For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

In this paper we examine solutions in the Gaussian integers to the Diophantine equation $ax^{4}+by^{4}=cz^{2}$ for different choices of $a,b$ and $c$. Elliptic curve methods are used to show that these equations have a finite number of solutions or have no solution.

If the centre of a group $G$ is trivial, then so is the centre of its automorphism group. We study the structure of the centre of the automorphism group of a group $G$ when the centre of $G$ is a cyclic group. In particular, it is shown that the exponent of $Z(\text{Aut}(G))$ is less than or equal to the exponent of $Z(G)$ in this case.

In a recent important paper, Hoffstein and Hulse [Multiple Dirichlet series and shifted convolutions, arXiv:1110.4868v2] generalized the notion of Rankin–Selberg convolution $L$-functions by defining shifted convolution$L$-functions. We investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and “mixed mock modular” forms.

We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise.

In previous work by Coates, Galkin and the authors, the notion of mutation between lattice polytopes was introduced. Such mutations give rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterization of such mutations in terms of T-singularities. We also show that the weights involved satisfy Diophantine equations, generalizing results of Hacking and Prokhorov.

Given a prime number p and the Galois orbit O(T) of an integral transcendental element T of , the topological completion of the algebraic closure of the field of p-adic numbers, we study the p-adic analytic continuation around O(T) of functions defined by limits of sequences of restricted power series with p-adic integer coefficients. We also investigate applications to generating elements for or for some classes of closed subfields of .

Given a finite field of q elements, we consider a trajectory of the map associated with a polynomial ]. Using bounds of character sums, under some mild condition on f, we show that for an appropriate constant C > 0 no N ⩾ Cq½ distinct consecutive elements of such a trajectory are contained in a small subgroup of , improving the trivial lower bound . Using a different technique, we also obtain a similar result for very small values of N. These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.

We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. Furthermore, we find a necessary and sufficient condition for a meromorphic Siegel modular function of degree g to have neither a zero nor a pole on a certain subset of the Siegel upper half-space .

Let $\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue $1/4+t_{j}^{2}$. Let ${\it\lambda}_{j}(n)$ be the $n$th Fourier coefficient of ${\it\phi}_{j}$ and $d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for $d_{3}(n)$ and ${\it\lambda}_{j}(n)$ is considered, leading to the estimate

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$

$t_{j}$

${\it\varepsilon}$

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval $[0,1)$, viewed as $\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

With analytic applications in mind, in particular beyond endoscopy, we initiate the study of the elliptic part of the trace formula. Incorporating the approximate functional equation into the elliptic part, we control the analytic behavior of the volumes of tori that appear in the elliptic part. Furthermore, by carefully choosing the truncation parameter in the approximate functional equation, we smooth out the singularities of orbital integrals. Finally, by an application of Poisson summation we rewrite the elliptic part so that it is ready to be used in analytic applications, and in particular in beyond endoscopy. As a by product we also isolate the contributions of special representations as pointed out in [Beyond endoscopy, in Contributions to automorphic forms, geometry and number theory (Johns Hopkins University Press, Baltimore, MD, 2004), 611–697].

We establish the existence of smooth transfer for Guo–Jacquet relative trace formulae in the $p$-adic case. This kind of smooth transfer is a key step towards a generalization of Waldspurger’s result on central values of L-functions of $\text{GL}_{2}$.

An asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

We discuss heuristic asymptotic formulae for the number of isogeny classes of pairing-friendly abelian varieties of fixed dimension $g\geqslant 2$ over prime finite fields. In each formula, the embedding degree $k\geqslant 2$ is fixed and the rho-value is bounded above by a fixed real ${\it\rho}_{0}>1$ . The first formula involves families of ordinary abelian varieties whose endomorphism ring contains an order in a fixed CM-field $K$ of degree $g$ and generalizes previous work of the first author when $g=1$ . It suggests that, when ${\it\rho}_{0}<g$ , there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when ${\it\rho}_{0}>g$ . The second formula involves families whose endomorphism ring contains an order in a fixed totally real field $K_{0}^{+}$ of degree $g$ . It suggests that, when ${\it\rho}_{0}>2g/(g+2)$ (and in particular when ${\it\rho}_{0}>1$ if $g=2$ ), there are infinitely many isogeny classes of $g$ -dimensional abelian varieties over prime fields whose endomorphism ring contains an order of $K_{0}^{+}$ . We also discuss the impact that polynomial families of pairing-friendly abelian varieties has on our heuristics, and review the known cases where they are expected to provide more isogeny classes than predicted by our heuristic formulae.

In this article we explain how the results in our previous article on ‘algebraic Hecke characters and compatible systems of mod $p$ Galois representations over global fields’ allow one to attach a Hecke character to every cuspidal Drinfeld modular eigenform from its associated crystal that was constructed in earlier work of the author. On the technical side, we prove along the way a number of results on endomorphism rings of ${\it\tau}$-sheaves and crystals. These are needed to exhibit the close relation between Hecke operators as endomorphisms of crystals on the one side and Frobenius automorphisms acting on étale sheaves associated to crystals on the other. We also present some partial results on the ramification of Hecke characters associated to Drinfeld modular eigenforms. An important phenomenon absent from the case of classical modular forms is that ramification can also result from places of modular curves of good but non-ordinary reduction. In an appendix, jointly with Centeleghe we prove some basic results on $p$-adic Galois representations attached to $\text{GL}_{2}$-type cuspidal automorphic forms over global fields of characteristic $p$.

Let $\mathbb{A}=(A,+)$ be a (possibly non-commutative) semigroup. For $Z\subseteq A$, we define $Z^{\times }:=Z\cap \mathbb{A}^{\times }$, where $\mathbb{A}^{\times }$ is the set of the units of $\mathbb{A}$ and

$$\begin{eqnarray}{\it\gamma}(Z):=\sup _{z_{0}\in Z^{\times }}\inf _{z_{0}\neq z\in Z}\text{\text{ord}}(z-z_{0}).\end{eqnarray}$$

${\it\gamma}(\cdot )$

$\mathbb{A}$

$X,Y\subseteq A$

$$\begin{eqnarray}|X+Y|\geqslant \min ({\it\gamma}(X+Y),|X|+|Y|-1).\end{eqnarray}$$

$\mathbb{A}$

${\it\gamma}(X+Y)$

$|S|$

$S$

$\mathbb{A}$

Let $\Vert \cdot \Vert$ denote the distance to the nearest integer and, for a prime number $p$, let $|\cdot |_{p}$ denote the $p$-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether $\inf _{q\geqslant 1}$$q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$ holds for every badly approximable real number ${\it\alpha}$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number ${\it\alpha}$ grows too rapidly or too slowly, then their conjecture is true for the pair $({\it\alpha},p)$ with $p$ an arbitrary prime.

Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$ . The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$ . We prove the non-triviality of the $p$ -adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ . The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ .

An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally, several gaps are equally common. Hence, there can be more than one jumping champion for the same $x$. In 1999, Odlyzko et al provided convincing heuristics and empirical evidence for the truth of the hypothesis that the jumping champions greater than 1 are 4 and the primorials $2,6,30,210,2310,\ldots \,$. In this paper, we prove that an appropriate form of the Hardy–Littlewood prime $k$-tuple conjecture for prime pairs and prime triples implies that all sufficiently large jumping champions are primorials and that all sufficiently large primorials are jumping champions over a long range of $x$.