Schmidt [‘Integer points on curves of genus 1’, Compos. Math. 81 (1992), 33–59] conjectured that the number of integer points on the elliptic curve defined by the equation $y^{2}=x^{3}+ax^{2}+bx+c$ , with $a,b,c\in \mathbb{Z}$ , is $O_{\unicode[STIX]{x1D716}}(\max \{1,|a|,|b|,|c|\}^{\unicode[STIX]{x1D716}})$ for any $\unicode[STIX]{x1D716}>0$ . On the other hand, Duke [‘Bounds for arithmetic multiplicities’, Proc. Int. Congress Mathematicians, Vol. II (1998), 163–172] conjectured that the number of algebraic number fields of given degree and discriminant $D$ is $O_{\unicode[STIX]{x1D716}}(|D|^{\unicode[STIX]{x1D716}})$ . In this note, we prove that Duke’s conjecture for quartic number fields implies Schmidt’s conjecture. We also give a short unconditional proof of Schmidt’s conjecture for the elliptic curve $y^{2}=x^{3}+ax$ .

Let $R$ be a finite commutative ring of odd characteristic and let $V$ be a free $R$ -module of finite rank. We classify symmetric inner products defined on $V$ up to congruence and find the number of such symmetric inner products. Additionally, if $R$ is a finite local ring, the number of congruent symmetric inner products defined on $V$ in each congruence class is determined.

Let $M$ be an irreducible holomorphic symplectic (hyperkähler) manifold. If $b_{2}(M)\geqslant 5$ , we construct a deformation $M^{\prime }$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real $(1,1)$ -classes is hyperbolic. If $b_{2}(M)\geqslant 14$ , similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).

We give an asymptotic formula for correlations

$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$

$f,\ldots ,f_{m}$

$f:\mathbb{N}\rightarrow \{-1,+1\}$

$1957$

$f(n)=n^{s}$

$\operatorname{Re}(s)<1$

$|f(n)|$

$n=a+b$

$a,b$

$\mathbb{N}$

This paper investigates interrelated price online inventory problems, in which decisions as to when and how much of a product to replenish must be made in an online fashion to meet some demand even without a concrete knowledge of future prices. The objective of the decision maker is to minimize the total cost while meeting the demands. Two different types of demand are considered carefully, that is, demands which are linearly and exponentially related to price. In this paper, the prices are online, with only the price range variation known in advance, and are interrelated with the preceding price. Two models of price correlation are investigated, namely, an exponential model and a logarithmic model. The corresponding algorithms of the problems are developed, and the competitive ratios of the algorithms are derived as the solutions by use of linear programming.

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro- $p$ extensions of imaginary quadratic number fields for $p$ an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of $\mathbb{F}_{q}(t)$ , the Galois groups of the maximal unramified pro- $p$ extensions, as $q\rightarrow \infty$ , have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an abelian variety, provided the geometric coniveau is maximal. This provides an affirmative answer to Mazur’s question for all uni-ruled threefolds, for instance. Concerning cohomology in degree three, we show that the image of the Abel–Jacobi map admits a distinguished model over the rationals.

We give a Rankin–Selberg integral representation for the Spin (degree eight) $L$ -function on $\operatorname{PGSp}_{6}$ that applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\unicode[STIX]{x1D70B}$ corresponds to a level-one Siegel modular form $f$ of even weight, and if $f$ has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$ -function $\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$ of  $\unicode[STIX]{x1D70B}$ .

We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at nonpositive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the $\unicode[STIX]{x1D709}$-function defined by Arakawa and Kaneko. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.

We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by revealing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov co-product on Hopf algebras of iterated integrals.

For any finite abelian group $G$ with $|G|=m$ , $A\subseteq G$ and $g\in G$ , let $R_{A}(g)$ be the number of solutions of the equation $g=a+b$ , $a,b\in A$ . Recently, Sándor and Yang [‘A lower bound of Ruzsa’s number related to the Erdős–Turán conjecture’, Preprint, 2016, arXiv:1612.08722v1] proved that, if $m\geq 36$ and $R_{A}(n)\geq 1$ for all $n\in \mathbb{Z}_{m}$ , then there exists $n\in \mathbb{Z}_{m}$ such that $R_{A}(n)\geq 6$ . In this paper, for any finite abelian group $G$ with $|G|=m$ and $A\subseteq G$ , we prove that (a) if the number of $g\in G$ with $R_{A}(g)=0$ does not exceed $\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$ , then there exists $g\in G$ such that $R_{A}(g)\geq 6$ ; (b) if $1\leq R_{A}(g)\leq 6$ for all $g\in G$ , then the number of $g\in G$ with $R_{A}(g)=6$ is more than $\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$ .

