We prove that if W and W′ are non-zero B-pairs whose tensor product is crystalline (or semi-stable or de Rham or Hodge–Tate), then there exists a character μ such that W(μ−1) and W′(μ) are crystalline (or semi-stable or de Rham or Hodge–Tate). We also prove that if W is a B-pair and if F is a Schur functor (for example Sym n or Λn) such that F(W)is crystalline (or semi-stable or de Rham or Hodge–Tate) and if the rank of W is sufficiently large, then there is a character μ such that W(μ−1)is crystalline (or semi-stable or de Rham or Hodge–Tate). In particular, these results apply to p-adic representations.

We consider the analogue of the André–Oort conjecture for Drinfeld modular varieties which was formulated by Breuer. We prove this analogue for special points with separable reflex field over the base field by adapting methods which were used by Klingler and Yafaev to prove the André–Oort conjecture under the generalized Riemann hypothesis in the classical case. Our result extends results of Breuer showing the correctness of the analogue for special points lying in a curve and for special points having a certain behaviour at a fixed set of primes.

We describe a probability distribution on isomorphism classes of principally quasi-polarized $p$-divisible groups over a finite field $k$ of characteristic $p$ which can reasonably be thought of as a ‘uniform distribution’, and we compute the distribution of various statistics ($p$-corank, $a$-number, etc.) of $p$-divisible groups drawn from this distribution. It is then natural to ask to what extent the $p$-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over $k$ are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for $\text{char~} k\not = p$, in which case the random $p$-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over ${\mathbf{F} }_{3} $ appear substantially less likely to be ordinary than hyperelliptic curves over ${\mathbf{F} }_{3} $.

We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general $K 3$ surface $X$ of degree $2$ over $ \mathbb{Q} $, together with a 2-torsion Brauer class $\alpha $ that is unramified at every finite prime, but ramifies at real points of $X$. With motivation from Hodge theory, the pair $(X, \alpha )$ is constructed from a double cover of ${ \mathbb{P} }^{2} \times { \mathbb{P} }^{2} $ ramified over a hypersurface of bidegree $(2, 2)$.

In his Tata Lecture Notes, Igusa conjectured the validity of a strong uniformity in the decay of complete exponential sums modulo powers of a prime number and determined by a homogeneous polynomial. This was proved for non-degenerate forms by Denef–Sperber and then by Cluckers for weighted homogeneous non-degenerate forms. In a recent preprint, Wright has proved this for degenerate binary forms. We give a different proof of Wright’s result that seems to be simpler and relies upon basic estimates for exponential sums mod $p$as well as a type of resolution of singularities with good reduction in the sense of Denef.

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis on factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${B}_{n} / n$ when $n$ is divisible by $p- 1$. Using these, we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${B}_{n} / n$ when $n$ is divisible by $p- 1$.

In this paper we investigate the zeros of the Estermann zeta function $E(s; k/ \ell , \alpha )= { \mathop{\sum }\nolimits}_{n= 1}^{\infty } {\sigma }_{\alpha } (n) \exp (2\pi ink/ \ell ){n}^{- s} $ as a function of a complex variable $s$, where $k$ and $\ell $ are coprime integers and ${\sigma }_{\alpha } (n)= {\mathop{\sum }\nolimits}_{d\vert n} {d}^{\alpha } $ is the generalized divisor function with a fixed complex number $\alpha $. In particular, we study the question on how the zeros of $E(s; k/ \ell , \alpha )$ depend on the parameters $k/ \ell $ and $\alpha $. It turns out that for some zeros there is a continuous dependency whereas for other zeros there is not.

For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation

$$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$

$x, y, z$

$f(n)\gt 0$

$n\geq 2$

$f(n)$

$f(p)$

$n$

$p$

$$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$

A small value estimate is a statement providing necessary conditions for the existence of certain sequences of non-zero polynomials with integer coefficients taking small values at points of an algebraic group. Such statements are desirable for applications to transcendental number theory to analyze the outcome of the construction of an auxiliary function. In this paper, we present a result of this type for the product $ \mathbb {G}_{\mathrm {a}}\times \mathbb {G}_{\mathrm {m}}$ whose underlying group of complex points is $\mathbb {C}\times \mathbb {C}^{*}$. It shows that if a certain sequence of non-zero polynomials in $ \mathbb {Z}[X_1,X_2]$ takes small values at a point $(\xi ,\eta )$ together with their first derivatives with respect to the invariant derivation $\partial /\partial X_1 + X_2 (\partial /\partial X_2)$, then both $\xi $ and $\eta $ are algebraic over $\mathbb {Q}$. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on the absolute values of their derivatives. It improves on a direct application of Philippon’s criterion for algebraic independence and compares favorably with constructions coming from Dirichlet’s box principle.

