We provide lower bounds for $p$ -adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb and Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers’ conjectures on the $p$ -adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the $p$ -Lucas property for almost all primes $p$ .

Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$ . Then $\min (S)=1$ . Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and  $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$ -entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$ , respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$ , where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$ . Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$ . In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

In this article we study various forms of $\ell$ -independence (including the case $\ell =p$ ) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of $\ell$ -independence for the unipotent fundamental group of smooth and projective varieties over finite fields. By then proving a certain ‘spreading out’ result we are able to deduce a much weaker form of $\ell$ -independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce $\ell$ -independence results for the cohomology of smooth and proper varieties over equicharacteristic local fields from the well-known results on $\ell$ -independence for smooth and proper varieties over finite fields. As another consequence of this ‘spreading out’ result we are able to deduce the existence of a Clemens–Schmid exact sequence for formal semistable families. Finally, by deforming to characteristic $p$ , we show a similar weak version of $\ell$ -independence for the unipotent fundamental group of a semistable curve in mixed characteristic.

In this paper, we consider the family of hyperelliptic curves over $\mathbb{Q}$ having a fixed genus $n$ and a marked rational non-Weierstrass point. We show that when $n\geqslant 9$ , a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as $n$ tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of $5/2$ on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2) 180 (2014), 1137–1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic representations and $L$ -functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23–91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.

We show that the automorphic étale cohomology of a (possibly noncompact) PEL-type or Hodge-type Shimura variety in characteristic zero is canonically isomorphic to the cohomology of the associated nearby cycles over most of their mixed characteristics models constructed in the literature.

Let $a_{1},a_{2},\ldots ,a_{m}$ and $b_{1},b_{2},\ldots ,b_{l}$ be two sequences of pairwise distinct positive integers greater than $1$ . Assume also that none of the above numbers is a perfect power. If for each positive integer $n$ and prime number $p$ the number $\prod _{i=1}^{m}(1-a_{i}^{n})$ is divisible by $p$ if and only if the number $\prod _{j=1}^{l}(1-b_{j}^{n})$ is divisible by $p$ , then $m=l$ and $\{a_{1},a_{2},\ldots ,a_{m}\}=\{b_{1},b_{2},\ldots ,b_{l}\}$ .

Let $\mathbb{N}$ be the set of all nonnegative integers. For a given set $S\subset \mathbb{N}$ the representation function $R_{S}(n)$ counts the number of solutions of the equation $n=s+s^{\prime }$ with $s<s^{\prime }$ and $s,s^{\prime }\in S$ . We obtain some results on a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers16 (2016), Article ID A36, 4 pages] about sets $A$ and $B$ such that $A\cup B=\mathbb{N}$ , $A\cap B=r+m\mathbb{N}$ and whose representation functions coincide.

Let $K$ be an algebraic number field. A cuboid is said to be $K$ -rational if its edges and face diagonals lie in $K$ . A $K$ -rational cuboid is said to be perfect if its body diagonal lies in $K$ . The existence of perfect $\mathbb{Q}$ -rational cuboids is an unsolved problem. We prove here that there are infinitely many distinct cubic fields $K$ such that a perfect $K$ -rational cuboid exists; and that, for every integer $n\geq 2$ , there is an algebraic number field $K$ of degree $n$ such that there exists a perfect $K$ -rational cuboid.

We describe the Schwarzian equations for the 328 completely replicable functions with integral $q$ -coefficients [Ford et al., ‘More on replicable functions’, Comm. Algebra 22 (1994) no. 13, 5175–5193].

Recently, Freiman et al. [‘Small doubling in ordered groups’, J. Aust. Math. Soc. 96(3) (2014), 316–325] proved two ‘structure theorems’ for ordered groups. We give elementary proofs of these two theorems.

For the groups $\operatorname{SO}(2n+1,F)$, where $F$ is a $p$-adic field, we consider the tempered irreducible representations of unipotent reduction. Lusztig has constructed and parametrized these representations. We prove that they satisfy the expected endoscopic identities which determine the parametrization.

This is an addendum to a recent paper by Zaïmi, Bertin and Aljouiee [‘On number fields without a unit primitive element’, Bull. Aust. Math. Soc.93 (2016), 420–432], giving the answer to a question asked in that paper, together with some historical connections.

Let $K$ be the field of fractions of a local Henselian discrete valuation ring ${\mathcal{O}}_{K}$ of characteristic zero with perfect residue field $k$ . Assuming potential semi-stable reduction, we show that an unramified Galois action on the second $\ell$ -adic cohomology group of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties.

Let $\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}$ be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$ over a nonarchimedean local field $F$ of residual characteristic $p$, different to $\ell$. Whenever the local Rankin–Selberg $L$-factor $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ such that the local Rankin–Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_{F}^{n}$ is equal to $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$. As an application we prove that the $L$-factor of a pair of banal $\ell$-modular cuspidal representations is the reduction modulo $\ell$ of the $L$-factor of any pair of $\ell$-adic lifts.

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a known case of the Colmez conjecture, due to Colmez and Obus, that is valid when the CM field is an abelian extension of the rationals. It links the height and the logarithmic derivatives of an $L$ -function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use the reduction theory of genus $2$ curves as developed by Igusa, Liu, Saito, and Ueno to relate the contribution at the finite places with the stable bad reduction of the curve. The subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang are used to bound the infinite places.

Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$ . Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$ -action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$ , show the existence of an ordinary lifting of $\overline{r}$ , and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.

In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4\sqrt{e}+\unicode[STIX]{x1D716}}]$ necessary to generate the finite field $\mathbb{F}_{p}^{\ast }$ with $p$ an odd prime. The general result depends on the distribution of the divisors of $p-1$ and can be used to deduce results which hold for almost all primes.

In characteristic two, some criteria are obtained for a symmetric square-central element of a totally decomposable algebra with orthogonal involution, to be contained in an invariant quaternion subalgebra.

We describe the ramification in cyclic extensions arising from the Kummer theory of the Weil restriction of the multiplicative group. This generalises the classical theory of Hecke describing the ramification of Kummer extensions.

Robin’s criterion states that the Riemann hypothesis is true if and only if $\unicode[STIX]{x1D70E}(n)<e^{\unicode[STIX]{x1D6FE}}n\log \log n$ for every positive integer $n\geq 5041$ . In this paper we establish a new unconditional upper bound for the sum of divisors function, which improves the current best unconditional estimate given by Robin. For this purpose, we use a precise approximation for Chebyshev’s $\unicode[STIX]{x1D717}$ -function.