Fix an integer $n\geqslant 2$ . To each non-zero point $\mathbf{u}$ in $\mathbb{R}^{n}$ , one attaches several numbers called exponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point  $\mathbf{u}$ . To avoid trivialities, one restricts to points  $\mathbf{u}$ whose coordinates are linearly independent over  $\mathbb{Q}$ . The resulting set of values is called the spectrum of these exponents. We show that, in an appropriate setting, any such spectrum is a compact connected set. In the case $n=3$ , we prove moreover that it is a semi-algebraic set closed under component-wise minimum. For $n=3$ , we also obtain a description of the spectrum of the exponents $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{2},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{3})$ recently introduced by Schmidt and Summerer.

Conditionally on the generalized Lindelöf hypothesis, we obtain an asymptotic for the fourth moment of Hecke–Maass cusp forms of large Laplacian eigenvalue for the full modular group. This lends support to the random wave conjecture.

We study the $1$ -level density of low-lying zeros of Dirichlet $L$ -functions attached to real primitive characters of conductor at most $X$ . Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$ , which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$ . We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$ , both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$ , and is analogous to a lower order term which was isolated by Rudnick in the function field case.

For the group $G=\operatorname{PGL}_{2}$ we perform a comparison between two relative trace formulas: on the one hand, the relative trace formula of Jacquet for the quotient $T\backslash G/T$, where $T$ is a nontrivial torus, and on the other the Kuznetsov trace formula (involving Whittaker periods), applied to nonstandard test functions. This gives a new proof of the celebrated result of Waldspurger on toric periods, and suggests a new way of comparing trace formulas, with some analogies to Langlands’ ‘Beyond Endoscopy’ program.

In this paper we prove two results concerning Vinogradov’s three primes theorem with primes that can be called almost twin primes. First, for any $m$ , every sufficiently large odd integer $N$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$ , the interval $[p_{i},p_{i}+H]$ contains at least $m$ primes, for some $H=H(m)$ . Second, every sufficiently large integer $N\equiv 3~(\text{mod}~6)$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$ , $p_{i}+2$ has at most two prime factors.

In a previous article, we proved that Shimura curves have no points rational over number fields under a certain assumption. In this article, we give another criterion of the nonexistence of rational points on Shimura curves and obtain new counterexamples to the Hasse principle for Shimura curves. We also prove that such counterexamples obtained from the above results are accounted for by the Manin obstruction.

After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge type Rapoport–Zink formal schemes, which uniformize certain formal completions of such integral models. In the second part, the general theory is applied to the special case of Shimura varieties associated with groups of spinor similitudes, and the reduced scheme underlying the Rapoport–Zink space is determined explicitly.

Let $b$ be an integer larger than 1. We give an asymptotic formula for the exponential sum

$$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ g(p)=k}}\exp \big(2\text{i}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FD}p\big),\end{eqnarray}$$

$p$

$\unicode[STIX]{x1D6FD}\in \mathbb{R}$

$k\in \mathbb{Z}$

$g:\mathbb{N}\rightarrow \mathbb{Z}$

$b$

$\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$

(Torsion in the cohomology of Kottwitz–Harris–Taylor Shimura varieties) When the level at $l$ of a Shimura variety of Kottwitz–Harris–Taylor is not maximal, its cohomology with coefficients in a $\overline{\mathbb{Z}}_{l}$-local system isn’t in general torsion free. In order to prove torsion freeness results of the cohomology, we localize at a maximal ideal $\mathfrak{m}$ of the Hecke algebra. We then prove a result of torsion freeness resting either on $\mathfrak{m}$ itself or on the Galois representation $\overline{\unicode[STIX]{x1D70C}}_{\mathfrak{m}}$ associated to it. Concerning the torsion, in a rather restricted case than Caraiani and Scholze (« On the generic part of the cohomology of compact unitary Shimura varieties », Preprint, 2015), we prove that the torsion doesn’t give new Satake parameters systems by showing that each torsion cohomology class can be raised in the free part of the cohomology of a Igusa variety.