Two arithmetic functions $f$ and $g$ form a Möbius pair if $f(n)= {\mathop{\sum }\nolimits}_{d\mid n} g(d)$ for all natural numbers $n$. In that case, $g$ can be expressed in terms of $f$ by the familiar Möbius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members $f$ and $g$ of a Möbius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero Möbius pair, one cannot have both ${\mathop{\sum }\nolimits}_{f(n)\not = 0} 1/ n\lt \infty $ and ${\mathop{\sum }\nolimits}_{g(n)\not = 0} 1/ n\lt \infty $.

Motivated by some earlier Diophantine works on triangular numbers by Ljunggren and Cassels, we consider similar problems for general polygonal numbers.

Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs mainly rely on a versatile Cassels–Gallagher type theorem and the cross fibering principle of Beresnevich, Haynes and Velani.

For a hyperbolic surface, embedded eigenvalues of the Laplace operator are unstable and tend to dissolve into scattering poles i.e. become resonances. A sufficient dissolving condition was identified by Phillips–Sarnak and is elegantly expressed in Fermi’s golden rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction $u_j$ into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the $L$-series $L(u_j\otimes F^n, s)$. This is the Rankin–Selberg convolution of $u_j$ with $F(z)^n$, where $F(z)$is the antiderivative of a weight two cusp form. In an example we show that the above-mentioned conditions force the embedded eigenvalue to become a resonance in a punctured neighborhood of the deformation space.

For $n= 1, 2, 3, \ldots $ let ${S}_{n} $ be the sum of the first $n$ primes. We mainly show that the sequence ${a}_{n} = \sqrt[n]{{S}_{n} / n}~(n= 1, 2, 3, \ldots )$ is strictly decreasing, and moreover the sequence ${a}_{n+ 1} / {a}_{n} ~(n= 10, 11, \ldots )$ is strictly increasing. We also formulate similar conjectures involving twin primes or partitions of integers.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$-torsion point over $\mathbb {Q}$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$. Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$. It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $.

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.

We prove upper bounds for Hecke–Laplace eigenfunctions on certain Riemannian manifolds $X$ of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are $d$-fold products of $2$-spheres or $3$-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (‘Weyl-type’) saving of $\mathrm{vol} \hspace{0.167em} (X)^{- 1/ 6+ \varepsilon } $.

We study the Rankin–Selberg integral for a pair of representations of ${\rm SO}_{2l}\times {\rm GL}_{n}$, where ${\rm SO}_{2l}$ is defined over a local non-Archimedean field and is either split or quasi-split. The integrals span a fractional ideal, and its unique generator, which contains any pole which appears in the integrals, is called the greatest common divisor (gcd) of the integrals. We describe the properties of the gcd and establish upper and lower bounds for the poles. In the tempered case we can relate it to the $L$-function of the representations defined by Shahidi. Results of this work may lead to a gcd definition for the $L$-function.

We prove analogues of several well-known results concerning rational maps between quadrics for the class of so-called quasilinear p-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternative approach, which is partly facilitated by the appearance of several non-traditional features in the study of these objects from an algebraic perspective. Our main results were previously known for the class of quasilinear quadrics. We provide new proofs here, because the original proofs do not immediately generalise for quasilinear hypersurfaces of higher degree.

Let π(f) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebra B over a totally real field F with associated Galois representation ρf. Let K be a totally complex quadratic extension of F embedding in B. Using families of CM points on towers of Shimura curves attached to B and K, we construct an Euler system for ρf. We prove that it extends to p-adic families of Galois representations coming from Hida theory and dihedral ℤdp-extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory.