Eisenstein classes of Siegel varieties are motivic cohomology classes defined as pull-backs by torsion sections of the polylogarithm prosheaf on the universal abelian scheme. By reduction to the Hilbert–Blumenthal case, we prove that the Betti realization of these classes on Siegel varieties of arbitrary genus have non-trivial residue on zero-dimensional strata of the Baily–Borel–Satake compactification. A direct corollary is the non-vanishing of a higher regulator map.

Modular curves like X0(N) and X1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here by Xnsp(p) and Xnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures of p-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc. (3) 77(1) (1998), 1–38; J. Algebra 231(1) (2000), 414–448).

We study a family of mixed Tate motives over  $\mathbb{Z}$ whose periods are linear forms in the zeta values  $\unicode[STIX]{x1D701}(n)$ . They naturally include the Beukers–Rhin–Viola integrals for  $\unicode[STIX]{x1D701}(2)$ and the Ball–Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over  $\mathbb{Z}$ which contains all the non-trivial extensions between simple objects. In a joint appendix with Don Zagier, we prove the compatibility between the structure of the motives considered here and the representations of their periods as sums of series.

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature $(1,n+1)$, with $n\geqslant 1$, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector-valued elliptic cusp forms, transforming under a Weil representation.

Inspired by the recent work of M. Nakasuji, O. Phuksuwan and Y. Yamasaki, we combine interpolated multiple zeta values and Schur multiple zeta values into one object, which we call interpolated Schur multiple zeta values. Our main result will be a Jacobi–Trudi formula for a certain class of these new objects. This generalizes an analogous result for Schur multiple zeta values and implies algebraic relations between interpolated multiple zeta values.

We show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J. 139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.

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.