX]{x1D709}>0$, these imply the asymptotic formula $\log N(d,\unicode[STIX]{x1D709})\sim \unicode[STIX]{x1D709}d^{2}$ as $d\rightarrow \infty$, which was conjectured in a question at Mathoverflow [https://mathoverflow.net/questions/177206/].

Let $G$ be an orthogonal, symplectic or unitary group over a non-archimedean local field of odd residual characteristic. This paper concerns the study of the “wild part” of an irreducible smooth representation of $G$, encoded in its “semisimple character”. We prove two fundamental results concerning them, which are crucial steps toward a complete classification of the cuspidal representations of $G$. First we introduce a geometric combinatorial condition under which we prove an “intertwining implies conjugacy” theorem for semisimple characters, both in $G$ and in the ambient general linear group. Second, we prove a Skolem–Noether theorem for the action of $G$ on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of $G$ which have the same characteristic polynomial must be conjugate under an element of $G$ if there are corresponding semisimple strata which are intertwined by an element of $G$.

We present a simple proof of the Chebotarev density theorem for finite morphisms of quasi-projective varieties over finite fields following an idea of Fried and Kosters for function fields. The key idea is to interpret the number of rational points with a given Frobenius conjugacy class as the number of rational points of a twisted variety, which is then bounded by the Lang–Weil estimates.