Let $f(x_{1},\ldots ,x_{n})$ be a regular indefinite integral quadratic form with $n\geqslant 9$ , and let $t$ be an integer. Denote by $\mathbb{U}_{p}$ the set of $p$ -adic units in $\mathbb{Z}_{p}$ . It is established that $f(x_{1},\ldots ,x_{n})=t$ has solutions in primes if (i) there are positive real solutions, and (ii) there are local solutions in $\mathbb{U}_{p}$ for all prime  $p$ .

In this paper, we investigate the nontrivial zeros of quadratic $L$ -functions near the real axis. Assuming the generalized Riemann hypothesis, we give an asymptotic formula for the weighted pair correlation function of quadratic $L$ -functions associated to the Kronecker symbols. From this formula, we obtain several results on the rate of simple zeros of quadratic $L$ -functions and on the average distance of such nontrivial zeros.

We improve the known upper bound for the number of Diophantine $D(4)$ -quintuples by using the most recent methods that were developed in the $D(1)$ case. More precisely, we prove that there are at most $6.8587\times 10^{29}$ $D(4)$ -quintuples.

We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic subgroup of GL2 (defined by F. Momose). We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$ , motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$ , and we complete its proof by reducing the positive characteristic case to characteristic $0$ . For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$ : we find an elliptic curve $E^{\prime }$ defined over a $p$ -adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$ .

We study the problem of Diophantine approximation on lines in $\mathbb{R}^{d}$ under certain primality restrictions.

For any given integer $k\geqslant 2$ we prove the existence of infinitely many $q$ and characters $\unicode[STIX]{x1D712}\,(\text{mod}\;q)$ of order $k$ such that $|L(1,\unicode[STIX]{x1D712})|\geqslant (\text{e}^{\unicode[STIX]{x1D6FE}}+o(1))\log \log q$ . We believe this bound to be the best possible. When the order $k$ is even, we obtain similar results for $L(1,\unicode[STIX]{x1D712})$ and $L(1,\unicode[STIX]{x1D712}\unicode[STIX]{x1D709})$ , where $\unicode[STIX]{x1D712}$ is restricted to even (or odd) characters of order $k$ and $\unicode[STIX]{x1D709}$ is a fixed quadratic character. As an application of these results, we exhibit large even-order character sums, which are likely to be optimal.

We show that the $\unicode[STIX]{x1D703}=\infty$ conjecture implies the Riemann hypothesis.

Diophantine problems involving recurrence sequences have a long history. We consider the equation $B_{m}B_{m+d}\cdots B_{m+(k-1)d}=y^{\ell }$ in positive integers $m,d,k,y$ with $\gcd (m,d)=1$ and $k\geq 2$ , where $\ell \geq 2$ is a fixed integer and $B=(B_{n})_{n=1}^{\infty }$ is an elliptic divisibility sequence, an important class of nonlinear recurrences. We prove that the equation admits only finitely many solutions. In fact, we present an algorithm to find all possible solutions, provided that the set of $\ell$ th powers in $B$ is given. We illustrate our method by an example.

We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras and apply this technique to sharpen recent results of P. Scholze.

Let $F/\mathbf{Q}$ be a totally real field and $K/F$ a complex multiplication (CM) quadratic extension. Let $f$ be a cuspidal Hilbert modular new form over $F$ . Let ${\it\lambda}$ be a Hecke character over $K$ such that the Rankin–Selberg convolution $f$ with the ${\it\theta}$ -series associated with ${\it\lambda}$ is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg $L$ -values $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$ , as ${\it\chi}$ varies over the class group characters of $K$ . Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters ${\it\chi}$ such that $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$ increases with the absolute value of the discriminant of $K$ . We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg $L$ -values. The Waldspurger formula plays an underlying role.

We study some questions on numerical semigroups of type 2. On the one hand, we investigate the relation between the genus and the Frobenius number. On the other hand, for two fixed positive integers g1, g2, we give necessary and sufficient conditions in order to have a numerical semigroup S such that {g1, g2} is the set of its pseudo-Frobenius numbers and, moreover, we explicitly build families of such numerical semigroups.

Let $G\subseteq \widetilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and having the same derived group. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of $G$ should be the restriction of those of $\widetilde{G}$ . Motivated by this, we hope to construct the L-packets of $\widetilde{G}$ from those of $G$ . The primary example in our mind is when $G=\text{Sp}(2n)$ , whose L-packets have been determined by Arthur [The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013)], and $\widetilde{G}=\text{GSp}(2n)$ . As a first step, we need to consider some well-known conjectural properties of L-packets. In this paper, we show how they can be deduced from the conjectural endoscopy theory. As an application, we obtain some structural information about L-packets of $\widetilde{G}$ from those of  $G$ .

Given a commutative complete local noetherian ring $A$ with finite residue field $\boldsymbol{k}$ , we show that there is a topologically finitely generated profinite group $\unicode[STIX]{x1D6E4}$ and an absolutely irreducible continuous representation $\overline{\unicode[STIX]{x1D70C}}:\unicode[STIX]{x1D6E4}\rightarrow \text{GL}_{n}(\boldsymbol{k})$ such that $A$ is a universal deformation ring for $\unicode[STIX]{x1D6E4},\overline{\unicode[STIX]{x1D70C}}$ .

Given $k\geqslant 2$ , we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$ ) for which

$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$

$\ell$

$(x,y,\ell )$

$y=0$

$\ell <\exp (3^{k})$

For natural integer n, let D n denote the random variable taking the values log d for d dividing n with uniform probability 1/τ(n). Then t↦ℙ(D n ≤n t ) (0≤t≤1) is an arithmetic process with respect to the uniform probability over the first N integers. It is known from previous works that this process converges to a limit law and that the same holds for various extensions. We investigate the generalized moments of arbitrary orders for the limit laws. We also evaluate the mean value of the two-dimensional distribution function ℙ(D n ≤n u , D{n/D n}≤n v ).

Let ${\it\lambda}$ and ${\it\mu}$ denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for $({\it\lambda}(n),{\it\lambda}(n+1),{\it\lambda}(n+2))$ occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand’s result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for $({\it\mu}(n),{\it\mu}(n+1))$ . A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.

We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give an affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

Let $F$ be a totally real number field, ${\wp}$ a place of $F$ above $p$ . Let ${\it\rho}$ be a $2$ -dimensional $p$ -adic representation of $\text{Gal}(\overline{F}/F)$ which appears in the étale cohomology of quaternion Shimura curves (thus ${\it\rho}$ is associated to Hilbert eigenforms). When the restriction ${\it\rho}_{{\wp}}:={\it\rho}|_{D_{{\wp}}}$ at the decomposition group of ${\wp}$ is semistable noncrystalline, one can associate to ${\it\rho}_{{\wp}}$ the so-called Fontaine–Mazur ${\mathcal{L}}$ -invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these ${\mathcal{L}}$ -invariants in the completed cohomology group of quaternion Shimura curves, which generalizes some of Breuil’s results [Breuil, Astérisque, 331 (2010), 65–115] in the $\text{GL}_{2}/\mathbb{Q}$ -case.

We present a ready to compute trace formula for Hecke operators on vector-valuedmodular forms of integral weight for SL2(ℤ) transforming under the Weil representation. As a corollary, we obtain a ready to compute dimension formula for the corresponding space of vector-valued cusp forms, which is more general than the dimension formulae previously published in the vector-valued setting.